Elastic and Annihilation Solitons of the (3+1)-Dimensional Generalized Shallow Water Wave System

Many dynamical problems in physics and other natural fields are usually characterized by the nonlinear evolution of partial differential equations known as governing equations. Searching for an analytical exact solution to a nonlinear system has long been an important and interesting topic in nonlinear science both for physicists and mathematicians, and various methods for obtaining exact solutions of a nonlinear system have been proposed, for example, the bilinear method, the standard Painlevé truncated expansion, the method of ‘coalescence of eigenvalue’ or ‘wavenumbers’, the homogenous balance method, the hyperbolic function method, the Jacobian elliptic method, the variable separation method, the (G′/G)-expansion method [1 – 12], and the mapping method [13 – 15], etc. The mapping approach is a kind of classic, efficient, and well-developed method to solve nonlinear evolution equations. The remarkable characteristic of which is that we can have many different ansatzes and, therefore, a large number of solutions [16 – 21]. In this paper, with the mapping approach and a linear variable separation approach, a new family of exact solutions with arbitrary functions of the (3+1)-dimensional generalized shallow water wave (GSWW) system is derived. Based on the derived solitary wave solution, we study some novel soliton excitations such as elastic and annihilation solitons. The GSWW system is given by