Study and application of solitary wave propagation at fractional order of time based on SPH method

A meshless particle method based on the smoothed particle hydrodynamics (SPH) method is first proposed for the numerical prediction of physical phenomena of nonlinear solitary wave propagation and complex phenomena arising from the inelastic interactions of solitary waves. The method is a fully discrete implicit scheme. This method does not rely on a grid, avoids the need to solve for derivatives of kernel functions, and makes the calculation more convenient. Additionally, the unique solvability of the proposed implicit scheme is proved. To verify the effectiveness and flexibility of the proposed method, we apply it to solving various time fractional nonlinear Schrödinger equations (TF-NLSE) on both regular and irregular domains. This mainly includes general or coupled TF-NLSE with or without analytical solutions. Moreover, the proposed method is compared with the existing methods. Through examples, it has been verified that this method can effectively predict complex propagation phenomena generated by the collision of nonlinear solitary waves, such as the collapse phenomenon of solitary waves with increasing fractional-order parameters. Research results indicate that this method provides a new and effective meshless method for predicting the propagation of nonlinear solitary waves, which can better simulate TF-NLSE in complex domains.