Exact solutions and Bäcklund transformation for a generalized (3+1)-dimensional variable-coefficient Fokas-typed equation

Abstract

This paper is concerned with a generalized $(3+1)$-dimensional variable-coefficient Fokas-typed equation, which is used to describe the interaction of nonlinear waves in ocean dynamics, shallow water waves, etc. By virtue of the Hirota bilinear method, the one-, two-, and three-soliton solutions are derived. Three kinds of bilinear Bäcklund transformation (BT) and the Bell-polynomial-typed BT are constructed based on the bilinear form. In addition, we obtain the lump solution and breather solution via the test function method. Three sets of variable coefficients are selected and the dynamic behavior of the corresponding lump solution is observed and analyzed. The main physics characteristics and spatial structure of these exact solutions are graphically discussed. Our results can be helpful in explaining certain related nonlinear phenomena.