### Abstract

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.

Original language | English |
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Article number | ARTN 036704 |

Number of pages | 9 |

Journal | Physical Review E: Statistical, Nonlinear, and Soft Matter Physics |

Volume | 74 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2006 |

### Keywords

- ROSSBY SOLITON MODEL
- QUANTUM-MECHANICS
- WAVES
- CELLULAR-AUTOMATA
- BLOCK-EDDY INTERACTION
- GAS MODEL

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## Cite this

Dong, P., Feng, S., Gao, S., & Zhong, L. (2006). Lattice Boltzmann schemes for the nonlinear Schrödinger equation.

*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*,*74*(3), [ARTN 036704]. https://doi.org/10.1103/PhysRevE.74.036704