The unsmooth boundary will greatly affect motion morphology of a shallow water wave, and a fractal space is introduced to establish a generalized KdV-Burgers equation with fractal derivatives. The semi-inverse method is used to establish a fractal variational formulation of the problem, which provides conservation laws in an energy form in the fractal space and possible solution structures of t...

Two classes of rational solutions to a KdV-like nonlinear differential equation are constructed. The basic object is a generalized bilinear differential equation based on a prime number p 1⁄4 3. A conjecture is made that the two presented classes of rational solutions contain all rations solutions to the considered KdV-like equation, which are generated from polynomial solutions to the correspo...

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived. © 2010 Published by Elsevier Ltd

Recent concept of integrable nonholonomic deformation found for the KdV equation is extended to the mKdV equation and generalized to the AKNS system. For the deformed mKdV equation we find a matrix Lax pair, a novel two-fold integrable hierarchy and exact N-soliton solutions exhibiting unusual accelerating motion. We show that both the deformed KdV and mKdV systems possess infinitely many gener...

The variational iterationmethod VIM 1–4 and homotopy perturbationmethod HPM 5– 8 , proposed byHe, are powerful analytical methods for various kinds of linear and nonlinear problems. For example, the variational iteration method has been applied to autonomous ordinary differential equation 9 and delay differential equation 10 . Abdou and Soliman applied this method to Schrodinger-KDV, generalize...

For a generalized super KdV equation, three Darboux transformations and the corresponding Bäcklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax representations. The reduction of one of the Bäcklund–Darboux transformations and the corresponding discrete system are considered for Kupershmidt’s super KdV equation....

It is renowned that Hirota–Satsuma–Ito (HSI) equation widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With aid symbolic calculations, various constraints on free p...

An analytic study was conducted on coupled partial differential equations. We formally derived new solitary wave solutions of generalized coupled system of Zakharov-Kuznetsov (ZK) and KdV equations by using modified extended tanh method. The traveling wave solutions for each generalized coupled system of ZK and KdV equations are shown in form of periodic, dark, and bright solitary wave solution...