Related papers: Modified method of simplest equation for obtaining…
We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation with steplike initial data leading to a rarefaction wave. In addition to the leading asymptotic we also compute the…
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related…
We apply an extension of a new method of group classification to a family of nonlinear wave equations labelled by two arbitrary functions, each depending on its own argument. The results obtained confirm the efficiency of the proposed…
A method for solving a quasilinear nonelliptical equation of the second order is developed and we give classification and parametrization of simple elements of the equation.We find exact solutions of an equation for potential stationary…
In this paper, we obtain some new explicit travelling wave solutions of the perturbed KdV equation through recent factorization techniques that can be performed when the coefficients of the equation fulfill a certain condition. The…
To obtain new types of exact travelling wave solutions to nonlinear partial differential equations, a number of approximate methods are known in the literature. In this study, we extend the class of auxiliary equations of Fibonnacci&Lucas…
In this paper we employ three recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a family of so-called short-pulse equations (SPE). A recent, novel application of phase-plane…
The bifurcation of plane waves to localised structures is investigated in the Dysthe equation, which incorporates the effects of mean flow and wave steepening. Through the use of phase modulation techniques, it is demonstrated that such…
In this communication we consider the widely used nonlinear Fokas-Lenells equation, the cubic focussing nonlinear Schr\"{o}dinger equation in (2+1)-dimensions and the coupled Drinfel'd-Sokolov-Wilson equation and attempt to construct almost…
We introduce the concept of soliton solutions of integrable nonlinear partial differential equations and point out that the inverse spectral method represents the rigorous mathematical formalism to construct such solutions. We work with the…
This article demonstrates how variation of parameters can be successfully implemented in combination with other classical techniques, such as the method of characteristics, to derive novel classes of solutions to nonlinear partial…
We establish the nonlinear stability of solitary waves (solitons) and periodic traveling wave solutions (cnoidal waves) for a Korteweg-de Vries (KdV) equation which includes a fifth order dispersive term. The traveling wave solutions which…
We emphasize that construction of travelling wave solutions for partial differential equations is a problem of considerable interest and thus introduce a simple algebraic method to generate such solutions for equations in the Burgers…
For the first time, Schr\"odinger equations with cubic and more complex nonlinearities containing the unknown function with constant delay are analyzed. The physical considerations that can lead to the appearance of a delay in such…
New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…
We discuss a new version of a method for obtaining exact solutions of nonlinear partial differential equations. We call this method the Simple Equations Method (SEsM). The method is based on representation of the searched solution as…
A new method named rational expansion method of exponent function is presented to find exact traveling wave solutions of differential-difference equations. This method generalizes the so-called tanh-method and other similar methods. Some…
In the framework of hyperbolic conservation laws regularised by including diffusive and dispersive terms, we study monotone travelling waves for the generalised Rosenau-Korteweg de Vries equation. We establish existence as well as linear…
We consider travelling internal waves in a two-layer fluid with linear shear currents from the viewpoint of the extended Korteweg-de Vries (eKdV) equation derived from a strongly-nonlinear long-wave model. Using an asymptotic…
The generalized Bretherton equation is studied. The classification of the meromorphic traveling wave solutions for this equation is presented. All possible exact solutions of the generalized Brethenton equation are given.