Computing the least action ground state of the nonlinear Schr\"odinger equation by a normalized gradient flow

TL;DR

This paper introduces a generalized normalized gradient flow method for computing the least action ground state of the nonlinear Schrödinger equation, demonstrating its superior accuracy, efficiency, and robustness through extensive numerical experiments.

Contribution

It extends the normalized gradient flow approach from energy to action minimization and develops a new discretization scheme with proven properties.

Findings

GFDN-BF scheme preserves positivity and reduces action unconditionally.

Numerical results show GFDN-BF outperforms other schemes in accuracy and efficiency.

The method successfully computes ground states for various potentials, confirming and extending existing results.

Abstract

In this paper, we generalize the normalized gradient flow method which was first applied to computing the least energy ground state to compute the least action ground state. A continuous normalized gradient flow (CNGF) will be presented and the action diminishing property will be proved to provide a mathematical justification of the gradient flow with discrete normalization (GFDN). Then we use backward-forward Euler method to further discretize the GFDN in time which leads to the GFDN-BF scheme. It is shown that the GFDN-BF scheme preserves the positivity and diminishes the action unconditionally. We compare it with other three schemes which are modified from corresponding ones designed for the least energy ground state and the numerical results show that the GFDN-BF scheme performs much better than the others in accuracy, efficiency and robustness for large time steps. Extensive numerical results of least action ground states for several types of potentials are provided. We also use our numerical results to verify some existing results and lead to some conjectures.