Bifurcation Method For Multi - Solution Computation Of Schr (?) Dinger Equation

In this master thesis, we discuss the multiple solution problem for the nonlinear Schr ?dinger equation（NLS）, its form is（?）where x0 is the center of domain ?, p > 1, λ, κ and r are given parameters. This paper includes two parts: in first part, we research the multiple solution of the Schr ?dinger equation on a square.We first compute the multiple non-trivial solutions of the equation（0.1） on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combined with Legendre pseudospectral methods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we take λ or r in the equation（0.1） as bifurcation parameters respectively and further obtain the D4 symmetric positive solution branch of the equation（0.1）numerically by the continuation method and pseudo-arclength algorithm. During continuation,we find the potential symmetry-breaking bifurcation points, and propose the extended systems,which can detect the fold and symmetry-breaking bifurcation points on the branch of the D4 symmetric positive solutions accurately. Because scientists are aim to the positive solutions, we compute the multiple positive solutions with various symmetries of the equation（0.1） by the branch switching method based on the Liapunov-Schmidt reduction. Finally, the bifurcation diagrams are constructed, showing the symmetry-breaking bifurcation positive solutions of the equation（0.1） on a square.In the second part, we research the multiple solution of the Schr ?dinger equation on a unit disk. Based on the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, we compute and visualize multiple solutions of the Schr ?dinger equation on a unit disk, using mixed Fourier-Legendre spectral and pseudospectral methods. Starting from the non-trivial solution branches of the corresponding nonlinear bifurcation problem, we take λ or r in the equation（0.1）as bifurcation parameters respectively and obtain multiple solutions of Schr ?dinger equation with various symmetries numerically. During continuation, we find the potential symmetry-breaking bifurcation points, and propose the extended systems, which can detect the O（2）-Σ₁, O（2）-Σ_d,O（2）-D₃, O（2）-D₄, O（2）-D₅, O（2）-D₆, O（2）-_D7 and O（2）-D₈symmetry-breaking bifurcation points on the branch of the O（2） symmetric positive solutions. The Σ₁（Σ_d, D₃,D₄, D₅, D₆, D₇ or D₈） symmetric positive solutions are computed by the branch switching method based on the Liapunov-Schmidt reduction. The bifurcation diagrams are constructed,showing the symmetry-breaking bifurcation positive solutions of the equation（0.1） on a unit disk.Numerical results demonstrate the effectiveness of these approaches. The final section is for some concluding discussions.