We have solved the Burgers-Fisher (BF) type equations, with time-dependent coefficients of convection and reaction terms, by using the auxiliary equation method. A class of solitary wave solutions are obtained, and some of which are derived for the first time. We have studied the effect of variable coefficients on physical parameters (amplitude and velocity) of solitary wave solutions. In some ...

Self-focusing of an intense optical beam in a plasma is studied, including the nonlinear effects of both the relativistic electron mass and the ponderomotive potential due to the electromagnetic wave. An exact steady-state asymptotic solution of beam propagation in a localized solitary waveform is obtained in slab geometry. Amplitude-width scaling relations are obtained, which imply that the wi...

In this paper, we study a class of nonlinear Schrödinger equations (NLS) which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we...

Double cnoidal waves of the Korteweg-de Vries equation are studied by direct solution of the nonlinear boundary value problem. These double cnoidal waves, which are the spatially periodic generalization of the well-known double soliton, are exact solutions with two independent phase speeds. The equation is written in terms of two phase variables and expanded in two-dimensional Fourier series. T...

By means of the coupled amplitude-phase method we find analytical dark solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic terms describing the effects of quintic nonlinearity on the ultra-short (femtosecond) optical soliton propagation in non-Kerr media. The dark solitary wave solution exists even for the coefficients of quintic terms much larger than...

The LL equation 1 has aroused considerable interest among physicists and mathematicians. For the one-dimensional case, there have been many contributions to the study of the soliton solution, the interaction of solitary waves, and other properties of the solitary waves 2–4 . However, we point out here that it will be a more challenging task study the high-dimensional dynamics 5, 6 about LL equa...

Using a variational approach we have studied the shape preserving coherent propagation of light pulses in a resonant dispersive medium in the presence of the Kerr nonlinearity. Within the framework of a combined nonintegrable system composed of one nonlinear Schrödinger and a pair of Bloch equations, we show the existence of a solitary wave. We have tested our analytical solution through numeri...

Abstract We prove the existence of a L 2 -normalized solitary wave solution for Maxwell-Dirac equations in (3+1)-Minkowski space. In addition, Coulomb-Dirac model, describing fermions with attractive Coulomb interactions mean-field limit, we (positive) energy minimizer.