Analysis of gradient flow for computing defocusing action ground states of rotating nonlinear Schr\"odinger equations

TL;DR

This paper proves the stability and convergence of a gradient flow method for computing ground states of rotating nonlinear Schrödinger equations, supported by numerical validation.

Contribution

It provides the first rigorous stability and convergence analysis of the direct gradient flow scheme for RNLS ground states, including exponential convergence rates.

Findings

Unconditional stability of the DGF scheme for arbitrary time steps

Global convergence to the ground state under minor assumptions

Numerical experiments confirm theoretical convergence results

Abstract

This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schr\"odinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.