# Journal of theKorean Mathematical SocietyJKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

## Article

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J. Korean Math. Soc. 2021; 58(4): 835-847

Published online July 1, 2021 https://doi.org/10.4134/JKMS.j200257

## Numerical solutions for one and two dimensional nonlinear problems related to dispersion managed solitons

Younghoon Kang, Eunjung Lee, Young-Ran Lee

Sogang University; Yonsei University; Sogang University

### Abstract

We study behavior of numerical solutions for a nonlinear eigenvalue problem on $\R^n$ that is reduced from a dispersion managed nonlinear Schr\"{o}dinger equation. The solution operator of the free Schr\"{o}dinger equation in the eigenvalue problem is implemented via the finite difference scheme, and the primary nonlinear eigenvalue problem is numerically solved via Picard iteration. Through numerical simulations, the results known only theoretically, for example the number of eigenpairs for one dimensional problem, are verified. Furthermore several new characteristics of the eigenpairs, including the existence of eigenpairs inherent in zero average dispersion two dimensional problem, are observed and analyzed.

Keywords: Schrodinger equation, numerical solution, eigenvalue problem

MSC numbers: 78M20, 65M06

Supported by: Young-Ran Lee and Eunjung Lee were supported by the National Research Foundation of Korea, NRF-2017R1D1A1B03033939 and NRF-2018R1D1A1B07042973, respectively.