Study on the dynamic evolution of fractional nonlinear solitary waves in Bose–Einstein condensates based on meshless method

Purpose In this paper, an efficient and reliable meshless coupling method is given for numerically solving the time-fractional nonlinear Schrödinger/fractional Ginzburg–Landau equations, and also numerically predicting the phenomenon of the evolution of the fractional nonlinear isolated wave dynamics in Bose–Einstein condensation. In conclusion, the method provides a new scheme for solving fractional partial differential problems of complex physical significance. Design/methodology/approach The method is discretised spatially by the improved smoothed particle hydrodynamics (SPH) method, while the time derivative is discretised by the L1 formulation, and finally a new treatment scheme for the nonlinear terms is proposed, which combines the above to present a method (IMSPH-L1) that can effectively predict the phenomenon of fractional nonlinear isolated wave dynamics evolution in Bose–Einstein condensation. The method is also employed to numerically simulate the time-fractional nonlinear Schrödinger/fractional Ginzburg-Landau problem with/without analytical solutions. Findings In order to verify the reliability and flexibility of the method, a number of numerical simulations are carried out and the results are compared with the analytical solution and other existing numerical results to analyse the numerical accuracy of the method and to demonstrate the superiority of the method in numerically predicting the problem studied in this paper. Originality/value We propose an efficient meshless scheme for time-fractional nonlinear Schrödinger/fractional Ginzburg–Landau problem, demonstrating the superiority of the new method in terms of numerical accuracy and convergence speed. Moreover, the method proposed in this paper can be used to simulate various other complex nonlinear problems.