Lattice Boltzmann Simulations For Necklace-ring Solitons In The Nonlinear Schr(?)dinger Equation

Since the birth of lattice Boltzmann method in the late 1980 s,it has rapidly become an alternative method for computational fluid dynamics,computational heat transfer,and even the numerical solutions for partial differential equations.Compared with the traditional numerical method,the lattice Boltzmann method has several advantages,such as its algorithmic parallelism,programming simplicity,and high boundary adaptability.The method has attracted continuous attention in various fields and has made much progress in the 21 st century.In this dissertation,the lattice Boltzmann method is applied in the system of nonlinear Schr(?)dinger equations to simulate the necklace-ring solitons as well as other types of nonlinear optical solitons associated with them in Kerr medium.In the part of the literature review,a brief discussion on the origin and development of the lattice Boltzmann method is proposed.We focus more on the applications of the lattice Boltzmann method in the field of complex flow problems and numerical simulations of partial differential equations.In addition,several recent research developments on the necklace-ring soliton are also introduced.In the part of the mathematical theory of the model,the spatiotemporal-swapping nonlinear Schr(?)dinger equation is firstly introduced.According to the characteristics of the equation,we select the lattice Boltzmann model based on the spatial evolution process.The mathematical theory of the lattice Boltzmann method is discussed in detail.Starting from the spatially evolving complex lattice Boltzmann equation,a series of partial differential equations are obtained through the Chapman-Enskog expansion and multi-scale expansion.And then the governing equation is recovered.In addition,based on the method of higher-order moments of the additional distribution function,the selection of additional distribution function and the elimination of truncation error are performed.Finally,the heuristic stability condition for the lattice Boltzmann model is given.For the purpose of studying the evolution of(2+1)-dimensional necklace-ring solitons in Kerr medium.A polar coordinate lattice Boltzmann model is proposed.Different from the general curvilinear coordinate lattice Boltzmann model,the discrete grids of the lattice Boltzmann model are uniform.In the iterative process,the macroscopic quantity is calculated in the uniform grids and then mapped to the twodimensional polar coordinate system.Thus,the complex interpolation process in the general curvilinear coordinate lattice Boltzmann model is avoided.In addition,the numerical results are also carried out.This model is verified by comparing it with the classic finite difference scheme.The influence of the number of beams,the nonlinear media,and the topological charge on the spatial propagation of necklace ring solitons is investigated subsequently.The numerical results are consistent with the conclusions in the literature.The computation cost for three-dimensional problems is significantly increased.To keep the stability of lattice Boltzmann model with a large iteration step,the evolution of(3+1)-dimensional necklace-ring solitons in Kerr medium is solved under the Cartesian coordinate system.By introducing double additional distribution functions,the order of the numerical convergence is improved.In addition,two types of ringshaped spatiotemporal solitons are simulated by using this model,and the results are acceptable.Finally,the conclusions are summarized.Several prospects for future work are presented at the end of this dissertation.