Numerical Investigation Of Nonlinear Solitary Wave Problems Based On Dimensionality Reduction Meshless FPM Coupled Absorption Boundary Method

The nonlinear diffusion of soliton wave is often described by the nonlinear Schrodinger(NLS)equation,which is an important equation in quantum dynamics or nonlinear optics.Due to the influence of nonlinear terms,it is difficult to obtain analytical solutions of high-dimensional NLS equations in most cases.Therefore,numerical study of NLS equations has become an international hot issue in the field of computing in recent years.At present,a variety of mesh methods(such as finite difference and finite element method)have been successfully applied to solve NLS equations.However,a pure meshless method,which can make up for the shortcomings of mesh methods,is still in the preliminary stage of numerical solution of NLS equations in unbounded regions.Based on the above analysis,this thesis introduces the idea of time splitting and local one dimension(dimension reduction),and studies the high-dimensional NLS equation on unbounded region by coupling the Finite Pointset Method(FPM)with the absorption boundary PML(Perfectly Matched Layer).Firstly,the NLS equation is divided into two equations with nonlinear term and linear derivative by time splitting,and the linear derivative equation is decomposed into multiple differential equations along different directions by local one-dimensional thought.Secondly,PML of absorbing boundary is used to deal with infinite region and boundary.Thirdly,FPM based on Taylor expansion and least square method is used to discretize the linear derivative equation.Finally,the dimensionally reduced FPM coupled PML method is used to numerically study the propagation process of nonlinear solitary waves in an infinite region.The main work of this thesis is as follows:(1)In order to reduce the computational complexity and improve the numerical stability,the time splitting and local one-dimensional ideas are introduced to decompose the nonlinear Schrodinger equation into multiple differential equations,and the SS-LOFP M method is obtained.The error and numerical convergence of the method are analyzed by numerical examples with analytical solutions.(2)In order to deal with the infinite region accurately and effectively,the SS-LOFPM-PML method,which can accurately solve the high-dimensional NLS equation,is extended by using the region truncation and absorption boundary method and combined with PML technology to deal with the infinite boundary,and the effectiveness and advantages of the coupling method are verified by two-dimensional/three-dimensional reference examples.Numerical results show that the proposed method can accurately capture the nonlinear diffusion process of isolated wavelet in high dimensions.(3)In order to further reflect the capacity of coupling method is proposed,using the proposed coupling method of infinite area without analytic solution in 2D/3D for the process of nonlinear quantum vortex solitary wave numerical prediction,and compared with other results,the results show that the proposed method to predict NLS equation under high dimensional description of quantum vortex process is reliable.