Lattice Boltzmann schemes for the nonlinear Schrödinger equation.

The lattice Boltzmann (LB) method is applied to solve the time-dependent nonlinear Schrödinger (NLS) equation. Through approximating the reaction term at different orders of accuracy, three diffusion-reaction LB schemes are constructed for the cubic NLS equation. A LB initial condition is proposed to include the first-order nonequilibrium distribution function. These LB schemes are used to solve the one-soliton propagation and the homoclinic orbit problems. Detailed simulation results confirm that the high-order reaction term and the LB initial condition are effective in reducing the truncation errors. Compared with the Crank-Nicolson finite difference scheme, the LB scheme is found to give at least comparable and generally more accurate approximation for the cubic NLS equation.