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

Numerical study of oscillatory regimes in the Kadomtsev-Petviashvili equation

The aim of this paper is the accurate numerical study of the KP equation. In particular we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey-Stewartson system. In a second step we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg-de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate.Comment: 39 pages, 36 figures (high resolution figures at http://www.mis.mpg.de/preprints/index.html

Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems