A splitting lattice Boltzmann scheme for (2+1)-dimensional soliton solutions of the Kadomtsev-Petviashvili equation

Recently, considerable attention has been given to (2+1)-dimensional Kadomtsev-Petviashvili equations due to their extensive applications in solitons that widely exist in nonlinear science. Therefore, developing a reliable numerical algorithm for the Kadomtsev-Petviashvili equations is crucial. The lattice Boltzmann method, which has been an efficient simulation method in the last three decades, is a promising technique for solving Kadomtsev-Petviashvili equations. However, the traditional higher-order moment lattice Boltzmann model for the Kadomtsev-Petviashvili equations suffers from low accuracy because of error accumulation. To overcome this shortcoming, a splitting lattice Boltzmann scheme for (2+1)-dimensional Kadomtsev-Petviashvili-Ⅰ type equations is proposed in this paper. The variable substitution method is applied to transform the Kadomtsev-Petviashvili-Ⅰ type equation into two macroscopic equations. Two sets of distribution functions are employed to construct these two macroscopic equations. Moreover, three types of soliton solutions are numerically simulated by this algorithm. The numerical results imply that the splitting lattice Boltzmann schemes have an advantage over the traditional high-order moment lattice Boltzmann model in simulating the Kadomtsev-Petviashvili-Ⅰ type equations.