Lump and Interaction Solutions to the ( 3 + 1 )-Dimensional Variable-Coefficient Nonlinear Wave Equation with Multidimensional Binary Bell Polynomials

Xuejun Zhou,Mostafa Abotaleb,Fangyuan Zhou,Onur Alp Ilhan,Sutarto Sutarto,Jalil Manafian,Giovanni Di Fratta

doi:10.1155/2021/4550582

Abstract

In this paper, we study the ( 3 + 1 )-dimensional variable-coefficient nonlinear wave equation which is taken in soliton theory and generated by utilizing the Hirota bilinear technique. We obtain some new exact analytical solutions, containing interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the generalized ( 3 + 1 )-dimensional nonlinear wave equation in liquid with gas bubbles by the Maple symbolic package. Making use of Hirota’s bilinear scheme, we obtain its general soliton solutions in terms of bilinear form equation to the considered model which can be obtained by multidimensional binary Bell polynomials. Furthermore, we analyze typical dynamics of the high-order soliton solutions to show the regularity of solutions and also illustrate their behavior graphically.

Highlights

It is known that there are a variety of useful and powerful tools to deal with the nonlocal equations, namely, the improved tan ðφ/2Þ-expansion method [1], the homotopy perturbation method [2], Lie symmetry analysis [3], the Bäcklund transformation method [4], the sine-Gordon expansion approach [5], the (G′/G, 1/G), modified (G′/G2), and (1/G′)-expansion methods [6], the multiple Expfunction method [7,8,9,10], Hirota’s bilinear method including the (2 + 1)-dimensional variable-coefficient Caudrey-DoddGibbon-Kotera-Sawada equation [11], the generalized unstable space time fractional nonlinear Schrödinger equation [12], the inverse Cauchy problems [13], a generalized hyperelastic rod equation [14], the Kadomtsev-Petviashvili equation [15], the bKP equation [16], the generalized Burgers equation [16], the inverse scattering transformation method [17, 18], and the KP hierarchy reduction method [19]
An improved Hirota bilinear method for the nonlocal complex MKdV equation was constructed in Ref. [21]
Under the multidimensional binary Bell polynomials, we derive that the lump-soliton and its interaction solutions of the (3 + 1)-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles has been successfully obtained

Summary

Introduction

It is known that there are a variety of useful and powerful tools to deal with the nonlocal equations, namely, the improved tan ðφ/2Þ-expansion method [1], the homotopy perturbation method [2], Lie symmetry analysis [3], the Bäcklund transformation method [4], the sine-Gordon expansion approach [5], the (G′/G, 1/G), modified (G′/G2), and (1/G′)-expansion methods [6], the multiple Expfunction method [7,8,9,10], Hirota’s bilinear method including the (2 + 1)-dimensional variable-coefficient Caudrey-DoddGibbon-Kotera-Sawada equation [11], the generalized unstable space time fractional nonlinear Schrödinger equation [12], the inverse Cauchy problems [13], a generalized hyperelastic rod equation [14], the Kadomtsev-Petviashvili equation [15], the bKP equation [16], the generalized Burgers equation [16], the inverse scattering transformation method [17, 18], and the KP hierarchy reduction method [19]. For equation (1), some solutions including the multisoliton, Bäcklund transformation, infinite conservation laws, lump solutions, and other soliton wave solutions have been investigated in Refs. The multisoliton solutions and periodic solutions for the (3 + 1 )-dimensional variable-coefficient nonlinear wave equation in liquid with gas bubbles were reported by Guo and Chen [31]. Two different types of bright solutions for the generalized (3 + 1)-dimensional nonlinear wave equation by the traveling wave method were obtained by Guo and Chen [33]. The major aim of this paper is to obtain some novel exact analytical solutions, including interaction between a lump-two kink solitons, interaction between two lumps, and interaction between two lumps-soliton, lump-periodic, and lump-three kink solutions for the (3 + 1)-dimensional variable-coefficient (VC) nonlinear wave equation in liquid with gas bubbles through the method of the bilinear analysis.

Multidimensional Binary Bell Polynomials

Þ φ1ðtÞðcðt

Conclusion