Commun. Comput. Phys., 2 (2007), pp. 123-140.


A Time-Splitting Spectral Method for Three-Wave Interactions in Media with Competing Quadratic and Cubic Nonlinearities

Weizhu Bao 1*, Chunxiong Zheng 2

1 Department of Mathematics, National University of Singapore, Singapore 117543.
2 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China.

Received 10 January 2006; Accepted (in revised version) 6 April 2006
Available online 30 August 2006

Abstract

This paper introduces an extension of the time-splitting spectral (TSSP) method for solving a general model of three-wave optical interactions, which typically arises from nonlinear optics, when the transmission media has competing quadratic and cubic nonlinearities. The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way, which allows us to develop an explicit and unconditionally stable numerical method for the problem. Furthermore, the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction, is time reversible and time transverse invariant, and conserves the total wave energy (or power or the norm of the solutions) in discretized level. Numerical examples are presented to demonstrate the efficiency and high resolution of the method. Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension (1D) and 2D.

AMS subject classifications: 65M70, 78M25, 42A10

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Key words: Nonlinear optics, three-wave, time-splitting spectral method, energy, continuous wave, soliton.

*Corresponding author.
Email: bao@cz3.nus.edu.sg (W. Bao), czheng@math.tsinghua.edu.cn (C. Zheng)
 

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