Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schr\"odinger equation

TL;DR

This paper rigorously analyzes the convergence of an $L^{p+1}$-normalized gradient flow method for computing the action ground state of the nonlinear Schrödinger equation, establishing global and local convergence rates.

Contribution

It provides the first comprehensive convergence analysis of the GFALM method for the $L^{p+1}$-constrained problem, including semi-discrete, fully discrete, and continuous cases.

Findings

Global convergence of the semi-discrete scheme is established.

Local exponential convergence rate is proven under non-degeneracy assumptions.

Extension of analysis to fully discrete Fourier pseudo-spectral discretization.

Abstract

This paper presents a rigorous convergence analysis of the $L^{p+1}$-normalized gradient flow with asymptotic Lagrange multiplier (GFALM) method for computing the action ground state of the nonlinear Schr\"odinger equation in the focusing case. First, a general global convergence theory is established for the semi-discrete GFALM scheme, guaranteeing the existence of an accumulation point and a convergent subsequence. Then, under additional non-degeneracy assumptions, a local exponential convergence rate is rigorously proven. This result is further extended to the fully discrete case using a Fourier pseudo-spectral discretization. The analysis is achieved by characterizing the local geometry of the $L^{p+1}$-constrained manifold near the ground state, establishing a quadratic growth property of the energy functional, and deriving a \L{}ojasiewicz-type gradient inequality. Finally, the paper also investigates the exponential convergence of the associated continuous-time gradient flow, providing a theoretical foundation for future numerical discretizations. This work extends existing convergence analyses for energy ground states, addressing the challenges posed by the $L^{p+1}$ constraint, especially the absence of an inner-product structure.