Abstract
In this paper, the polynomial solutions in terms of Jacobi’s elliptic functions of the KdV equation with a self-consistent source (KdV-SCS) are presented. The extended (G′/G)-expansion method is utilized to obtain exact traveling wave solutions of the KdV-SCS, which finally are expressed in terms of the hyperbolic function, the trigonometric function, and the rational function. Meanwhile we find the Lie point symmetry and Lie symmetry group and give several group-invariant solutions for the KdV-SCS. Finally, we supplement the results of the Painlevé property in our previous work and get the Bäcklund transformations of the KdV-SCS.
Highlights
It is well known that the soliton equations with self-consistent sources (SESCSs) can exhibit abundant nonlinear dynamics compared to soliton equations themselves and have important physical applications [1]
We get the polynomial solutions of the KdV equation with a self-consistent source (KdV-SCS) which are expressed in terms of Jacobi’s elliptic functions
The extended (G/G)-expansion method has been successfully applied in this paper to deal with the new exact traveling wave solutions of the KdV-SCS
Summary
It is well known that the soliton equations with self-consistent sources (SESCSs) can exhibit abundant nonlinear dynamics compared to soliton equations themselves and have important physical applications [1] These SESCSs are usually used to describe interactions between different solitary waves and are relevant in some problems related with hydrodynamics, solid state physics, or plasma physics [2,3,4]. The soliton solutions for the KdV hierarchy with self-consistent sources are obtained by the inverse scattering method [5]. The Jacobi’s elliptic function solutions, the group-invariant solutions by the Lie group approach [26], and the extended (G/G)expansion method for the KdV-SCS have not been presented.
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