Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations

Yongyi Gu,Jianming Qi

doi:10.1186/s13660-017-1587-5

Abstract

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the exp(-phi (z))-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

Highlights

In 1998, Yu et al [1] extended the Bogoyavlenskii Schiff equation ut + Φ(u)us = 0, Φ(u) = ∂x2 + 4u + 2ux∂x–1, (1)to the (3 + 1)-dimensional nonlinear evolution equations (NLEEs)–4ut + Φ(u)us x + 3uyy = 0, Φ(u) = ∂x2 + 4u + 2ux∂x–1. (2)Setting u := ux, equation (2) is changed into the (3 + 1)-dimensional potential Yu-TodaSasa-Fukuyama (YTSF) equation uxxxs – 4uxt + 4uxuxs + 2uxxus + 3uyy = 0. (3)The generalized (3 + 1)-dimensional Zakharov-Kuznetsov equation is given by a1u2ux + a2uxxx + a3uxyy + a4uxss + a5uux + a6uxxt + ut = 0. (4)Here ai (i = 1, 2, . . . , 6) are arbitrary constants.Gu and Qi Journal of Inequalities and Applications (2017) 2017:314
We study symmetries, symmetry reduction of the two higher-dimensional NLEEs, and we obtain their exact solutions via the exp(–φ(z))-expansion method and complex method
We can reduce the dimension of the NLEEs, which is relevant in the fields of mathematical physics and engineering

Summary

Introduction

If a4 = a5 = a6 = 0, equation (4) is the modified Zakharov-Kuznetsov equation [6] In recent years, it has aroused widespread interest in the study of NLEEs [7,8,9,10,11,12,13]. Over the past few years, many powerful methods for constructing the solutions of NLEEs have been used, for instance, the Bäcklund transform method [18], direct algebraic method [19], modified simple equation method [20], Lie group method [21, 22], exp(–φ(z))-expansion method [8, 9, 23, 24], and so on. We study symmetries, symmetry reduction of the two higher-dimensional NLEEs, and we obtain their exact solutions via the exp(–φ(z))-expansion method and complex method

Description of the methods

Exact solutions

Exact solutions of gZK equation via the complex method

Exact solutions of YTSF equation via the complex method

Conclusions