New Interaction Patterns for the Truncated M-Fractional Kadomtsev–Petviashvili Equation in High-Dimensional Space

A truncated M-fractional Kadomtsev–Petviashvili (KP) equation in high-dimensional space has been proposed. The model provides the oretical support for studying the interaction patterns among waves. According to the attributes of the truncated M-fractional derivative, the truncated M-fractional KP equation can be reduced to the new extended KP equation. New interaction patterns which are compositions of cnoidal functions and soliton or trigonometric functions have been derived by the consistent Riccati expansion method. Applying the simple direct method, the finite symmetry transformation group and Bäcklund transformation have been constructed. Based on the known dark soliton solution and lump solution, new interaction patterns have been derived, including compositions of a dark soliton and an exponential function, compositions of a dark soliton and a trigonometric sine function, and compositions of a lump and a trigonometric sine function. The innovative aspect lies in the way that we find two effective ways to construct new interplay patterns of fractional differential equations.