2021/06/25 by Younghun Hong, Hong, Younghun, Chulkwang Kwak +3
Engineering · Mathematics · Physics and Astronomy · #35Q55 #81T27 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Numerical Analysis (math.NA) #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2106.13417
openalex publication_date 2021/06/25 · openalex created_date 2021/07/05 · openalex updated_date 2026/07/28
In this study, we consider the nonlinear Schödinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, \emphFinite difference scheme for two-dimensional periodic nonlinear Schrödinger equations, Journal of Evolution Equations 21 (2021), no.~1, 391--418.], the loss of regularity in Strichartz estimates can be reduced as the domain expands, depending on the speed of expansion. This allows us to address the physically important three-dimensional case.