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 Zhu [Line soliton and rational solutions to (2+1)-dimensional Boussinesq equation by Dbar problem, 2017. arXiv:1704.02779v2 ; 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.