A Bifurcation Method For Computing Multiple Solutions Of A Class Of Partial Differential Equations On A Unit Circle

In this master thesis,we discuss the multiple solution of Neumann boundary value problem for a class of nonlinear elliptic equations on a unit disk.It is divided into two parts.In the first part,we research the multiple solution of the Schr(?)dinger equation on a unit disk.The form of the Schr(?)dinger equation iswhere?is an unit disk,?>0,p>1,??R,??R and l?0,they are given parame-ters.Firstly,we compute the multiple non-trivial solutions of the equation(0.1)on a unit disk by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory,combined with mixed Fourier-Legendre pseudospectral methods.Then,starting from the non-trivial solution branches of the corresponding nonlinear problem,we take?,?and l in the equation(0.1)as bi-furcation parameters respectively and further obtain the O(2)symmetric positive solution branch of the equation(0.1)numerically by the continuation method.During continuation,we find the potential symmetry-breaking bifurcation points,and propose the extended systems,which can de-tect symmetry-breaking bifurcation points on the branch of the O(2)symmetric positive solutions accurately.Since the scientists prefer to the positive solutions,so we compute the multiple posi-tive solutions with various symmetries of the equation(0.1)on a unit disk by the branch switching method based on the Liapunov-Schmidt reduction.Finally,the bifurcation diagrams are construct-ed,showing the symmetry-breaking bifurcation positive solutions of the equation(0.1)on a unit disk.In the second part,we research the multiple solution of the Concave-convex equation on a unit disk.The form of the equation iswhere 0<q<1<p,??R,??R and l?0,they are given parameters.Based on the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory,we compute and visualize multiple non-trivial solutions of the equation(0.2)on a unit disk,by using mixed Fourier-Legendre spectral and pseudospectral methods.Starting from the non-trivial solution branches of the corresponding nonlinear bifurcation problem,we take?and?in the equation(0.2)as bifurcation parameters respectively and obtain multiple solutions of Concave-convex equation with various symmetries numerically.During continuation,we find the potential symmetry-breaking bifurcation points and propose the extended systems which can detect symmetry-breaking bifurcation points on the branch of the O(2)symmetric positive solutions.The 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(0.2)on a unit disk.Numerical results demonstrate the effectiveness of these approaches.The final section is for some concluding discussions.