Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

Younghun Hong; Chulkwang Kwak; Shohei Nakamura; Changhun Yang

doi:10.1007/s00028-020-00585-y

Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

Younghun Hong, Chulkwang Kwak, Shohei Nakamura, Changhun Yang

Department of Mathematics

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

5 Scopus citations

Abstract

A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in L² to those of the NLS as the grid size h> 0 approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

Original language	English
Pages (from-to)	391-418
Number of pages	28
Journal	Journal of Evolution Equations
Volume	21
Issue number	1
DOIs	https://doi.org/10.1007/s00028-020-00585-y
State	Published - Mar 2021

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Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

Keywords

Continuum limit
Periodic nonlinear Schrödinger equation
Uniform Strichartz estimate

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10.1007/s00028-020-00585-y

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title = "Finite difference scheme for two-dimensional periodic nonlinear Schr{\"o}dinger equations",

abstract = "A nonlinear Schr{\"o}dinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schr{\"o}dinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in L2 to those of the NLS as the grid size h> 0 approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.",

keywords = "Continuum limit, Periodic nonlinear Schr{\"o}dinger equation, Uniform Strichartz estimate",

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

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year = "2021",

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doi = "10.1007/s00028-020-00585-y",

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T1 - Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

AU - Hong, Younghun

AU - Kwak, Chulkwang

AU - Nakamura, Shohei

AU - Yang, Changhun

PY - 2021/3

Y1 - 2021/3

N2 - A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in L2 to those of the NLS as the grid size h> 0 approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

AB - A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in L2 to those of the NLS as the grid size h> 0 approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

KW - Continuum limit

KW - Periodic nonlinear Schrödinger equation

KW - Uniform Strichartz estimate

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U2 - 10.1007/s00028-020-00585-y

DO - 10.1007/s00028-020-00585-y

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VL - 21

SP - 391

EP - 418

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

IS - 1

ER -

Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

Abstract

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Keywords

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