Bilinear approach to soliton and periodic wave solutions of two nonlinear evolution equations of Mathematical Physics

Rui Cao,Lin Gao,Qiulan Zhao

doi:10.1186/s13662-019-2051-2

Abstract

In the present paper, the potential Kadomtsev–Petviashvili equation and (3+1)-dimensional potential-YTSF equation are investigated, which can be used to describe many mathematical and physical backgrounds, e.g., fluid dynamics and communications. Based on Hirota bilinear method, the bilinear equation for the (3+1)-dimensional potential-YTSF equation is obtained by applying an appropriate dependent variable transformation. Then N-soliton solutions of nonlinear evolution equation are derived by the perturbation technique, and the periodic wave solutions for potential Kadomtsev–Petviashvili equation and (3+1)-dimensional potential-YTSF equation are constructed by employing the Riemann theta function. Furthermore, the asymptotic properties of periodic wave solutions show that soliton solutions can be derived from periodic wave solutions.

Highlights

The construction of analytic solutions for nonlinear evolution equations (NLEEs) is a key topic in the study of nonlinear phenomena [1,2,3,4,5,6,7]
With a motivation to further expand the area of applications of this method, in the present paper, we study the potential Kadomtsev–Petviashvili equation and (3 + 1)dimensional potential-YTSF equation to illustrate the efficiency of using the combination of the Hirota method and the Riemann theta function
5 Discussion and conclusion In the present paper, we investigate the (2 + 1)-dimensional potential KP equation and (3 + 1)-dimensional potential-YTSF equation based on the Hirota method and the Riemann theta function

Summary

Introduction

The construction of analytic solutions for nonlinear evolution equations (NLEEs) is a key topic in the study of nonlinear phenomena [1,2,3,4,5,6,7]. Ma and Lee et al [9] investigated a 3 + 1 dimensional Jimbo–Miwa equation via a transformed rational function method and obtained exact solutions. The Hirota method is one of the most effective methods of constructing multiple soliton solutions of NLEEs. It can transform the given nonlinear evolution equations to the corresponding bilinear forms through the dependent variable transformation.

Results

Conclusion