TY - JOUR
T1 - On the continuum limit for the discrete nonlinear Schrödinger equation on a large finite cubic lattice
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
UR - http://www.scopus.com/inward/record.url?scp=85141500668&partnerID=8YFLogxK
U2 - 10.1016/j.na.2022.113171
DO - 10.1016/j.na.2022.113171
M3 - Article
AN - SCOPUS:85141500668
SN - 0362-546X
VL - 227
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
M1 - 113171
ER -