Soliton and lump-soliton solutions in the Grammian form for the Bogoyavlenskii\u2013Kadomtsev\u2013Petviashvili equation

Wenjuan Rui,Yufeng Zhang

doi:10.1186/s13662-020-02602-3

Wenjuan Rui, Yufeng Zhang

Open Access

https://doi.org/10.1186/s13662-020-02602-3

Copy DOI

Abstract

This paper investigates the Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation by using Hirota’s direct method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Soliton solutions in the Grammian determinant form for the BKP-II equation are obtained and soliton collisions are shown graphically. Lump-soliton solutions for the BKP-I equation are presented in terms of the Grammian determinants. Various evolution processes of the lump-soliton solutions are demonstrated graphically through the study of three kinds of lump-soliton solutions. The fusion of lumps and kink solitons into kink solitons and the fission of kink solitons into lumps and kink solitons are observed in the interactions of lumps and solitons.

Highlights

Investigation of explicit solutions for nonlinear evolution equations (NLEEs) is very important to understand complex nonlinear phenomena in plasma physics, nonlinear optics, fluid mechanics, and other scientific fields [1,2,3,4]
4 Discussion on the interactions between lumps and kink solitons we investigate the dynamics of lump-soliton solutions for the BKP-I equation
N -soliton solutions in the Grammian determinant form for the BKP-II equation are derived and soliton collisions are illustrated through graphs

Summary

Introduction

Investigation of explicit solutions for nonlinear evolution equations (NLEEs) is very important to understand complex nonlinear phenomena in plasma physics, nonlinear optics, fluid mechanics, and other scientific fields [1,2,3,4]. One of the advantages of deriving solutions in determinant form is that soliton solutions and many other types of solutions, such as complexiton solutions [18, 19], rogue waves [20,21,22,23,24], lumps [25], and semi-rational solutions [26,27,28,29,30], Rui and Zhang Advances in Difference Equations (2020) 2020:195 can be derived Another advantage is that one can get solutions of any order based on determinant solutions, and the expression of the solutions is very simple [6, 15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. We investigate the Grammian determinant solutions and the interactions of the obtained nonlinear waves for the following Bogoyavlenskii–Kadomtsev–Petviashvili (BKP) equation [32, 33]:

Objectives

Conclusion