The Homotopy Method For Nonlinear Eigenvalue Problem

This paper focuses on homotopy method for solving nonlinear eigenvalue problems derived from GP(Gross-Pitaevskii)equations.This paper first converts the GP equation to a nonlinear eigenvalue problem of differential equation by transformation,and then discusses the cases in 1D and 2D,respectively.For the case of 1D,with the finite difference discretization,the nonlinear eigenvalue problem is reduced to a nonlinear algebraic eigenvalue problem,which is di cult to solve.Based on the idea of homotopy method,an equation that is easy to solve is chosen as initial equation,and an appropriate parameter is selected as homotopy parameter,then a feasible homotopy function is constructed to solve the nonlinear algebraic eigenvalue problem.By tracking the homotopy paths which connect eigen-pairs of a linear eigenvalue problem to eigen-pairs of the nonlinear eigenvalue problem,several approximate solutions to the nonlinear eigenvalue problem are obtained.Meanwhile,the feasibility of the homotopy method is discussed by analyzing the rank of the Jacobi matrix of the homotopy function.For the 1D case,some numerical experiments will be conducted to show the effectiveness of the homotopy method.For the case of 2D,because of the complexity of the eigenvalues of the discrete matrix,this paper focuses on the feasibility of homotopy function in a certain bound of parameters.First,we consider selecting parameters and constructing homotopy equations similar to the steps in 1D.However,since the initial equation in 2D may have multiple eigenvalues,the situation is more complicated than that of 1D case.At the starting point,if a solution corresponding to a multiple eigenvalue is chosen,the Jacobian of the homotopy function will be rank-decient and bifurcation point will be encountered,and the usual path tracking algorithm will fail.In order to tackle this di culty,a new homotopy parameter will be considered,and a new homotopy funciton is constructed.The feasibility of the new homotopy are also analysed.Finally,some numerical experiments will be conducted to show the effectiveness of the homotopy method discussed previously.