the johnson equation, fredholm and wronskian

Research ArticleThe Johnson Equation Fredholm and WronskianRepresentations of Solutions and the Case of Order Three

Pierre Gaillard

Universite de Bourgogne Institut de Mathematiques de Bourgogne 9 avenue Alain Savary BP 47870 21078 Dijon Cedex France

Correspondence should be addressed to Pierre Gaillard pierregaillardu-bourgognefr

Received 10 November 2017 Accepted 8 May 2018 Published 1 August 2018

Academic Editor Giampaolo Cristadoro

Copyright copy 2018 PierreGaillardThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians oforder 2119873 giving solutions of order 119873 depending on 2119873 minus 1 parameters We obtain 119873 order rational solutions that can be writtenas a quotient of two polynomials of degree 2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2 parameters This methodgives an infinite hierarchy of solutions to the Johnson equation In particular rational solutions are obtainedThe solutions of order3 with 4 parameters are constructed and studied in detail by means of their modulus in the (119909 119910) plane in function of time 119905 andparameters 1198861 1198862 1198871 and 1198872

1 Introduction

The Johnson equation was introduced in 1980 by Johnson [1]to describe waves surfaces in shallow incompressible fluids[2 3] This equation was derived for internal waves in astratified medium [4] The Johnson equation is dissipativeit is well known that there is no solution with a linear frontlocalized along straight lines in the (119909 119910) planeThis Johnsonequation is for example able to explain the existence ofthe horseshoe-like solitons and multisoliton solutions quitenaturally

We consider the Johnson equation (J) in the followingnormalization

(119906119905 + 6119906119906119909 + 119906119909119909119909 + 1199062119905)119909 minus 31199061199101199101199052 = 0 (1)

where as usual subscripts 119909 119910 and 119905mean partial derivativesThe first solutions were constructed in 1980 by Johnson

[1] Other types of solutions were found in [5] A newapproach to solve this equation was given in 1986 [6] bygiving a link between solutions of the Kadomtsev-Petviashvili(KP) [7] and solutions of the Johnson equation In 2007other types of solutions were obtained by using the Darbouxtransformation [8] More recently in 2013 other extensionshave been considered as the elliptic case [9]

Here we consider the famous Kadomtsev-Petviashvili(KPI) which can be written in the following form

(4119906119905 minus 6119906119906119909 + 119906119909119909119909)119909 minus 3119906119910119910 = 0 (2)

The KPI equation first appeared in 1970 [7] in a paper writtenby Kadomtsev and Petviashvili This equation is consideredas a model for surface and internal water waves [10] and innonlinear optics [11]

In the following wewill use theKPI equation to constructsolutions to the Johnson equation but in anotherway differentfrom this used in [6] Indeed these last authors consideranother representation of KPI equation given by

(119906119905 + 6119906119906119909 + 119906119909119909119909)119909 minus 3119906119910119910 = 0 (3)

and so the transformations between solutions of (3) and (1)are different from those we use to transform solutions to (2)in solutions to (1)

In fact to obtain solutions to (1) from solutions to (2) weuse the following transformation

(119909 119910 119905) 997891997888rarr (1199091 = minus119894119909 minus 119894119910211990512 1199101 = 119910119905 1199051 = 4119894119905) (4)

In this paper we give solutions by means of Fredholmdeterminants of order 2119873 depending on 2119873 minus 1 parameters

HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 1642139 18 pageshttpsdoiorg10115520181642139

2 Advances in Mathematical Physics

and then by means of Wronskians of order 2119873 with 2119873 minus 1parameters So we construct an infinite hierarchy of solutionsto the Johnson equation depending on 2119873 minus 1 real parame-ters

New rational solutions depending a priori on 2119873 minus 2parameters at order 119873 are constructed when one parametertends to 0

We obtain families depending on 2119873 minus 2 parameters forthe119873th order as a ratio of two polynomials of degree 2119873(119873+1) in 119909 119905 and of degree 4119873(119873 + 1) in 119910

In this paper we construct only rational solutions oforder 3 depending on 4 real parameters we constructthe representations of their modulus in the plane of thecoordinates (119909 119910) according to the four real parameters 119886119894 and119887119894 for 1 le 119894 le 2 and time 1199052 Solutions to Johnson Equation Expressed byMeans of Fredholm Determinants

Some notations are given We define first real numbers 120582119895such that minus1 lt 120582] lt 1 ] = 1 2119873 they depend on aparameter 120598 and can be written as

120582119895 = 1 minus 212059821198952120582119873+119895 = minus120582119895

1 le 119895 le 119873(5)

Then we define 120581] 120575] 120574] and 119909119903] they are functions of120582] 1 le ] le 2119873 and are defined by the following formulas

120581119895 = 2radic1 minus 1205822119895120575119895 = 120581119895120582119895120574119895 = radic 1 minus 1205821198951 + 120582119895

119909119903119895 = (119903 minus 1) ln 120574119895 minus 119894120574119895 + 119894 119903 = 1 3120591119895 = minus121198941205822119895radic1 minus 1205822119895 minus 4119894 (1 minus 1205822119895)radic1 minus 1205822119895

120581119873+119895 = 120581119895120575119873+119895 = minus120575119895120574119873+119895 = 120574minus1119895

119909119903119873+119895 = minus119909119903119895120591119873+119895 = 120591119895

119895 = 1 119873

(6)

119890] 1 le ] le 2119873 are defined by

119890119895 = 2119894(12119872minus1sum119896=1

119886119896 (119895119890)2119896+1 minus 11989412119872minus1sum119896=1

119887119870 (119895119890)2119896+1) 119890119873+119895 = 2119894(12119872minus1sum

119896=1

119886119896 (119895119890)2119896+1 + 11989412119872minus1sum119896=1

119887119896 (119895119890)2119896+1) 1 le 119895 le 119873

119886119896 119887119896 isin R 1 le 119896 le 119873

(7)


120598119895 = 1120598119873+119895 = 0

1 le 119895 le 119873(8)

As usual 119868 is the unit matrix and 119863119903 = (119889119895119896)1le119895119896le2119873 is thematrix defined by the following

119889]120583 = (minus1)120598] prod120578 =120583

(120574120578 + 120574]120574120578 minus 120574120583)sdot exp (120581]119909 + (120581]11991012 minus 2120575])119910119905 + 4119894120591]119905 + 119909119903] + 119890])

(9)

Then we get the following theorem

Theorem 1 The function V defined by

V (119909 119910 119905) = minus2 1003816100381610038161003816119899 (119909 119910 119905)10038161003816100381610038162119889 (119909 119910 119905)2 (10)

with

119899 (119909 119910 119905) = det (119868 + 1198633 (119909 119910 119905)) (11)

119889 (119909 119910 119905) = det (119868 + 1198631 (119909 119910 119905)) (12)

and 119863119903 = (119889119895119896)1le119895119896le2119873 is the matrix

119889]120583 = (minus1)120598] prod120578 =120583


(13)

is a solution to (1) depending on 2119873minus1 parameters 119886119896 119887119896 1 le119896 le 119873 minus 1 and 120598

Advances in Mathematical Physics 3

Proof The solution V to the KPI equation can be written asfollows by using [12]

V (119909 119910 119905) = minus2 1003816100381610038161003816119899 (119909 119910 119905)10038161003816100381610038162119889 (119909 119910 119905)2 (14)

where

119899 (119909 119910 119905) = det (119868 + 1198633 (119909 119910 119905)) (15)

119889 (119909 119910 119905) = det (119868 + 1198631 (119909 119910 119905)) (16)

and119863119903 = (119889119895119896)1le119895119896le2119873 is the matrix

119889]120583 = (minus1)120598] prod120578 =120583

(120574120578 + 120574]120574120578 minus 120574120583)sdot exp (119894120581]119909 minus 2120575]119910 + 120591]119905 + 119909119903] + 119890])

(17)

where 120581] 120575] 119909119903] 120574] 120591] 119890] and 120598] are defined in (6) (5) (7)and (8)

The connection between the solutions to the Johnsonequation and these to the KPI equation was already explainedin [6] but with another expression of the KPI equation (3)

Here the knowledge of a solution 119906 to the KPI equation(2) gives a solution to the Johnson equation (1) Let usconsider 119906(119909 119910 119905) a solution of the KPI equation (2) then thefunction

(1199091 1199101 1199051) (18)

for

1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905

(19)

is a solution to the KPI equation (2) Using this crucialtransformation the solution to the Johnson equation takesthe form

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162det (119868 + 1198631 (119909 119910 119905))2 (20)

with the matrix119863119903 defined in (17)So we get the solutions to (14) by means of Fredholm

determinants

3 Solutions to the Johnson Equation byMeans of Wronskians

We use the following notations

120601119903] = sinΘ119903] 1 le ] le 119873120601119903] = cosΘ119903] 119873 + 1 le ] le 2119873 119903 = 1 3 (21)

with

Θ119903] = minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 1 le ] le 2119873 (22)

119882119903(119908) is the Wronskian of the functions 1206011199031 1206011199032119873defined by

119882119903 (119908) = det [(120597120583minus1119908 120601119903])]120583isin[12119873]] (23)

We consider the matrix119863119903 = (119889]120583)]120583isin[12119873] defined in (17)

Then we have the following result

Theorem 2

det (119868 + 119863119903) = 119896119903 (0) times 119882119903 (1206011199031 1206011199032119873) (0) (24)

where

119896119903 (119910) = 22119873 exp (119894 sum2119873]=1Θ119903])prod2119873]=2prodVminus1120583=1 (120574] minus 120574120583) (25)

Proof First we remove the factor (2119894)minus1119890119894Θ119903] in each row ] inthe Wronskian119882119903(119908) for 1 le ] le 2119873

Then

119882119903 = 2119873prod]=1

119890119894Θ119903] (2119894)minus119873 (2)minus119873 times 119903 (26)

with

119903 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1 minus 119890minus2119894Θ1199031) 1198941205741 (1 + 119890minus2119894Θ1199031) (1198941205741)2119873minus1 (1 + (minus1)2119873 119890minus2119894Θ1199031)(1 minus 119890minus2119894Θ1199032) 1198941205742 (1 + 119890minus2119894Θ1199032) (1198941205742)2119873minus1 (1 + (minus1)2119873 119890minus2119894Θ1199032)

(1 minus 119890minus21198941205791199032119873) 1198941205742119873 (1 + 119890minus2119894Θ1199032119873) (1198941205742119873)2119873minus1 (1 + (minus1)2119899 119890minus2119894Θ1199032119873)

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(27)


The determinant 119903 can be written as

119903 = det (120572119895119896119890119895 + 120573119895119896) (28)

where 120572119895119896 = (minus1)119896(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 1 le 119895 le 119873 1 le 119896 le 2119873120572119895119896 = (minus1)119896minus1(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 119873 + 1 le 119895 le 2119873 1 le 119896 le 2119873We have to calculate 119903 So we use the following lemma

Lemma 3 Let 119860 = (119886119894119895)119894119895isin[1119873] let 119861 = (119887119894119895)119894119895isin[1119873] andlet (119867119894119895)119894119895isin[1119873] be the matrix formed by replacing in 119860 thejth row of 119860 by the ith row of 119861 Then

det (119886119894119895119909119894 + 119887119894119895) = det (119886119894119895)times det(120575119894119895119909119894 + det (119867119894119895)

det (119886119894119895) ) (29)

Proof Let 119860 = (119886119895119894)119894119895isin[1119873] be the transposed matrix incofactors of 119860 Then 119860 times 119860 = det119860 times 119868

So det(119860) = (det(119860))119873minus1Then the general term of the product (119888119894119895)119894119895isin[1119873] =(119886119894119895119909119894 + 119887119894119895)119894119895isin[1119873] times (119886119895119894)119894119895isin[1119873] can be expressed by

119888119894119895 = 119873sum119904=1

(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1

119886119894119904119886119895119904 + 119899sum119904=1

119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)

(30)

We obtain

det (119888119894119895) = det (119886119894119895119909119894 + 119887119894119895) times (det (119860))119873minus1= (det (119860))119873 times det(120575119894119895119909119894 + det (119867119894119895)

det (119860) ) (31)

So det(119886119894119895119909119894+119887119894119895) = det(119860)timesdet(120575119894119895119909119894+det(119867119894119895) det(119860))We use the notations 119880 = (120572119894119895)119894119895isin[12119873] and 119881 =(120573119894119895)119894119895isin[12119873]Using the preceding lemma we get

119903 = det (120572119894119895119890119894 + 120573119894119895)= det (120572119894119895) times det(120575119894119895119890119894 + det (119867119894119895)

det (120572119894119895) )= det (119880) times det(120575119894119895119890119894 + det (119867119894119895)

det (119880) ) (32)

where (119867119894119895)119894119895isin[1119873] is the matrix obtained by replacing in 119880the jth row of 119880 by the ith row of 119881 defined previously119880 is the classical Vandermonde determinant that is equalto

det (119880) = 119894119873(2119873minus1) prod2119873ge119897gt119898ge1

(120574119897 minus 120574119898) (33)

We have to compute det(119867119894119895) to evaluate the determinant 119903To do that we study two cases

(1) For 1 le 119895 le 119873 the matrix 119867119894119895 is a Vandermondematrix where the 119895th row of 119880 in 119880 is replaced by the 119894throw of 119881 Then we have

det (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (34)

with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Thus we get

det (119867119894119895) = minus (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1

(119898119897 minus 119898119896)= minus (119894)119873(2119873minus1) times prod

2119873ge119897gt119898ge1119897 =119895119898 =119895

(120574119897 minus 120574119898)times prod119897lt119895

(minus120574119894 minus 120574119897) times prod119897gt119895

(120574119897 + 120574119894) = (minus1)119895 (119894)119873(2119873minus1)

times prod2119873ge119897gt119898ge1119897 =119895119898 =119895

(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894)

(35)

To compute 119903 we have to simplify the quotient 119902119894119895 fldet(119867119894119895) det(119880)119902119894119895= (minus1)119895 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898)= (minus1)119895prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895) = (minus1)119895prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895)= minus prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)

(36)

119902119894119895 is equal to 119903119894119895 defined byminusprod119897 =119895(120574119897+120574119894)prod119897 =119894(120574119897minus120574119894) becausedet(120575119894119895119909119894 + det(119902119894119895) det(119860)) = det(120575119894119895119909119894 + det(119903119894119895) det(119860))

Thus 119903119894119895 can be written as

119903119894119895 = (minus1)120598119894 (prod119897 =119895

120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)

with the notations given in (17)(2) We can do the same estimations for 119873 + 1 le 119895 le 2119873

are made det119867119894119895 is first as followsdet (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (38)


with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get

det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1

(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod

2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894)

= (minus1)119895minus1 (119894)119873(2119873minus1)times prod2119873ge119897gt119898ge1119897 =119895119898 =119895

(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894) (39)

Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as

119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)

(40)

119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed

Then 119903119894119895 can be written as

119903119894119895 = (minus1)120598119894prod119897 =119895

(120574119897 + 120574119894120574119897 minus 120574119894) = (minus1)120598(119894) prod119897 =119895 (120574119897 + 120574119894)prod119897 =119894 (120574119897 minus 120574119894)= 119888119894119895119890minus2119894Θ119903119894(0)

(41)

with notations given in (17)119890119894 is replaced by 119890minus2119894Θ119903119894 Then det 119903 can be rewritten as

det 119903 = det (119880) times det(120575119894119895119890119894 + det (119867119894119895)det (119880) )

= det (119880) times det (120575119894119895119890119894 + 119903119894119895) = det (119880)sdot 2119873prod119894=1

119890minus2119894Θ119894 det(120575119894119895 + (minus1)120598(119894)prod119897 =119894

10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894)

(42)

We compute the two members of the last relation (42) in 119910 =0 Using (33) we getdet 119903 (0)= 119894119873(2119873minus1) prod

2119873ge119897gt119898ge1

(120574119897 minus 120574119898) 2119873prod119894=1

119890minus2119894Θ119903119894(0)

times det(120575119894119895 + (minus1)120598(119894)prod119897 =119894

10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894(0))

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1

(120574119895 minus 120574119894) 119890minus2119894sum2119873119894=1 Θ119903119894(0) det (120575119894119895 + 119888119894119895)

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1

(120574119895 minus 120574119894) 119890minus2119894sum2119873119894=1 Θ119903119894(0) det (119868 + 119862119903)= 119894119873(2119873minus1) 2119873prod

119895=2

119895minus1prod119894=1

(120574119895 minus 120574119894) 119890minus2119894sum2119873119894=1 Θ119903119894(0) det (119868 + 119863119903) (43)

Thus the Wronskian119882119903 given by (26) can be rewritten as

119882119903 (1206011199031 1206011199032119873) (0) = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

times 119903 = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

sdot 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1

(120574119895 minus 120574119894) 119890minus2119894sum2119873119894=1 Θ119903119894(0) det (119868 + 119863119903)= (2)minus2119873 2119873prod

119895=2

119895minus1prod119894=1


(44)

Then

det (119868 + 119863119903) = 119896119903 (0)119882119903 (1206011 1206012119873) (0) (45)

This finishes the proof of Theorem 2Then the solution V to the Johnson equation can be

rewritten as

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162(det (119868 + 1198631 (119909 119910 119905)))2 (46)

With (24) the following link between Fredholm determi-nants and Wronskians is obtained

det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)


and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)

As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem


V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)

is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)

120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873

120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3

(50)

where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)

4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0

We get the following statement

Theorem 5 The function V

V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)

is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters

We only give the expression without parameters and wegive it in the appendix because of the length of the solutions

We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905

The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894

The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle

5 Conclusion

We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters

Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]

We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero

The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5

In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix

The solutions to the Johnson equation can be written as

V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)


Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0

Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102


Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0

Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0

+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106


Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0

+ 40633270272011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102minus 1818884491444224011990541199104+ 15743974368984297835069441199056+ 2588379624530509824011990541199102minus 2437996216320011990521199104+ 77651388735915294720001199054minus 1845632792788992011990521199102minus 17140311501727334401199052+ 376147987660800

a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104

minus 2407347121029120011990551199106+ 35781759929509767806976011990571199102minus 303004757633531904011990551199104minus 406332702720011990531199106+ 323876044161962698321428481199057+ 77651388735915294720011990551199102minus 571744941244416011990531199104+ 2594177594889458166005761199055minus 72220413630873600011990531199102minus 526920137850853785601199053+ 1880739938304001199051199102+ 78238781433446400119905

a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106


minus 384506831831040011990561199108+ 3478782215369005203456011990581199104minus 104425368449974272011990561199106minus 42326323200011990541199108+ 490721279033276815638528011990581199102minus 10938985317956321280011990561199104minus 94733567262720011990541199106+ 4858140662429440474821427201199058+ 1490906663729573658624011990561199102minus 10231225264373760011990541199104+ 59039903883691116881510401199056+ 1386631941712773120011990541199102+ 3918208204800011990521199104minus 2495937495082991616001199054+ 3410408421457920011990521199102+ 96053150129061888001199052+ 493694233804800

a3

= 380160119905911991018 + 268738560119905911991016+ 171992678400119905911991014+ 75951966781440119905911991012 + 134369280119905711991014+ 27104945362698240119905911991010minus 511678218240119905711991012+ 780622426445709312011990591199108minus 288534917283840119905711991010+ 181433086062329069568011990591199106minus 10021696903839744011990571199108minus 28217548800119905511991010+ 34077866599533112197120011990591199104minus 2050845760706052096011990571199106minus 9171509575680011990551199108+ 4416491511299491340746752011990591199102

minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2

= 95041199051011991020 + 49766401199051011991018+ 39414988801199051011991016 + 16511297126401199051011991014+ 4199040119905811991016 + 6538473662054401199051011991012minus 23648993280119905811991014+ 2054269543278182401199051011991010minus 10990332149760119905811991012+ 54232715942544015360119905101199108minus 4695812902748160119905811991010minus 1175731200119905611991012+ 11359288866511037399040119905101199106minus 123042186185932800011990581199108minus 5224277606400119905611991010+ 2249139195569185405009920119905101199104minus 21200061686630842368011990581199106+ 37614798766080011990561199108+ 235546213935972871506493440119905101199102minus 1313417775190338699264011990581199104


+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1

= 1441199051111991022 + 414721199051111991020 + 597196801199051111991018+ 171992678401199051111991016 + 77760119905911991018+ 99067782758401199051111991014 minus 604661760119905911991016+ 28531521434419201199051111991012minus 180592312320119905911991014+ 8217078173112729601199051111991010minus 121358033879040119905911991012 minus 27993600119905711991014+ 236651851385646612480119905111199108minus 31028029559930880119905911991010minus 162855567360119905711991012+ 34077866599533112197120119905111199106minus 688180296998191104011990591199108+ 37614798766080119905711991010+ 9814425580665536312770560119905111199104minus 124242221977464471552011990591199106

+ 4721075187351552011990571199108+ 9069926400119905511991010+ 565310913446334891615584256119905111199102minus 1703893329976655609856011990591199104+ 2859286420550320128011990571199106+ 2234829864960011990551199108+ 16280954307254444878528826572811990511+ 368040959274957611728896011990591199102+ 42523379545858375680011990571199104+ 1621615769026560011990551199106+ 52997898135593896088961024001199059+ 18849319962866752684032011990571199102+ 345053087347507200011990551199104+ 114281072640011990531199106+ 101211263800613343225446401199057+ 65402806584119132160011990551199102minus 418464636272640011990531199104+ 1643990830094663811072001199055minus 67556178583879680011990531199102+ 3323583435292803072001199053minus 411411861504001199051199102minus 59243308056576000119905

a0

= 1199051211991024 + 4976641199051211991020 + 6481199051011991020+ 1031956070401199051211991016 minus 66355201199051011991018minus 4478976001199051011991016+ 114126085737676801199051211991012minus 16511297126401199051011991014 minus 291600119905811991016minus 1139279501721601199051011991012minus 2149908480119905811991014+ 709955554156939837440119905121199108minus 1369513028852121601199051011991010


+ 935210188800119905811991012minus 6162808629834547200119905101199108+ 594406696550400119905811991010 + 125971200119905611991012+ 23554621393597287150649344119905121199104minus 3786429622170345799680119905101199106+ 30466315231690752011990581199108+ 386983526400119905611991010+ 153350399697899004887040119905101199104+ 27445041098196516864011990581199106+ 72791601315840011990561199108+ 32561908614508889757057653145611990512minus 2622398328167196524544011990581199104minus 4779029840265216011990561199106+ 2380855680011990541199108+ 1385011737943520484458181427211990510+ 17515857652607877120011990561199104minus 20897110425600011990541199106+ 450850175111823074367897601199058+ 6814547709788160011990541199104+ 7321416652243442073600001199056minus 171421608960011990521199104+ 58397426461924392960001199054minus 6296717313441792001199052 minus 30855889612800

(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4

= 15049359360119905711991013 + 11557907988480119905711991011+ 564686361722880011990571199109+ 191731824039297024011990571199107+ 270888468480011990551199109+ 46837345586742558720011990571199105+ 7951502206557726179328011990571199103+ 7275537965776896011990551199105+ 85876223830823442736742401199057119910+ 9860493807735275520011990551199103minus 20897110425600011990531199105+ 186363332966196707328001199055119910minus 16048980806860800011990531199103minus 173328992714096640001199053119910+ 4513775851929600119905119910

b3

= 716636160119905811991015 + 481579499520119905811991013+ 257576235171840119905811991011+ 9700717287702528011990581199109+ 180592312320119905611991011+ 2793806578858328064011990581199107minus 8668430991360011990561199109+ 615294813602681192448011990581199105minus 3281124964958208011990561199107+ 95418026478692714151936011990581199103+ 41085390865563648011990561199105minus 2321901158400011990541199107+ 1177731069679864357532467201199058119910+ 266233332808852439040011990561199103minus 227360561430528sss0011990541199105+ 3578175992950976780697601199056119910minus 757511894083829760011990541199103

+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2

= 22394880119905911991017 + 11466178560119905911991015+ 7430083706880119905911991013+ 2853152143441920119905911991011+ 7524679680119905711991013+ 92442129447518208011990591199109minus 8668430991360119905711991011+ 23665185138564661248011990591199107minus 335839783550976011990571199109+ 5111679989929966829568011990591199105minus 61628086298345472011990571199107minus 145118822400011990551199109+ 654295038711035754184704011990591199103+ 3697685177900728320011990571199105minus 156031757844480011990551199107+ 10599579627118779217792204801199059119910+ 5395662211592742764544011990571199103minus 80565883650441216011990551199105+ 42938111915411721368371201199057119910minus 8011651218784911360011990551199103+ 18807399383040011990531199105+ 326135832690844237824001199055119910+ 12036735605145600011990531199103+ 340880352337723392001199053119910+ 3949553870438400119905119910

b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015


+ 171189128606515201199051011991011minus 361184624640119905811991013+ 4930246903867637760119905101199109minus 113927950172160119905811991011+ 946607405542586449920119905101199107minus 3281124964958208011990581199109minus 48372940800119905611991011+ 272622932796264897576960119905101199105minus 287597736058945536011990581199107minus 5108182548480011990561199109+ 19628851161331072625541120119905101199103+ 11832592569282330624011990581199105minus 1658394683375616011990561199107+ 565310913446334891615584256011990510119910+ 64747946539112913174528011990581199103minus 1997777130838032384011990561199105+ 1044855521280011990541199107+ 294432767419966089383116801199058119910+ 203372684784540057600011990561199103+ 300918390128640011990541199105+ 6709079986783081463808001199056119910minus 67405719388815360011990541199103+ 10630844886464593920001199054119910+ 65825897840640011990521199103minus 2437438960041984001199052119910

b0

= 34561199051111991021 + 14332723201199051111991017+ 1866240119905911991017 + 2377626786201601199051111991013minus 5733089280119905911991015 minus 1031956070400119905911991013+ 19720987615470551040119905111199109minus 475525357240320119905911991011 minus 671846400119905711991013

minus 19686749789749248011990591199109minus 619173642240119905711991011+ 817868798388794692730880119905111199105+ 3944197523094110208011990591199107+ 66499249176576011990571199109minus 709955554156939837440011990591199105minus 17118912860651520011990571199107+ 21767823360011990551199109+ 1356746192271203739877402214411990511119910+ 327147519355517877092352011990591199103minus 12233175130221576192011990571199105minus 11145125560320011990551199107+ 883298302259898268149350401199059119910+ 3549777770784699187200011990571199103+ 22227838417502208011990551199105+ 45238367910880206441676801199057119910minus 7703510787293184000011990551199103+ 2742745743360011990531199105+ 156966735801885917184001199055119910minus 1203673560514560011990531199103minus 162134828601311232001199053119910minus 987388467609600119905119910

(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108minus 13544423424119905511991010+ 1129848248803000320011990571199106minus 1950396973056011990551199108+ 240201629156431311667211990571199104minus 53496602689536011990551199106+ 35781759929509767806976011990571199102+ 374904191648268288011990551199104+ 243799621632011990531199106+ 323876044161962698321428481199057+ 139772499724647530496011990551199102+ 150459195064320011990531199104+ 1520724797004165131796481199055+ 74146291327696896011990531199102+ 1026107636867452108801199053minus 1629974613196801199051199102minus 61387351586242560119905

c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108


minus 2167107747840119905611991010+ 17748888853923495936011990581199106minus 50153065021440011990561199108+ 3478782215369005203456011990581199104+ 22254586718846976011990561199106+ 25395793920011990541199108+ 490721279033276815638528011990581199102+ 16793653516299141120011990561199104+ 16717688340480011990541199106+ 4858140662429440474821427201199058+ 4472719991188720975872011990561199102+ 13842245945917440011990541199104+ 37570847925985256197324801199056+ 3235474530663137280011990541199102minus 3395780444160011990521199104+ 44094895746466185216001199054minus 1805510340771840011990521199102minus 57054126768390144001199052+ 117546246144000

c3

= 380160119905911991018 + 268738560119905911991016+ 171992678400119905911991014+ 75951966781440119905911991012 minus 44789760119905711991014+ 27104945362698240119905911991010minus 150493593600119905711991012+ 780622426445709312011990591199108minus 55725627801600119905711991010+ 181433086062329069568011990591199106+ 106993205379072011990571199108+ 16930529280119905511991010+ 34077866599533112197120011990591199104+ 866216990748966912011990571199106

+ 1044855521280011990551199108+ 4416491511299491340746752011990591199102+ 375684814074713997312011990571199104+ 1225963811635200011990551199106+ 51820167065914031731428556801199059+ 89454399823774419517440011990571199102+ 866644963570483200011990551199104minus 377308938240011990531199106+ 613401598791596019548160001199057+ 24959374950829916160011990551199102minus 175535727575040011990531199104+ 1187400664327481878118401199055minus 2407347121029120011990531199102minus 3335620170897948672001199053+ 391820820480001199051199102minus 12789031580467200119905

c2

= 95041199051011991020 + 49766401199051011991018+ 39414988801199051011991016 + 16511297126401199051011991014minus 1399680119905811991016 + 6538473662054401199051011991012minus 6449725440119905811991014+ 2054269543278182401199051011991010minus 2941074800640119905811991012+ 54232715942544015360119905101199108minus 416084687585280119905811991010 + 705438720119905611991012+ 11359288866511037399040119905101199106+ 18188844914442240011990581199108+ 348285173760119905611991010+ 2249139195569185405009920119905101199104+ 14297716021216149504011990581199106+ 48759924326400011990561199108+ 235546213935972871506493440119905101199102


+ 4224235547233792032768011990581199104+ 93084088679792640011990561199106minus 23581808640011990541199108+ 3731052028745810284662856089611990510+ 1104122877824872835186688011990581199102+ 25614173367749836800011990561199104minus 4179422085120011990541199106+ 6403912691384262444082790401199058minus 1890256662942852317184011990561199102+ 5717449412444160011990541199104+ 19360487961859749366988801199056+ 3505097408218398720011990541199102+ 489776025600011990521199104minus 111623871307878236160001199054+ 188073993830400011990521199102minus 21936950640377856001199052minus 105791621529600

c1

= 1441199051111991022 + 414721199051111991020 + 597196801199051111991018+ 171992678401199051111991016 minus 25920119905911991018+ 99067782758401199051111991014 minus 156764160119905911991016+ 28531521434419201199051111991012minus 77396705280119905911991014+ 8217078173112729601199051111991010minus 12383472844800119905911991012 + 16796160119905711991014+ 236651851385646612480119905111199108minus 1069932053790720119905911991010+ 4837294080119905711991012+ 34077866599533112197120119905111199106+ 174612911178645504011990591199108+ 4179422085120119905711991010+ 9814425580665536312770560119905111199104

+ 136074814546746802176011990591199106+ 2474217874391040011990571199108minus 7860602880119905511991010+ 565310913446334891615584256119905111199102+ 18742826629743211708416011990591199104+ 2376533077879947264011990571199106+ 261213880320011990551199108+ 16280954307254444878528826572811990511+ 7728860144774109846306816011990591199102+ 317631156781672562688011990571199104minus 317636078469120011990551199106+ 38865125299435523798571417601199059minus 50158359901187799515136011990571199102+ 556899633998069760011990551199104+ 27209779200011990531199106+ 174819455655604865571225601199057+ 40443431633289216000011990551199102+ 57989481431040011990531199104minus 1886263162950719530598401199055minus 84407608431083520011990531199102minus 1136749310549950464001199053minus 176319369216001199051199102minus 16362437463244800119905

c0

= 1199051211991024 + 4976641199051211991020 minus 2161199051011991020+ 1031956070401199051211991016 minus 16588801199051011991018minus 8062156801199051011991016+ 114126085737676801199051211991012 + 174960119905811991016minus 1733686198272001199051011991012+ 709955554156939837440119905121199108+ 684756514426060801199051011991010minus 148343685120119905811991012


minus 11093055533702184960119905101199108minus 534966026895360119905811991010 minus 109175040119905611991012+ 23554621393597287150649344119905121199104+ 7572859244340691599360119905101199106+ 66630018649817088011990581199108+ 116095057920119905611991010minus 51116799899299668295680119905101199104+ 12161275696206839808011990581199106minus 40052794982400011990561199108+ 32561908614508889757057653145611990512+ 235546213935972871506493440119905101199102+ 1175863886572431605760011990581199104+ 16405624824791040011990561199106+ 566870400011990541199108+ 1045825189875719549488830873611990510minus 368040959274957611728896011990581199102+ 1993283416212111360011990561199104+ 2786281390080011990541199106+ 671674750676797641405235201199058minus 612336665460360609792011990561199102+ 2501384117944320011990541199104minus 12422447308861054805606401199056minus 3428062300345466880011990541199102minus 73466403840011990521199104minus 4971653274349338624001199054+ 188073993830400011990521199102minus 11261870750564352001199052 + 4407984230400

(A5)

Conflicts of Interest

The author declares that there are no conflicts of interest

References

[1] R S Johnson ldquoWater waves and Kortewegde Vries equationsrdquoJournal of Fluid Mechanics vol 97 no 4 pp 701ndash719 1980

[2] R S Johnson AModern Introduction to the Mathematical The-ory of Water Waves Cambridge University Press CambridgeUK 1997

[3] M J AblowitzNonlinear DispersiveWaves Asymptotic Analysisand Solitons CambridgeUniversity Press CambridgeUK 2011

[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985

[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984

[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986

[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970

[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007

[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013

[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979

[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995

[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017

[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010

[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


and then by means of Wronskians of order 2119873 with 2119873 minus 1parameters So we construct an infinite hierarchy of solutionsto the Johnson equation depending on 2119873 minus 1 real parame-ters

New rational solutions depending a priori on 2119873 minus 2parameters at order 119873 are constructed when one parametertends to 0

We obtain families depending on 2119873 minus 2 parameters forthe119873th order as a ratio of two polynomials of degree 2119873(119873+1) in 119909 119905 and of degree 4119873(119873 + 1) in 119910

In this paper we construct only rational solutions oforder 3 depending on 4 real parameters we constructthe representations of their modulus in the plane of thecoordinates (119909 119910) according to the four real parameters 119886119894 and119887119894 for 1 le 119894 le 2 and time 1199052 Solutions to Johnson Equation Expressed byMeans of Fredholm Determinants

Some notations are given We define first real numbers 120582119895such that minus1 lt 120582] lt 1 ] = 1 2119873 they depend on aparameter 120598 and can be written as

120582119895 = 1 minus 212059821198952120582119873+119895 = minus120582119895

1 le 119895 le 119873(5)

Then we define 120581] 120575] 120574] and 119909119903] they are functions of120582] 1 le ] le 2119873 and are defined by the following formulas

120581119895 = 2radic1 minus 1205822119895120575119895 = 120581119895120582119895120574119895 = radic 1 minus 1205821198951 + 120582119895

119909119903119895 = (119903 minus 1) ln 120574119895 minus 119894120574119895 + 119894 119903 = 1 3120591119895 = minus121198941205822119895radic1 minus 1205822119895 minus 4119894 (1 minus 1205822119895)radic1 minus 1205822119895

120581119873+119895 = 120581119895120575119873+119895 = minus120575119895120574119873+119895 = 120574minus1119895

119909119903119873+119895 = minus119909119903119895120591119873+119895 = 120591119895

119895 = 1 119873

(6)


119890119895 = 2119894(12119872minus1sum119896=1

119886119896 (119895119890)2119896+1 minus 11989412119872minus1sum119896=1

119887119870 (119895119890)2119896+1) 119890119873+119895 = 2119894(12119872minus1sum

119896=1

119886119896 (119895119890)2119896+1 + 11989412119872minus1sum119896=1

119887119896 (119895119890)2119896+1) 1 le 119895 le 119873

119886119896 119887119896 isin R 1 le 119896 le 119873

(7)


120598119895 = 1120598119873+119895 = 0

1 le 119895 le 119873(8)

As usual 119868 is the unit matrix and 119863119903 = (119889119895119896)1le119895119896le2119873 is thematrix defined by the following

119889]120583 = (minus1)120598] prod120578 =120583


(9)

Then we get the following theorem


V (119909 119910 119905) = minus2 1003816100381610038161003816119899 (119909 119910 119905)10038161003816100381610038162119889 (119909 119910 119905)2 (10)

with

119899 (119909 119910 119905) = det (119868 + 1198633 (119909 119910 119905)) (11)

119889 (119909 119910 119905) = det (119868 + 1198631 (119909 119910 119905)) (12)

and 119863119903 = (119889119895119896)1le119895119896le2119873 is the matrix

119889]120583 = (minus1)120598] prod120578 =120583


(13)

is a solution to (1) depending on 2119873minus1 parameters 119886119896 119887119896 1 le119896 le 119873 minus 1 and 120598



V (119909 119910 119905) = minus2 1003816100381610038161003816119899 (119909 119910 119905)10038161003816100381610038162119889 (119909 119910 119905)2 (14)

where

119899 (119909 119910 119905) = det (119868 + 1198633 (119909 119910 119905)) (15)

119889 (119909 119910 119905) = det (119868 + 1198631 (119909 119910 119905)) (16)


119889]120583 = (minus1)120598] prod120578 =120583


(17)




(1199091 1199101 1199051) (18)

for

1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905

(19)


V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162det (119868 + 1198631 (119909 119910 119905))2 (20)


determinants




with



119882119903 (119908) = det [(120597120583minus1119908 120601119903])]120583isin[12119873]] (23)



Theorem 2

det (119868 + 119863119903) = 119896119903 (0) times 119882119903 (1206011199031 1206011199032119873) (0) (24)

where



Then

119882119903 = 2119873prod]=1


with

119903 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(27)



119903 = det (120572119895119896119890119895 + 120573119895119896) (28)




det (119886119894119895) ) (29)



119888119894119895 = 119873sum119904=1

(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1

119886119894119904119886119895119904 + 119899sum119904=1

119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)

(30)

We obtain


det (119860) ) (31)




det (119880) ) (32)



(120574119897 minus 120574119898) (33)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894) = (minus1)119895 (119894)119873(2119873minus1)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894)

(35)


(36)



119903119894119895 = (minus1)120598119894 (prod119897 =119895

120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894) (39)



(40)



119903119894119895 = (minus1)120598119894prod119897 =119895


(41)





10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894)

(42)


2119873ge119897gt119898ge1

(120574119897 minus 120574119898) 2119873prod119894=1

119890minus2119894Θ119903119894(0)


10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894(0))

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1



119882119903 (1206011199031 1206011199032119873) (0) = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

times 119903 = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

sdot 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1


(44)

Then

det (119868 + 119863119903) = 119896119903 (0)119882119903 (1206011 1206012119873) (0) (45)


rewritten as

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162(det (119868 + 1198631 (119909 119910 119905)))2 (46)


det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)


and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)



V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)




(50)





V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)






5 Conclusion







V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)




= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










V (119909 119910 119905) = minus2 1003816100381610038161003816119899 (119909 119910 119905)10038161003816100381610038162119889 (119909 119910 119905)2 (14)

where

119899 (119909 119910 119905) = det (119868 + 1198633 (119909 119910 119905)) (15)

119889 (119909 119910 119905) = det (119868 + 1198631 (119909 119910 119905)) (16)


119889]120583 = (minus1)120598] prod120578 =120583


(17)




(1199091 1199101 1199051) (18)

for

1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905

(19)


V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162det (119868 + 1198631 (119909 119910 119905))2 (20)


determinants




with



119882119903 (119908) = det [(120597120583minus1119908 120601119903])]120583isin[12119873]] (23)



Theorem 2

det (119868 + 119863119903) = 119896119903 (0) times 119882119903 (1206011199031 1206011199032119873) (0) (24)

where



Then

119882119903 = 2119873prod]=1


with

119903 =1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816(27)



119903 = det (120572119895119896119890119895 + 120573119895119896) (28)




det (119886119894119895) ) (29)



119888119894119895 = 119873sum119904=1

(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1

119886119894119904119886119895119904 + 119899sum119904=1

119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)

(30)

We obtain


det (119860) ) (31)




det (119880) ) (32)



(120574119897 minus 120574119898) (33)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894) = (minus1)119895 (119894)119873(2119873minus1)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894)

(35)


(36)



119903119894119895 = (minus1)120598119894 (prod119897 =119895

120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894) (39)



(40)



119903119894119895 = (minus1)120598119894prod119897 =119895


(41)





10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894)

(42)


2119873ge119897gt119898ge1

(120574119897 minus 120574119898) 2119873prod119894=1

119890minus2119894Θ119903119894(0)


10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894(0))

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1



119882119903 (1206011199031 1206011199032119873) (0) = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

times 119903 = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

sdot 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1


(44)

Then

det (119868 + 119863119903) = 119896119903 (0)119882119903 (1206011 1206012119873) (0) (45)


rewritten as

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162(det (119868 + 1198631 (119909 119910 119905)))2 (46)


det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)


and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)



V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)




(50)





V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)






5 Conclusion







V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)




= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










119903 = det (120572119895119896119890119895 + 120573119895119896) (28)




det (119886119894119895) ) (29)



119888119894119895 = 119873sum119904=1

(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1

119886119894119904119886119895119904 + 119899sum119904=1

119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)

(30)

We obtain


det (119860) ) (31)




det (119880) ) (32)



(120574119897 minus 120574119898) (33)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894) = (minus1)119895 (119894)119873(2119873minus1)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894)

(35)


(36)



119903119894119895 = (minus1)120598119894 (prod119897 =119895

120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)







2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894) (39)



(40)



119903119894119895 = (minus1)120598119894prod119897 =119895


(41)





10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894)

(42)


2119873ge119897gt119898ge1

(120574119897 minus 120574119898) 2119873prod119894=1

119890minus2119894Θ119903119894(0)


10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894(0))

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1



119882119903 (1206011199031 1206011199032119873) (0) = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

times 119903 = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

sdot 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1


(44)

Then

det (119868 + 119863119903) = 119896119903 (0)119882119903 (1206011 1206012119873) (0) (45)


rewritten as

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162(det (119868 + 1198631 (119909 119910 119905)))2 (46)


det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)


and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)



V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)




(50)





V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)






5 Conclusion







V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)




= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018












2119873ge119897gt119898ge1119897 =119895119898 =119895



(120574119897 + 120574119894)


(120574119897 minus 120574119898)times prod119897 =119895

(120574119897 + 120574119894) (39)



(40)



119903119894119895 = (minus1)120598119894prod119897 =119895


(41)





10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894)

(42)


2119873ge119897gt119898ge1

(120574119897 minus 120574119898) 2119873prod119894=1

119890minus2119894Θ119903119894(0)


10038161003816100381610038161003816100381610038161003816 120574119897 + 120574119894120574119897 minus 12057411989410038161003816100381610038161003816100381610038161003816 1198902119894Θ119903119894(0))

= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


= 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1



119882119903 (1206011199031 1206011199032119873) (0) = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

times 119903 = 2119873prod119895=1

119890119894Θ119903119895(0) (2)minus2119873 (119894)minus119873

sdot 119894119873(2119873minus1) 2119873prod119895=2

119895minus1prod119894=1


119895=2

119895minus1prod119894=1


(44)

Then

det (119868 + 119863119903) = 119896119903 (0)119882119903 (1206011 1206012119873) (0) (45)


rewritten as

V (119909 119910 119905) = minus2 1003816100381610038161003816det (119868 + 1198633 (119909 119910 119905))10038161003816100381610038162(det (119868 + 1198631 (119909 119910 119905)))2 (46)


det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)


and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)



V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)




(50)





V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)






5 Conclusion







V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)




= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018









and

det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)



V (119909 119910 119905) = minus2 10038161003816100381610038161198823 (12060131 12060132119873) (0)10038161003816100381610038162(1198821 (12060111 12060112119873) (0))2 (49)




(50)





V (119909 119910 119905) = lim120598997888rarr0

minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)






5 Conclusion







V3 (119909 119910 119905) = minus2 10038161003816100381610038161198993 (119909 119910 119905)10038161003816100381610038162(1198893 (119909 119910 119905))2 1198993 (119909 119910 119905) = 1198603 (119909 119910 119905) + 1198941198613 (119909 119910 119905) 1198893 (119909 119910 119905) = 1198623 (119909 119910 119905)(A1)

with

1198603 (119909 119910 119905) = 12sum119896=0

119886119896 (119910 119905) 1199091198961198613 (119909 119910 119905) = 12sum

119896=0

119887119896 (119910 119905) 1199091198961198623 (119909 119910 119905) = 12sum

119896=0

119888119896 (119910 119905) 119909119896(A2)




= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018











= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052+ 40122452017152

a9

= 113515167744011990531199106 + 80244904034304011990531199104+ 23110532361879552011990531199102

+ 271163579712720076801199053+ 334353766809601199051199102+ 13909116699279360119905

a8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102+ 1253826625536011990521199104+ 14642833304486884147201199054+ 641959232274432011990521199102




+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018











+ 19643952507597619201199052minus 125382662553600

a7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104+ 278628139008011990531199106+ 62121110988732235776011990551199102+ 80244904034304011990531199104

+ 562284798892296351252481199055+ 53924575511052288011990531199102+ 1553027774718305894401199053minus 835884417024001199051199102minus 24073471210291200119905

a6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106




a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018











a5

= 197074944119905711991014 + 180592312320119905711991012+ 100553799499776119905711991010+ 3923084197232640011990571199108+ 40633270272119905511991010+ 1129848248803000320011990571199106minus 4550926270464011990551199108+ 240201629156431311667211990571199104


a4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010+ 2821754880119905611991012+ 69759569907154944011990581199108minus 6501323243520119905611991010+ 17748888853923495936011990581199106



a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










a3


minus 180447036681555542016011990571199104minus 378934269050880011990551199106+ 51820167065914031731428556801199059+ 17890879964754883903488011990571199102+ 815288224988528640011990551199104+ 435356467200011990531199106+ 907834366211562108931276801199057+ 135889930287851765760011990551199102+ 593477936087040011990531199104+ 292856666089737682944001199055+ 142033480140718080011990531199102+ 5261497867721244672001199053+ 1645647446016001199051199102+ 119615060076134400119905

a2



+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018









+ 103355436396183552011990561199106+ 27209779200011990541199108+ 3731052028745810284662856089611990510+ 122680319758319203909632011990581199102+ 24689752073274654720011990561199104+ 51546205716480011990541199106+ 9053807598163957248530841601199058+ 2795449994492950609920011990561199102+ 14143164336046080011990541199104+ 8625959983006819024896001199056+ 4429518702693580800011990541199102+ 2057059307520011990521199104+ 136305919870365597696001199054minus 1015599566684160011990521199102+ 83504853260697600001199052minus 246847116902400

a1



a0




(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










(A3)b12 = 0b11 = 0b10 = 213986410758144119905119910b9

= 17832200896512011990521199103+ 513567385819545601199052119910

b8

= 6687075336192011990531199105

+ 3423782572130304011990531199103+ 55465277668510924801199053119910+ 802449040343040119905119910

b7

= 1486016741376011990541199107+ 998603250204672011990541199105+ 287597736058945536011990541199103+ 3549777770784699187201199054119910+ 53496602689536011990521199103+ 1540702157458636801199052119910

b6

= 216710774784011990551199109+ 166433875034112011990551199107+ 63339977584410624011990551199105+ 13804691330829385728011990551199103+ 15603175784448011990531199105+ 149090666372957365862401199055119910+ 5991619501228032011990531199103+ 129418981226525491201199053119910minus 2006122600857600119905119910

b5

= 216710774784119905611991011 + 17336861982720011990561199109+ 7703510787293184011990561199107+ 2218611106740436992011990561199105+ 2600529297408011990541199107+ 414140739924881571840011990561199103+ 748952437653504011990541199105+ 4293811191541172136837121199056119910+ 297869083775336448011990541199103+ 6212111098873223577601199054119910minus 100306130042880011990521199103minus 4429518702693580801199052119910


b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018









b4


b3


+ 3327916660110655488001199054119910+ 150459195064320011990521199103+ 8184980211499008001199052119910

b2


b1

= 4147201199051011991019 + 1194393601199051011991017+ 1375941427201199051011991015+ 396271131033601199051011991013 + 179159040119905811991015



b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










b0



(A4)

c12 = 8916100448256c11 = 89161004482561199051199102 + 2567836929097728119905c10

= 408654603878411990521199104+ 213986410758144011990521199102+ 3389544746409000961199052minus 13374150672384

c9

= 113515167744011990531199106 + 80244904034304011990531199104


+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018









+ 23110532361879552011990531199102+ 271163579712720076801199053minus 111451255603201199051199102 + 1069932053790720119905

c8

= 21284093952011990541199108 + 17832200896512011990541199106+ 6954558349639680011990541199104+ 1479074071160291328011990541199102minus 417942208512011990521199104+ 14642833304486884147201199054+ 5777633090469888001199052+ 75229597532160

c7

= 28378791936119905511991010 + 2600529297408011990551199108+ 1212589660962816011990551199106+ 349225822357291008011990551199104minus 92876046336011990531199106+ 62121110988732235776011990551199102minus 26748301344768011990531199104+ 562284798892296351252481199055+ 23110532361879552011990531199102+ 665583332022131097601199053+ 501530650214401199051199102+ 14444082726174720119905

c6

= 2759049216119905611991012 + 2600529297408119905611991010+ 135227523465216011990561199108+ 46791695152447488011990561199106minus 13544423424011990541199108+ 11216311706298875904011990561199104minus 10402117189632011990541199106+ 1789087996475488390348811990561199102+ 2888816545234944011990541199104

+ 15743974368984297835069441199056+ 2588379624530509824011990541199102+ 1462797729792011990521199104+ 40378722142675953254401199054+ 722204136308736011990521199102+ 15214433804904038401199052minus 325994922639360

c5


c4

= 10264320119905811991016 + 8599633920119905811991014+ 5056584744960119905811991012+ 2139864107581440119905811991010 minus 940584960119905611991012+ 69759569907154944011990581199108



c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










c3



c2




c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










c1



c0




(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018










(A5)



References






















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








the johnson equation, fredholm and wronskian

Documents