Convergence of a normalized gradient algorithm for computing ground states

摘要

We consider the approximation of the ground state of the one-dimensional cubic nonlinear Schrodinger equation by a normalized gradient algorithm combined with a linearly implicit time integrator, and a finite difference space approximation. We show that this method, also called the imaginary time evolution method in the physics literature, is locally convergent, and we provide error estimates: for initial data in a neighborhood of the ground state, the algorithm converges exponentially toward a modified soliton that is a space discretization of the exact soliton, with error estimates depending on the discretization parameters.