Families of exact solutions of a new extended (2+1)-dimensional Boussinesq equation

TL;DR

This paper derives various exact solutions, including solitons, breathers, rogue waves, and lumps, for a newly introduced extended (2+1)-dimensional Boussinesq equation, expanding the understanding of its nonlinear wave phenomena.

Contribution

It presents the first derivation of one-soliton solutions and constructs multi-soliton, breather, rational, and semi-rational solutions for the extended equation using Hirota and traveling wave methods.

Findings

Derived one-soliton solutions of both bright and dark types.

Obtained N-soliton, breather, and rational solutions including rogue waves and lumps.

Constructed semi-rational solutions combining lumps with solitons and breathers.

Abstract

A new variant of the $(2+1)$-dimensional [$(2+1)d$] Boussinesq equation was recently introduced by J. Y. Zhu, arxiv:1704.02779v2, 2017; see eq. (3). First, we derive in this paper the one-soliton solutions of both bright and dark types for the extended $(2+1)d$ Boussinesq equation by using the traveling wave method. Second, $N$-soliton, breather, and rational solutions are obtained by using the Hirota bilinear method and the long wave limit. Nonsingular rational solutions of two types were obtained analytically, namely: (i) rogue-wave solutions having the form of W-shaped lines waves and (ii) lump-type solutions. Two generic types of semi-rational solutions were also put forward. The obtained semi-rational solutions are as follows: (iii) a hybrid of a first-order lump and a bright one-soliton solution and (iv) a hybrid of a first-order lump and a first-order breather.