# Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles

Research paper by Jian-Min Tu, Shou-Fu Tian, Mei-Juan Xu, Xiao-Qiu Song, Tian-Tian Zhang

Indexed on: 18 Sep '15Published on: 18 Sep '15Published in: Nonlinear Dynamics

#### Abstract

A generalized (3+1)-dimensional nonlinear wave is investigated, which describes many nonlinear phenomena in liquid containing gas bubbles. In this paper, a lucid and systematic approach is proposed to systematically study the complete integrability of the equation by using Bell’s polynomials scheme. Its bilinear equation, N-soliton solution and Bäcklund transformation with explicit formulas are successfully structured, which can be reduced to the analogues of (3+1)-dimensional KP equation, (3+1)-dimensional nonlinear wave equation and Korteweg-de Vries equation, respectively. Moreover, the infinite conservation laws of the equation are found by using its Bäcklund transformation. All conserved densities and fluxes are presented with explicit recursion formulas. Furthermore, by employing Riemann theta function, the one- and two-periodic wave solutions for the equation are constructed well. Finally, an asymptotic relation is presented, which implies that the periodic wave solutions can be degenerated to the soliton solutions under some special conditions.