bäcklund transformation and the henon-heiles system
TRANSCRIPT
Volume 105A, number 8 PHYSICS LETTERS 5 November 1984
B~,CKLUND TRANSFORMATION AND THE HENON-HEILES SYSTEM ~"
John WEISS Center for Studies of Nonlinear Dynamics, La Jolla Institute, 8950 Villa La Jolla Drive, Suite 2150, La Jolla, Ca 92037, USA and Institute for Pure and Applied Physical Science, University of Ulaifornia, San Diego, La Jolla, CA 92093, USA
Received 18 July 1984 Revised manuscript received 29 August 1984
A B~cklund transformation and linearization for an instance of the H6non-Heiles system is examined. This provides a special form of solution depending on the three parameters. In addition, a direct formulation in terms of the schwarzian de- rivative is defined for the H6non-Heiles system and second Painlev6 transcendent. This provides (I) a classification of the H~non-Heiles system as equations of"Novikov" type, and; (2) a simple method for deriving the B//cklund transformations and special solutions of the second Painlev6 transcendent. As equations of Novikov type the integrable occurrences of the H6non- Heiles system can be completely integrated by known methods.
In ref. [1 ] B/icklund transformations and the conse- quent lineadzations of the H6non-Heiles system were found using the methods of refs. [2,3]. It can be'shown that the solution found by this method is "special" in that it will depend on three (not four) arbitrary param- eters. To fred the complete solution a direct formulation in terms of the schwarzian derivative is presented. Herein we present this result and in addition, show how a "direct" formulation can provide a different form of Backlund transformation. B/icklund transforms are derived for the second Painlev6 transcendent, and i t is found that the three integrable instances of the H6non-Heiles system can be transformed into a "canonical" class of "Novikov" equations [4-6] considered in ref. [3].
We consider the H6non-Heiles system:
= - A X - 2dXY, ~ = - B Y + c Y 2 - d X 2 , (1)
with hamiltonian
H = }(J/2 + Y2 +AX2 + B y 2 ) + d X 2 y - ~ c y 3 . (2)
TMS sytems is known to be integrable [7] when:
O) a l e = - 1 , B = A ,
¢' Work supported by Department of Energy contract DOE DE-AC03-81ER 10923 and U.S. Air Grant No. AFOSR 83- 0095.
0.375-9601/84/$ 03.00 ©Elsevier Science Publishers B.V. (North.Holland Physics Publishing Division)
(ii)
(rio d/c = _ 1 B = 16,4. (3)
In ref. [I] we found the B~ckhmd transformations for cases (li) and (iii). Case (i) is separable [7]. The BT ofref. [1] for case (ii) may be "improved" to the extent that the BT will depend on an additional arbitrary param- eter. Therefore we first present this result. For case (ii),
cllc = -~-, (4)
the solutions (X, Y) of eqs. (1)have (meromorphic)ex- pansions of the form:
jo--® X = 9 - 1 ~ _ j X # / , Y = 9 -2 . (5) "= ,=
As explained in ref. [1], to define the BT we let
X=X0 9 -1 +X1 , Y= Y0 9 - 2 + Y19-1 + Y2, (6)
where (9, X0, X, Y0, Y1, Y2)are functions of t. Their results, after evaluation:
Y 0 = - 9 2, Y l = g t t ,
Y2 = ~ (4X -- B - 3 V - 392trip2),
X 2 = g l V , X l f - ½ ( V t / V + 9 t t / g t ) V 1 / 2 , (7)
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where h is arbitrary and
v - - t ) + x, (8)
Vtt+~V2+2(~B-2A +~X)V+~B 2 - ~ h 2 = 0 , (9)
~V;1 2+½V3+(~B _ 2 / l + ~ ) k ) V 2 + ( ~ s 2-~]k2)V--0.
(lO) The BT (6) is well def'med and the integration of eqs. (1) is reduced to eq. (I0), which defines Vas a Weier- strass elliptic function, depending on two parameters (compare ref. [i]). Eq. (8) then determines SO and by (6), (7) (X, X 1), (Y, I"2). If
so -'- U I / U 2 , (11)
and (U 1 , U2) are linearly independent solutions of
U n -- -½(V + •)V, (12)
then SO satisfies (8). Since V depends on two arbitrary parameters (~, V0) and two parameters can be introduced through the Moebius group (see ref. [1 ]), a "general'" four parameter solution should be found by the above procedure. However due to the homogeneous dependence of (X 1 , Y2) on ~o the Moebius transformation intro- duces only one arbitrary parameter and the solution ob- tained is "special" (depends on three parameters). A similar conclusion applies to case Oil) as well.
With this in mind it is interesting to compare the above results with a direct formulation of the H6non- Heiles system in terms of the schwarzian derivative. From eqs. (1):
Y = - ( 2 d ) - l ( y ] X + A) , (13)
and letting
X = ~ot 1/2 , (14)
find
Y = (4d) -1 ((¢; t ) - 2,,1) . (15)
By eqs. (1), (14) and (15) with
S = (SO; t ) , (16)
the H6non-Hefles system is
ff~ -- (c[4d)S 2 + (Z + cA /d )S
= A (2B + cA/d) - 4d2~o t . (17)
For the integrable cases (3):
(i) S +¼S 2 =A 2 -4dZ]~o t , (18)
(ii) S + ~S 2 + (B - 6A)S = 2A(B -3A)-4d2]so t ,
(19) Cfii) S + 4S 2 = 16,42 - 4d2/~o t. (20)
Remarkably, eqs. (18), (19) and (20) are precisely those equations, (17), in the class examined in ref. [3]. That is, (19) is a form of higher order KdV equation and (18), (20) are (related to) Caudrey-Dodd-Gibbon/Kuper- schmidt equations. Therefore, with
V = ~ottJ~ot, (21)
applying
a/at + v , (22)
obtains the "modified" equations of "Novikov" type:
(i) v . . - ' _ 2 ~ VtVtt V2 Vtt - ; VV;
+ ~ p'5 = A 2 V , (23)
(ii) Vttt t __ S V2 Vt t s 2 - vv; +]v5
+ ( B - 6 A ) ( V n - ½ V 3 ) = 2 A ( B - 3A)V , (24)
(iii) Vtttt + 5 V t Vtt - 5 V 2 Vtt - 5 V V 2 + V 5
=16A2V. (25)
Eqs. (23), (24) and (25) are equivalent to integrable hamiltonian systems (Novikov equations) and may be integrated by the methods of refs. [4--6]. Note that when
V--> - 2 V , A ->4,4 , (26)
then eqs. (i) -+ Off) and, since (iXdlc = - 1 ) is "separable" [7], Off) (d/c = - ~ ) [dual to (i)] is "indirectly" separa- ble. Case (ii) (die = - ~ ) has recently been shown to be separable as well [8].
The previous results indicate that integrable ham- iltonian ordinary differential equations might be "ca. nonicaUy" classified as occurrences of equations of "Novikov" type. For these (Novikov) equations the in- tegrals can be found by algorithmic methods [4].
Finally, we consider the dkect schwarzian formula. tion of the second Palnlev6 transcendent:
Vxx - ½ v3 _ = u , (27)
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Following ref. [3], let
V = ~Oxx]~O x , (28)
and find that
{~o; x ) + ½ (~oAo x - x ) = latpAo x , (29)
where
{~0 ; x ) = (a/i)x)(~Oxx/~Ox) - - ½(~Oxx/~Ox) 2 . (30)
Eq. (29) is the formulation of eq. (27) in terms the schwarzian derivative, (30). Applying the operator
a / a x + ~OxxAO x (31)
to eq. (29) obtains (27). Although eq. (29) is not in- variant under the full Moebius group (see ref. [3]), it does possess certain partial invariances that define Bticklund transformations for eqs. (27), (29). That is, if
(½ - la)~oAo x + (~ + la)~/~O x = ] x + (~Oxx/~)2 . (36)
Eqs. (24) and (36) imply that (~0, ~) satisfy eqs. (29), (35), respectively and thus uniquely define a B~cklund transformation between these equations. If a solution of eq. (29) is known for a particular param. eter value /a0, then recursive application of (32) and (36) obtains a solution for each parameter in the se- quence
4 -}/dO 4 /a0 -" ~ - /a0 " tao - ~ ' * ~ - # o - ~ .... (37)
When gO = 0, a particular solution is
= x , ( 38 )
and when/a 0 = ~ eq. (29) is integrated by use of the Airy functions. In this way the known exact solutions of the second Painlev6 transcendent [9] are found nearly by inspection.
= 1 / ~ , (32 )
then ~k satisfies eq. (29) with
/z - , ~ - / a . ( 3 3 )
On the other hand, letting
~x = ~°x 1' (34)
we claim that
( ~ ; x ) + ~ (~0/~ x - x) = -u¢,/%,, (35)
i.e., # -+ -/a. Now, substituting (34) into (29), and using (35), obtains
R e f e r e n c e s
[1] J. Weiss, Phys. Lett. 102A (1984) 329. [2] J. Weiss, J. Math. Phys. 24 (1983) 1405. [3] J. Weiss, J. Math. Phys. 25 (1984) 13. [4] I.M. Gel'land and L.A. Dikii, Russian Math. Surveys 80
(1975) 77. [5] O.I. Bogoyavlenskii arid S.P. Novikov, Funct. An. Appl.
10 (1976) 8. [6] I.M. Gel'fand and L.A. Dikii, Funct. An. Appl. 13(1979) 6. [7] Y.F. Chang, M. Tabor and J. Weiss, J. Math. Phys. 23
(1982) 531. [8] S. Wojcieehowski, Phys. Lett. 100A (1984) 277. [9] H. Aixault, Stud. Appl. Math. 61 (1979) 31.
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