1-Soliton solution of the coupled KdV equation and Gear–Grimshaw model

qt þ aqmqx þ bqxxx þ crnrx ¼ 0;
ð1Þ
rt þ k1qlrx þ k2rxxx ¼ 0:
ð2Þ
Here, the dependent variables are q and r, while the independent variables are x and t which are the spatial and time variables respectively. The coefficients a, b, c, kj for j = 1, 2 are constants. The exponents l, m and n are arbitrary. In (1) the first term represents the evolution term, while a and c represents the coefficients of nonlinear terms, while b is the coefficient of dispersion term. For (2), k1 is the coefficient of nonlinear term while k2 is the coefficient of dispersion term.
1-Soliton solution of the coupled KdV equation and Gear–Grimshaw model
Anjan Biswas a,*, M.S. Ismail b
a Applied Mathematics Research Center, Center for Research and Education in Optical Sciences and Applications, Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA b Department of Mathematics, College of Science, P.O. Box 80203, King Abdulaziz University, Jeddah-21589, Saudi Arabia
p2vA2B tanh coshp2 s
s
À
k1p2Al1A2B tanh
s coshlp1 þp2
s
À
k2p32A2B3 tanh
coshp2
s
þ
bp2ðp2
þ
1Þðp2 þ 2ÞA2B3
coshp2þ2s
tanh
s
¼
0:
ð10Þ
Now from (9), equating the exponents mp1 + p1 and p1 + 2 gives mp1 þ p1 ¼ p1 þ 2;
2. Coupled KdV equation
The dimensionless form of the coupled KdV equation with power law nonlinearity that is going to be studied in this paper is given by [2]
In order to seek soliton solutions to (1) and (2), the starting assumptions are [1,7]
qðx;
tÞ
¼
A1 coshp1
s
ð3Þ
and
rðx;
tÞ
¼
A2 coshp2
s
;
ð4Þ
where A1 and A2 are the amplitudes of the solitons of the q and r equations respectively. Also
s ¼ Bðx À vtÞ;
ð5Þ
where B is the inverse width of the solitons and v is the soliton velocity. The unknown exponents p1 and p2 will be deter-
mined in terms of l, m and n during the course of derivation of the soliton solution to (1) and (2). From (3), it is possible to obtain
ð11Þ
which yields
2 p1 ¼ m :
ð12Þ
Again equating the exponents np2 + p2 and p1 + 2 gives np2 þ p2 ¼ p1 þ 2;
ð13Þ
which implies
2ðm þ 1Þ
p2 ¼ mðn þ 1Þ :
article info
Keywords: Solitons Integrability Conserved quantities
abstract
This paper carries out the integration of the coupled KdV equation with power law nonlinearity. The solitary wave ansatz is used to carry out the integration. The domain restrictions of the coefficients of nonlinear and dispersion terms fall out. The results are then supplemented by numerical simulations.
qt
¼
p1
vA1B tanh coshp1 s
s
;
ð6Þ
qx
¼
À
p1A1B tanh
coshp1 s
s
ð7Þ
and
qxxx
¼
À
p31A1B3 tanh
coshp1 s
s
þ
p1ðp1
þ
1Þðp1 þ 2ÞA1B3
coshp1 þ2 s
tanh
s
;
ð8Þ
with similar expressions for the r variable. Substituting these into (1) and (2) yields
Applied Mathematics and Computation 216 (2010) 3662–3670
Contents lists available at ScienceDirect
Applied Mathematics and Computation
journal homepage: /locate/amc
A. Biswas, M.S. Ismail / Applied Mathematics and Computation 216 (2010) 3662–3670
3663
This coupled KdV equations given by (1) and (2) describes the interaction with two long waves with different dispersion relations. So, it is associated with most types of long waves with weak dispersion, for example internal acoustic and planetary waves that arises in geophysical fluid dynamics. Thus, the dependent variables q and r are the wave variables in the two wave modes. In this paper, the search is going to be conducted for soliton solutions to (1) and (2). Solitons are nonlinear waves that are the result of a delicate balance between dispersion and nonlinearity.
p1vA1B tanh coshp1 s
s
À
ap1Am1 þ1B tanh
s coshmp1 þp1
s
À
cp32An2þ1B tanh
s coshnp2þp2
s
À
bp31A1B3 tanh
coshp1 s
s
þ
bp1 ðp1
þ
1Þðp1 þ 2ÞA1B3
coshp1 þ2 s
tanh
s
¼
0
ð9Þ
and
* Corresponding author. E-mail address: biswas.anjan@ (A. Biswas).
0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.017
There are various methods of integration that are available to carry out the integration of NLEEs. They are F-expansion method, G0/G method, exponential function method, Lie group analysis, Adomian decomposition method, variational iteration method, homotopy perturbation method and many more. These methods have been developed in the past decade. Indeed, one has to be careful in applying these methods as pointed out by Kudryashov in 2009 [3]. In this paper, one such modern method of integrability will be applied to carry out the integration of the coupled Korteweg–de Vries (KdV) equation and then subsequently integrate the Gear–Grimshaw model. This is the solitary wave ansatz method.