boussinesq solitons
TRANSCRIPT
-
7/27/2019 Boussinesq Solitons
1/11
SOOCHOW JOURNAL OF MATHEMATICS
Volume 30, No. 1, pp. 91-101, January 2004
TRAVELLING AND PERIODIC WAVE SOLUTIONS
OF A CLASSICAL BOUSSINESQ SYSTEM
BY
M. INC
Abstract. In this paper, we use of the modified extended tanh-function and theJacobi elliptic function methods to obtain travelling wave and Jacobi doubly peri-odic wave solutions for the classical Boussinesq system. In addition, the propertiesof this equation shown with figures.
1. Introduction
The investigation of the travelling wave solutions play an important role
in nonlinear science. These solutions may well describe various phenomena in
nature, such as vibrations, solitons and propagation with a finite speed. Thewave phenomena observed in fluid dynamics, plasma and elastic media. Vari-
ous methods for obtaining explicit travelling solitary wave solutions to nonlinear
evolution equations have been proposed. In recent years, directly searching for
exact solutions of nonlinear PDEs has become more and more attractive partly
due to the availability of computer symbolic systems like Maple or Mathematica
which allow us to perform some complicated and tedious algebraic calculation
on a computer, as well as help us to find new exact solutions of PDEs ([1-8]).
One of most effectively straightforward methods to construct exact solution of
PDE is extended tanh-function method ([9-13]). Recently, Elwakil et al. [14, 15]developed a modified extended tanh-function method for solving nonlinear PDEs.
Let us simply describe the modified extended tanh-function method.
Received April 1, 2003; revised September 1, 2003.AMS Subject Classification. 35C05, 35Q51, 35B10.Key words. modified extended tanh-function method, Jacobi elliptic function method, soli-
tary wave solution, nonlinear equation, periodic wave solution.
91
-
7/27/2019 Boussinesq Solitons
2/11
92 M. INC
For given a nonlinear equation
H(u, ut, ux, uxx, uxt, . . .) = 0, (1)
when we look for its travelling wave solutions, the first step is to introduce the
wave transformation u = U() , = x + t and change (1) to an ordinary differ-
ential equation
H
U, U
, U
, . . .
= 0. (2)
The next crucial step is to introduce a new variable = () which is a solutionof the Riccati equation
= b + 2, (3)
where b is a parameter to be determined, = () ,
= d/d. Then we
propose the following series expansion as a solution of (1):
u (x, t) = U() =mi=0
aii +
mi=0
bii, (4)
where the positive integer m can be determined by balancing the highest deriva-
tive term with nonlinear terms in (2). Substituting (3) and (4) into (2) will get a
system of algebraic equations with respect to ai, bi, b and (where i = 0, 1, . . . , m)
because all the coefficients ofi have to vanish. With the aid of Mathematica, one
can determine ai, bi, b and . The Riccati equation (3) has the general solutions
=
b tanh
bb coth
b for b < 0, (5)
=
1
for b = 0, (6)
=
b tan
b
b cot
b for b > 0. (7)
We now describe the Jacobi elliptic function method.
We again consider Eq.(2). The fact that the solutions of many nonlinear
equations can be expressed as a finite series of Jacobi elliptic sine, cosine and the
-
7/27/2019 Boussinesq Solitons
3/11
TRAVELLING AND PERIODIC WAVE SOLUTIONS 93
third kind of Jacobi elliptic functions expansions can be written as, respectively
[16-18]:
u () =n
j=0
ajsnj,
u () =n
j=0
bjcnj,
u () =n
j=0
cjdnj.
(8)
Notice that the highest power order of U() is equal to n,
O (U()) = n, (9)
and the highest power order of dU/d can be taken as
O
dU
d
= n + 1. (10)
We have
O
dpU
dp
= n + p, p = 1, 2, 3, . . . , (11)
and
O
Uq
dpU
dp
= (q+ 1) n + p, q = 0, 1, 2, . . . , (12)
so n can be obtained by balancing the derivative term of the highest order with
the nonlinear term in Eq.(2). c, a0, . . . , an; b0, b1, . . . , bn are parameters to be
determined. Substituting (8) into (2) will yield a set of algebraic equations for
c, a0, . . . , an; b0, b1, . . . , bn because all coefficients of snj and cnj have to vanish.
From these relations, c, a0, . . . , an; b0, b1, . . . , bn can be obtained. Therefore, the
travelling solitary wave solutions are obtained.
It is known that there are the following relations between elliptic functions:
cn2= 1 sin2 , sn2+ cn2 = 1,dn2= 1 m2sn2 , d
dsn = cndn, (13)
d
dcn=sndn , d
ddn = m2sncn,
where m is the modulus 0 < m < 1.
-
7/27/2019 Boussinesq Solitons
4/11
94 M. INC
When m 1, the Jacobi functions degenerate to the hyperbolic functions,i.e.,
sn tanh , cn sec h , dn sec h. (14)When m 0, the Jacobi functions degenerate to the triangular functions, i.e.,
sn sin , cn cos , dn 1. (15)
In [19], three sets of model equations are derived for modelling nonlinear and
dispersive long gravity waves travelling in two horizontal directions on shallow
waters of uniform depth. Omitting the higher order terms, one of these equations,
the Wu-Zhang (WZ) equation, can be written as
ut + uux + vuy + wx = 0,
vt + uvx + vvy + wy = 0,
wt + (uw)x + (uw)y +1
3(u3x + uxyy + vxxy + v3y) = 0,
(16)
where w 1 is the elevation of the water wave, u is the surface velocity of wateralong the x direction, and v is the surface velocity of water along the y direction.
By scaling transformation and symmetry reduction, Eq.(16) can be reduced to
the (1+1)-dimensional dispersive long wave equation ([19-23])vt + vvy + wy = 0,
wt + (wu)y +1
3vyyy = 0.
(17)
A good understanding of all solutions of Eq.(16) is very helpful for coastal and
civil engineers to apply the nonlinear water wave model in a harbor and coastal
design. In [20], some special type soliton solutions for Eq.(16) is derived directly
by using the standard and nonstandard truncation of the WTCs approach and
the modified Contes invariant Painleve expansion for the WZ equation. In [21],
Zheng et al. obtain known solitary wave solutions, other new and more generalsolutions of Eq.(17) by using the generalized extended tanh-function method with
a new ansatze. Zhang and Li [22] study bidirectional solitons on water of Eq.(17)
by using the Darboux transformation method. In this paper, we consider the
following classical Boussinesq systemt + [(1 + ) u]x +
1
4uxxx = 0,
ut + uux + x = 0,(18)
-
7/27/2019 Boussinesq Solitons
5/11
TRAVELLING AND PERIODIC WAVE SOLUTIONS 95
where is the elevation of water wave, and u is the surface velocity of water along
the xdirection ([24-26]). Li et al. [24] gave two basic Darboux transformationsof a spectral problem associated with the Broer-Kaup system and used them to
generate new solutions of the Eq.(18). Recently, Li and Zhang [25] presented
the third kind of Darboux transformation of Eq.(18) and they discussed its rela-
tionship with the two basic Darboux transformations. Thus, they obtained the
solutions of multiple soliton interactions. More recently, Zhang et al. [26] maked
a simple ansatz to the solutions for Eq.(18) and they obtained the general ex-
plicit solutions. Here we use the modified extended tanh-function and the Jacobielliptic function methods for obtaining new travelling wave and Jacobi doubly
periodic wave solutions of Eq.(18).
2. Travelling and Periodic Wave Solutions of Eq.(18)
To seek the travelling wave and Jacobi doubly periodic wave solutions of
Eq.(18). We make the travelling wave transformation (x, t) = (x) , u (x, t) =
U() , = x t and we change Eq.(18) into the form
() + U() () + U() + 14
U
() = 0, (19)
() U() + 12
U2 () = 0, (20)
where the prime denotes d/d ([21]). Inserting Eq.(20) into Eq.(19) leads to an
ordinary differential equation
3
2U2 () 1
2U3 () 2U() + U() + 1
4U
() = 0. (21)
Balancing U
with U3 yields m = 1. Therefore, we have
U = a0 + a1 + b0 + b11. (22)
Substituting Eq.(22) into Eq.(21) and making use of Eq.(3), with the help of
Mathematica we get a system of algebraic equations for a0, a1, b0, b1, b and :
3
2a20 + 3a0b0 + 3a1b1 +
3
2b20
3
2a20b0
1
2a30 3a1b1a0 3a0a1b1
12
b30 3
2a0b
20 2a0 2b0 + a0 + b0 = 0,
-
7/27/2019 Boussinesq Solitons
6/11
96 M. INC
3a0a1 + 3a1b0 32
a1b20
3
2a21b13a0a1b0
3
2a20a1 + a1
12
+
1
2a1b = 0,
3a0b1 + 3b0b1 3
2b1a
20
1
2b31
3
2a1b
20
3
2b20b1
3
2a1b
21 3a0b0b1
2b1 + b1 +1
2bb1 = 0,
a21 a21b0 a0a21 = 0, b21 b21b0 a0b21 = 0,1
2a31 +
1
2a1 = 0,
1
2b2b1 = 0.
From the out put of symbolic computation software Mathematica, we obtain
b = 0, a1 = i, a0 = b0 and b1 = 13
i
2 + 1
, (23)
b = 0, b0 = b1 = 0, a1 = i, a0 =1
3
2 1
, b =
1
3
44 162 + 6 + 1
,
(24)
b = 0, b0 = b1 = 0, a1 = i, a0 = and b = 2 2, (25)where b0 and b are arbitrary constants. Since b is a arbitrary parameter, according
to (5)-(7) and (23)-(25), we obtain three kinds of travelling wave solutions for the
Eq.(18): Soliton solutions with b < 0,
u1 =1
3
2 1
ib tanh
b (x t)
, (26)
u2 =1
3
2 1
ib coth
b (x t)
, (27)
where b = 13
44 162 + 6 + 1 .
u3 = i
2 + 2 tanh
2 + 2 (x t)
, (28)
u4 = i
2 + 2 coth
2 + 2 (x t)
. (29)
Periodic solutions with b > 0,
u5 =1
3
2 1
i
b tan
b (x t)
, (30)
u6 =1
3
2 1
i
b cot
b (x t)
, (31)
-
7/27/2019 Boussinesq Solitons
7/11
TRAVELLING AND PERIODIC WAVE SOLUTIONS 97
where b = 13
44 162 + 6 + 1 .
u7 =
2 + 2 tan
2 + 2i (x t)
, (32)
u8 =
2 + 2 cot
2 + 2i (x t)
. (33)
A rational solution with b = 0,
u9 = b0 i
2 + 4
3 (x t) . (34)
According to the Jacobi elliptic function method, we get the following Jacobidoubly periodic wave solutions for Eq.(18):
u10 =
1
2
1 + m2
2
1/2msn
x
1
2
1 + m2
2
1/2t
, (35)
u11 =
1
2
1 2m2
2
1/2mcn
x
1
2
1 2m2
2
1/2t
, (36)
u12 =
1
2
2m2
2
1/2 dn
x
1
2
2m2
2
1/2t
, (37)
u13 =
1
2
2m2
2
1/2
cs
x
1
2
2m2
2
1/2
t
. (38)
And (35)-(38) corresponding travelling wave solutions are
u14 = i + tanh (x it) , (39)
u15 =
3i + sec h
x
3it
, (40)
u16 = i
5
2 csc h
x
3
2it
. (41)
Remark. It is easily seen that u1, . . . , u8 are like the solutions of Zheng etal. [21] but these solutions are not the same. We knowledge, the obtained solu-
tions of Eq.(18), u9, u14, u15 and u16 were not found by the modified extended
tanh-function method ([14, 15]) and the generalized extended tanh-function
method ([21]). In addition, we obtain some new complex formal solutions and
Jacobi doubly periodic wave solutions in the paper. To compare the new for-
mal solutions for Eq.(18) with the known formal solutions, we draw some plots
-
7/27/2019 Boussinesq Solitons
8/11
98 M. INC
for some formal solutions of WZ equation (18). The properties of some formal
solutions are shown in Figure 1.
Figure 1. The soliton and periodic wave solutions of Eq.(18), where = 1, 1.5.
-
7/27/2019 Boussinesq Solitons
9/11
TRAVELLING AND PERIODIC WAVE SOLUTIONS 99
Figure 2. The Jacobi doubly periodic wave solutions of Eq.(18).
3. Conclusions
In this paper, the modified extended tanh-function mathod and the Jacobi
elliptic function expansion method are applied to the classical Boussinesq system.
The aim to obtain travelling wave and Jacobi doubly periodic wave solutions of
this equation by using these methods have been achieved. In the fact, the present
methods are readily applicable to a large variety of such nonlinear equations. The
properties of the Jacobi doubly periodic wave solutions are shown in Figure 2.
Our present methods are very easy applied to both differential equations and
linear or nonlinear differential systems.
-
7/27/2019 Boussinesq Solitons
10/11
100 M. INC
References
[1] R. X. Yao and Z. B. Li, New solitary wave solutions for nonlinear evolution equations, Chin.
Phys., 11:9 (2002), 864-868.
[2] R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys.
Lett. A, 85:8 (1981), 407-408.
[3] M. J. Ablowitz and P. A. Clarkson, Solitons: Nonlinear Evolution Equations and Inverse
Scattering, Cambridge University Press, Cambridge, 1991.
[4] Z. F. Zhang and F. M. Wu, Backlund transformation and multiple soliton solutions for the
(3 + 1)-dimensional Jimbo-Miwa equation, Chin. Phys., 11:5 (2002), 425-428.
[5] W. Hereman and M. Takaoka, Solitary wave solutions of nonlinear evolution and wave
equations using a direct method and MACSYMA, J. Phys. A, 23:21 (1990), 4805-4822.
[6] B. Q. Lu, B. Z. Xiu, Z. L. Pang and X. F. Jiang, Exact travelling wave solution of one class
of nonlinear diffusion equation, Phys. Lett. A, 175:2 (1993), 113-115.
[7] E. G. Fan, Darboux transformationans soliton-like solutions for the Gerdjikov-Ivanov equa-
tion, J. Phys. A, 33:39 (2000), 6925-6933.
[8] P. J. Olver, Applications of Lie Groups to Differential Equations, New York, Springer, 1990.
[9] X. Feng, Exploratory approach to explicit solution of nonlinear evolution equations, Intern.
J. Theor. Phys., 39:1 (2000), 207-222.
[10] E. G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys.
Lett. A, 277:4 (2000), 212-218.
[11] E. G. Fan, Soliton solutions for a generalized Hirota-Satsuma coupled KdV equation and a
coupled MKdV equation, Phys. Lett. A, 282:1 (2001), 18-22.
[12] E. J. Parkers and B. R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers
equation, Phys. Lett. A, 229:4 (1997), 217-220.
[13] E. G. Fan, Travelling wave solutions for two generalized Hirota-Satsuma KdV systems, Z.
Naturforsch, 56a:3 (2001), 312-318.
[14] S. A. Elwakil, S. K. El-labany, M. A. Zahran and R. Sabry, Modified extended tanh-function
method for solving nonlinear PDEs, Phys. Lett. A, 299 (2002), 179-188.
[15] S. A. Elwakil, S. K. El-labany, M. A. Zahran and R. Sabry, Two new applications of the
modified extended tanh-function method, Z. Naturforsch, 58a:1 (2003), 1-8.
[16] E. G. Fan, A new algebraic method for finding the line soliton solutions and doubly periodic
wave solution to a two-dimensional perturbed KdV equation, Chaos, Solitons and Fractals,
15:3 (2003), 567-574.
[17] E. G. Fan, Uniformly constructing a series of explicit exact solutions to nonlinear equations
in mathematical physics, Chaos, Solitons and Fractals, 16:5 (2003), 819-839.
[18] E. G. Fan, Multiple travelling wave solutions of nonlinear evolution equations using a unified
algebraic method, J. Phys. A, 35:32 (2002), 6853-6872.[19] T. Y. Wu and J. E. Zhang, In Mathematics is for Solving Problems, Philadephia, SIAM,
1996.
[20] C. L. Chen, X. Y. Tang and S. Y. Lou, Solutions of a (2 + 1)-dimensional dispersive long
wave equation, Phys. Rev. E, 66:036605 (2002), 1-9.
[21] X. Zheng, Y. Chen and H. Zhang, Generalized extended tanh-function method and its ap-
plication to (1 + 1)-dimensional dispersive long wave equation, Phys. Lett. A, 311 (2003),
145-157.
[22] E. J. Zhang, and Y. Li, Bidirectional solitons on water, Phys. Rev. E, 67:016306 (2003),
1-8.
-
7/27/2019 Boussinesq Solitons
11/11
TRAVELLING AND PERIODIC WAVE SOLUTIONS 101
[23] C. L. Chen and S. Y. Lou, Soliton excitations and periodic waves without dispersion relation
in shallow water system, Chaos, Solitons and Fractals, 16 (2003), 27-35.[24] Y. Li, W. X. Ma and J. E. Zhang, Darboux transformations of classical Boussinesq system
and its new solutions, Phys. Lett A, 275:1 (2000), 60-66.[25] Y. Li and J. E. Zhang, Darboux transformations of classical Boussinesq system and its multi
soliton solutions, Phys. Lett A, 284:6 (2001), 253-258.
[26] S. Q. Zhang, G. Q. Xu and Z. B. Li, General explicit solutions of a classical Boussinesq
system, Chin. Phys., 11:10 (2002), 993-995.
Department of Mathematics, Firat University, Elazig 23119 / TURKIYE.
E-mail: [email protected]