Auto-Bäcklund transformation and new exact solutions of the (3+1)-dimensional KP equation with variable coefficients

The (3+1)-dimensional variable coefficient Kadomtsev-Petviashvilli equation describes the dynamics of solitons and nonlinear waves in plasmas and superfluids. Based on computerized symbolic computation and extended variable coefficient homogeneous balance method, several families of exact soliton-like solutions, rational solutions, and auto-Backlund transformation are presented. With the use of the auto-BT and the ε-expansion method, we can obtain a soliton-like solution including N-solitary wave of the (3+1)-dimensional Kadomtsev-Petviashvilli equation with variable coefficients. Especially, we get a soliton-like solution including two-solitary waves as an illustrative example in detail.