Multiple-soliton solutions and multiple-singular soliton solutions for two higher-dimensional shallow water wave equations

Abstract

Two (3 + 1)-dimensional shallow water wave equations are studied for complete integrability. The Hirota’s bilinear method is used to determine the multiple-soliton solutions for these equations. Moreover, multiple-singular soliton solutions will also be determined for each model. The analysis highlights the capability of the direct method in handling completely integrable equations.

Introduction

The (3 + 1)-dimensional shallow water wave equations [1]

$$u_{\mathit{yzt}}+u_{\mathit{xxxyz}}-6u_x{u}_{\mathit{xyz}}-6u_{\mathit{xz}}{u}_{\mathit{xy}}=0$$

and

$$u_{\mathit{xzt}}+u_{\mathit{xxxyz}}-2(u_{\mathit{xx}}{u}_{\mathit{yz}}+u_y{u}_{\mathit{xxz}})-4(u_x{u}_{\mathit{xyz}}+u_{\mathit{xz}}{u}_{\mathit{xy}})=0$$

will be studied. Both equations reduce to the potential KdV equation for $z=y=x$. The difference between the first terms of the two models is that x replaces y in the term $u_{\mathit{yzt}}$. The generalized shallow water wave equations studied by Ablowitz [2] and Hirota and Satsuma [3], and by Mansfield and Clarkson [1] arise as reduction of these two equations.

A variety of distinct methods are used for classification of integrable equations. The Painlevé analysis, the inverse scattering method, the Bäcklund transformation method, the conservation laws method, and the Hirota bilinear method [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] are mostly used in the literature for investigating complete integrability. The Hirota’s bilinear method is rather heuristic and possesses significant features that make it ideal for the determination of multiple-soliton solutions for a wide class of nonlinear evolution equations.

Two goals are set for this work. We first aim to use Hirota’s bilinear method to emphasize its power and reliability for handling integrable equations. We second aim to determine multiple-soliton solutions and multiple-singular soliton solutions [14], [15], [16], [17], [18], [19], [20], [21], [22] for these Eqs. (1), (2). The Hereman’s simplified form [12], [13] will be combined with the Hirota’s method to achieve these two goals.