Lattice Boltzmann schemes for the nonlinear Schrödinger equation

Area of Science:

• Computational physics
• Numerical analysis
• Quantum mechanics

Background:

• The nonlinear Schrödinger (NLS) equation models various physical phenomena.
• Solving the time-dependent NLS equation numerically presents challenges in accuracy and efficiency.
• Existing methods like finite difference schemes have limitations.

Purpose of the Study:

• To apply the lattice Boltzmann (LB) method for solving the time-dependent cubic NLS equation.
• To develop and evaluate novel LB schemes with improved accuracy for reaction terms.
• To introduce an LB initial condition enhancing numerical stability and precision.

Main Methods:

• Construction of three diffusion-reaction LB schemes with varying reaction term approximation orders.
• Implementation of a novel LB initial condition incorporating the first-order nonequilibrium distribution function.
• Application of these schemes to simulate one-soliton propagation and homoclinic orbit problems.

Main Results:

• The proposed LB schemes effectively solve the cubic NLS equation.
• High-order approximation of the reaction term and the new LB initial condition significantly reduce truncation errors.
• LB method demonstrates comparable or superior accuracy to the Crank-Nicolson finite difference scheme.

Conclusions:

• The developed LB schemes provide an accurate and efficient approach for the cubic NLS equation.
• The LB method offers a promising alternative for simulating complex NLS problems.
• The study highlights the effectiveness of advanced LB techniques in computational physics.