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Numerical solution of the nonlinear Schrödinger equation in a Cartesian coordinate system

A.A. Degtyarev, A.E. Derkach

Samara State Aerospace University

 PDF, 598 kB

Pages: 9-13.

Full text of article: Russian language.

Abstract:
To solve the Schrödinger equation describing the propagation of an electromagnetic wave in a nonlinear medium, the authors develop a Peaceman-Rachford difference scheme with the iterative improvement, which has a quadratic order of accuracy regarding all three spatial variables. Particular cases of the problem providing for the analytical solution are considered to investigate the scheme. The analytical and numerical solutions are compared. The results of numerical simulation of the process of propagation of an electromagnetic wave in a nonlinear medium are presented, which allow to observe both the phenomena of self-focusing and self-defocusing of a light beam, which is fully consistent with the theory of electromagnetic waves [1]. The authors emphasize the possibility of using a difference scheme to analyze the processes of propagation of an electromagnetic wave in media with a spatially dependent refractive index.

Citation:
Degtyarev AA, Derkach AE. Numerical solution of the nonlinear Schrödinger equation in a Cartesian coordinate system. Computer Optics 2001; 21: 9-13.

References:
  1. Vinogradova MB, Rudenko OV, Sukhorukov AP. Theory of waves [In Russian]. Moscow: "Nauka" Publisher; 1979.
  2. Samarskii AA. The theory of difference schemes. CRC Press; 2001.
  3. Adams MJ. An introduction to optical waveguides. New York: John Wiley & Sons Inc; 1981.

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