Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method

Abstract

In this paper, a coupled pure meshless finite pointset method (CFPM) is developed for the first time to numerically predict the inelastic collision process of solitary wave described by the time fractional coupled nonlinear Schrödinger (TF-CNLS) equation. Its construction process is as follows: 1) a high-precision difference scheme is used for the Caputo time fractional derivative; 2) FPM discrete scheme based on Taylor expansion and weighted least square method is adopted for spatial derivatives; 3) The region is locally refined and the double cosine kernel function with good stability is used to improve the numerical accuracy. In the numerical study, the one-dimensional TF-CNLS equations with analytical solutions are solved by CFPM, and the error and convergence rate are analyzed when the nodes are uniformly distributed or locally refined, showing that the proposed method has the approximate second-order accuracy and the flexibility of easy local refinement. Secondly, the inelastic collision process of solitary waves, which is described by the one-dimensional TF-CNLS equation without analytical solution, is numerically predicted by CFPM, and the wave collapse phenomenon depicted above is completely different from the multi-wave phenomenon under the integer order. Meanwhile, by comparing the result with that obtained by finite difference method, it suggests that the CFPM is reliable to predict the complex propagation of the inelastic collision process of the solitary waves under the time fractional order.