On the continuum limit for the discrete nonlinear Schrödinger equation on a large finite cubic lattice

Younghun Hong, Chulkwang Kwak, Changhun Yang

Research output: Contribution to journalArticlepeer-review

Abstract

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 (Hong et al., 2021), 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.

Original languageEnglish
Article number113171
JournalNonlinear Analysis, Theory, Methods and Applications
Volume227
DOIs
StatePublished - Feb 2023

Bibliographical note

Funding Information:
This research of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT ( NRF-2020R1A2C4002615 ). C. K. was supported by the National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIT) (No. 2020R1F1A1A0106876811 ) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, South Korea (No. 2019R1A6A1A11051177 ). C. Yang was supported by the National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIT) (No. 2021R1C1C1005700 ).

Publisher Copyright:
© 2022 Elsevier Ltd

Keywords

  • Continuum limit
  • Dirichlet boundary condition
  • Nonlinear Schrödinger equation
  • Strichartz estimate

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