Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schrödinger equation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908850341281792 |
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| author | Liu, Wei Wang, Tingfeng Zhao, Xiaofei |
| author_facet | Liu, Wei Wang, Tingfeng Zhao, Xiaofei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20820 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schrödinger equation Liu, Wei Wang, Tingfeng Zhao, Xiaofei Numerical Analysis 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. |
| title | Convergence analysis of $L^{p+1}$-normalized gradient flow for action ground state of nonlinear Schrödinger equation |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.20820 |