Families of exact solutions of a new extended $$\varvec{(2+1)}$$(2+1)-dimensional Boussinesq equation

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.

The families of explicit solutions for the Hirota equation

“…In order to seek dark one soliton solution of Equation , we let where A , η 1 , and η 2 are free unknown parameters. η 2 is the velocity of the soliton.…”

Rational and semi‐rational solutions for a (3 + 1)‐dimensional generalized KP–Boussinesq equation in shallow water wave

“…35 So far, the attention is mainly focused on the (2 + 1)-dimensional cases, and it has not been applied to the (3 + 1)-dimensional case yet. [36][37][38][39] Motivated by above open problem, we are going to propose two types of "long wave" limits and apply them to a new extended (3 + 1)-dimensional generalized KP-Boussinesq equation (1.2) in the form…”

Painlevé Analysis and Higher-Order Rogue Waves of a Generalized(3+1)-dimensional Shallow Water Wave Equation

“…Further, various nonlinear wave solutions including periodic waves, lump solutions, multi-waves, and interaction waves, etc., of the above mentioned works have been reported. Apart from the above listed models, several types of higher-dimensional nonlinear equations under different physical situations have been reported with interesting results on rogue waves in the recent years, see for example [33,34,35,36,37,38,39,40,41,42]. From these different possible and existing models, it is clear that the considered (3+1)D Hirota-Satsuma-Ito equation (1.1) is more general with much physical importance.…”