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

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 Schrödinger 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 Łojasiewicz-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.

Figures (1)

• Figure 1: Error of GFALM \ref{['eq:GFALM-disc.']} along iteration (on logarithmic scale). (a): $\|u_{h}^{n} - \mathcal{I}_{h}u^{*}\|_{H^{1}}$ under different $\tau$ in Example \ref{['ex:1']} (a); (b): $\|u_{h}^{n} - u_{h}^{*}\|_{H^{1}}$ under different initial data $u_{0}$ in Example \ref{['ex:2']}.

Theorems & Definitions (46)

• Lemma 2.1: Energy-decaying property
• proof
• Theorem 2.2: Accumulation point & unconditional convergent subsequence
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• proof : Proof of Theorem \ref{['thm:convergent-subsequences']}