Solitons, breath-wave transitions, quasi-periodic waves and asymptotic behaviors for a (2+1)-dimensional Boussinesq-type equation

In this paper, the N-soliton solutions of a (2+1)-dimensional Boussinesq-type equation are obtained by the Hirota’s bilinear method, and one-, two-, three-soliton solutions and their clear images are given in detail. Then, one breath-wave solution and two breath-wave solution are obtained by taking the complex conjugate of soliton solutions. The transformation mechanism of the breath-waves is analyzed systematically. Through the multi-dimensional Riemann theta function and bilinear method, the quasi-periodic wave solutions are obtained. Among these periodic waves, the high-dimensional complex three-periodic waves are firstly presented, the one-periodic waves are often applied to one-dimensional models of periodic waves in shallow water, the two-periodic waves and three-periodic waves are the generalization of one-periodic waves. The asymptotic behaviors of one-, two-, three-periodic waves and the relations between periodic wave solutions and soliton wave solutions are strictly established and proved by a limiting procedure. The characteristic line method is developed to analyze the dynamical characteristics of the quasi-periodic waves.