Exact traveling wave solutions of some nonlinear evolution equations

Hitender Kumar,Fakir Chand

doi:10.1007/s40094-014-0114-z

Abstract

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV-Zakharov- Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyper- bolic functions, trigonometric functions and rational func- tions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolu- tion equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.

Highlights

The effort in finding exact solutions to nonlinear equations is significant for the understanding of most nonlinearH
Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV–Zakharov– Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations
The rest of the paper is structured as follows: in ‘‘Description of methods’’ we give brief descriptions of extended F-expansion and projective Riccati equation methods; in ‘‘Traveling wave reduction of some nonlinear evolution equations’’, a few NLEEs of physical interest are transformed into ellipticlike equations

Summary

Introduction

Equating each coefficient of FpGq or (Fp) to zero yields a system of algebraic equations for cji (j = 0, 1, 2, Á Á Án; i = 0, Á Á Á, j) and ki, x, or (ai, bj , i = 1, 2, Á Á Án; j = 1, 2, Á Á Án; ki, x) Solving these equations by use of Mathematica, cij, ki and x can be expressed in terms of Pi, Qi, Ri, l, m and the parameters of nonlinear ODE Eq (4).

ÀkÀk2 s2 and

Conclusions