Numerical Simulation for the Wave of the Variable Coefficient Nonlinear Schrödinger Equation Based on the Lattice Boltzmann Method

The variable coefficient nonlinear Schrödinger equation has a wide range of applications in various research fields. This work focuses on the wave propagation based on the variable coefficient nonlinear Schrödinger equation and the variable coefficient fractional order nonlinear Schrödinger equation. Due to the great challenge of accurately solving such problems, this work considers numerical simulation research on this type of problem. We innovatively consider using a mesoscopic numerical method, the lattice Boltzmann method, to study this type of problem, constructing lattice Boltzmann models for these two types of equations, and conducting numerical simulations of wave propagation. Error analysis was conducted on the model, and the convergence of the model was numerical validated. By comparing it with other classic schemes, the effectiveness of the model has been verified. The results indicate that lattice Boltzmann method has demonstrated advantages in both computational accuracy and time consumption. This study has positive significance for the fields of applied mathematics, nonlinear optics, and computational fluid dynamics.