The KdV equation models the propagation of long waves in dispersive media, while the NLS equation models the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A system that couples the two equations to model the interaction of long and short waves is mathematically attractive and such a system has been studied over the last decades. We evaluate the validity of this...

We construct a one-parameter family of N=3 supersymmetric extensions of the KdV equation as a Hamiltonian flow on N=3 superconformal algebra and argue that it is non-integrable for any choice of the parameter. Then we propose a modified N=3 super KdV equation which possesses the higher order conserved quantities and so is a candidate for an integrable system. Upon reduction to N=2, it yields th...

In order to find higher dimensional integrable models, we study differential equations of hyperelliptic functions up genus four. For two, can be written in the Hirota form. If is more than one, have KdV equation. and another equations. becomes three, there appear which cannot form, means that form not enough characterize We shown some are satisfied for general genus. obtain step by step.

In this paper, using the Lie group analysis method, we study the invariance properties of the general time fractional fifth-order Korteweg-de Vries (KdV) equation. A systematic research to derive Lie point symmetries of the equation is performed. In the sense of point symmetry, all of the geometric vector fields and the symmetry reductions of the equation are obtained, the exact power series so...

In this paper, tanh method is applied to obtain exact solutions for two systems of nonlinear wave equations, namely, two component evolutionary system of homogeneous KdV equations of order 3 (type I as well as type II). Moreover, traveling wave hypothesis is used to obtain sech solution of type II coupled KdV system, in a more general setting. The results show that this method presents exact so...

in this paper, we consider a coupled system of nonlinear fractional differential equations (fdes), such that bothequations have a particular perturbed terms. using emph{leray-schauder} fixed point theorem, we investigate the existence and multiplicity of positive solutions for this system.

The aim of this paper are in two fold: First, introduce a new simplified bilinear method based on a transformation method combined with the Hirota bilinear sense. Second, apply this new technique to study the time-space fractional Boussinesq equation. To the best of my knowledge, the proposed work present new N-soliton solutions of the fractional Boussinesq equation. These new exact solutions e...

Abstract In this paper, the KdV-Sawada-Kotera-Ramani(KdVSKR) equation in various dimensions are studied. The bilinear form of (1+1)-dimensional and (2+1)-dimensional KdVSKR obtained by independent transformation. Based on Hirota method, we constructed new interaction solutions studying unknown nonlinear differential equations for corresponding parameters. Three dimensional plots, density plots ...