Bäcklund transformations and Riemann–Bäcklund method to a (3 + 1)-dimensional generalized breaking soliton equation

Abstract

In this paper, the bilinear method is employed to investigate the N-soliton solutions of a (3 + 1)-dimensional generalized breaking soliton equation. Three sets of bilinear Backlund transformations are obtained by means of gauge transformation. The Riemann–Backlund method is further extended to the (3 + 1)-dimensional nonlinear integrable systems. The quasiperiodic wave solutions of the (3 + 1)-dimensional generalized breaking soliton equation are systematically analyzed. The asymptotic properties of the quasiperiodic solutions are discussed by using a limiting procedure. The one-periodic and two-periodic waves tend to the 1-soliton and 2-soliton under a small amplitude limit, respectively. The dynamical characteristics of the one- and two-periodic waves are summarized by selecting different parameters. Furthermore, we obtain some new types of the quasiperiodic wave solutions of the variable coefficient (3 + 1)-dimensional generalized breaking soliton equation. These solutions present the dynamical behaviors of C-type, anti-C-type and Z-type periodic waves moving on the background of the periodic waves of bell type.