Dynamics of breather, multi-wave, interaction and other wave solutions to the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

Abstract

Purpose The purpose of this paper is to study the new (3 + 1)-dimensional integrable fourth-order nonlinear equation which is used to model the shallow water waves. Design/methodology/approach By means of the Cole–Hopf transform, the bilinear form of the studied equation is extracted. Then the ansatz function method combined with the symbolic computation is implemented to construct the breather, multiwave and the interaction wave solutions. In addition, the subequation method tis also used to search for the diverse travelling wave solutions. Findings The breather, multiwave and the interaction wave solutions and other wave solutions like the singular periodic wave structure and dark wave structure are obtained. To the author’s knowledge, the solutions obtained are all new and have never been reported before. Originality/value The solutions obtained in this work have never appeared in other literature and can be regarded as an extension of the solutions for the new (3 + 1)-dimensional integrable fourth-order nonlinear equation.

On traveling wave solutions with stability and phase plane analysis for the modified Benjamin-Bona-Mahony equation

“…The study on solutions, structures, interactions, and further properties of solitary wave and soliton attained much concentration, and different meaningful results have been successfully derived [ 19 – 26 ]. In recent years, significant advancements have been made in the study of soliton solutions through the utilization of the Hirota bilinear method [ 27 ].…”

Soliton molecules, interaction and other wave solutions of the new (3+1)-dimensional integrable fourth-order equation for shallow water waves

“…With the development of soliton theory, many and varied effective approaches for solving the NPDEs have been proposed, such as the Kudryashov approach [18,19], extended F-Expansion approach [20][21][22][23], Bäcklund transformation [24][25][26][27], G'/G-expansion technique [28,29], variational technique [30,31], exp-function approach [32][33][34][35], trial equation method [32,36], general integral method [37,38] and so on [39][40][41][42][43][44][45][46][47][48][49]. In this research, we are going to plumb the exact solutions of the new IFNE in the (3+1)-dimensional as [50,51] where , a b and g are the non zero real numbers, x y z t , , , ( ) y y = is like the Boussinesq equation that usually describes the both right and left travelling waves. In [50], the lumps and multiple soliton solutions are investigated by using the simplified Hirota's method and the lump schemes.…”

“…In [50], the lumps and multiple soliton solutions are investigated by using the simplified Hirota's method and the lump schemes. In [51], the breather, multi-wave, interaction and other wave solutions like the singular periodic wave and dark wave solutions are constructed. Hereby, we will look for some new kinds of the exact solutions to equation (1.1).…”