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

Younghun Hong; Chulkwang Kwak; Changhun Yang

doi:10.1016/j.na.2022.113171

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

Younghun Hong, Chulkwang Kwak, Changhun Yang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

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 language	English
Article number	113171
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	227
DOIs	https://doi.org/10.1016/j.na.2022.113171
State	Published - 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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10.1016/j.na.2022.113171

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title = "On the continuum limit for the discrete nonlinear Schr{\"o}dinger equation on a large finite cubic lattice",

abstract = "In this study, we consider the nonlinear Sch{\"o}dinger 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.",

keywords = "Continuum limit, Dirichlet boundary condition, Nonlinear Schr{\"o}dinger equation, Strichartz estimate",

author = "Younghun Hong and Chulkwang Kwak and Changhun Yang",

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: {\textcopyright} 2022 Elsevier Ltd",

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doi = "10.1016/j.na.2022.113171",

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AU - Hong, Younghun

AU - Kwak, Chulkwang

AU - Yang, Changhun

N1 - 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

PY - 2023/2

Y1 - 2023/2

N2 - 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.

AB - 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.

KW - Continuum limit

KW - Dirichlet boundary condition

KW - Nonlinear Schrödinger equation

KW - Strichartz estimate

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JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

M1 - 113171

ER -

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

Abstract

Bibliographical note

Keywords

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