A Time-Splitting Spectral Method for Three-Wave Interactions in Media with Competing Quadratic and Cubic Nonlinearities (Communications in Computational Physics, 2007)
This paper introduces an extension of the time-splitting spectral (TSSP) methodfor solving a general model of three-wave optical interactions, which typicallyarises from nonlinear optics, when the transmission media has competing quadraticand cubic nonlinearities. The key idea is to formulate the terms related to quadraticand cubic nonlinearities into a Hermitian matrix in a proper way, which allows us todevelop an explicit and unconditionally stable numerical method for the problem. Furthermore,the method is spectral accurate in transverse coordinates and second-orderaccurate in propagation direction, is time reversible and time transverse invariant, andconserves the total wave energy (or power or the norm of the solutions) in discretizedlevel. Numerical examples are presented to demonstrate the efficiency and high resolutionof the method. Finally the method is applied to study dynamics and interactionsbetween three-wave solitons and continuous waves in media with competingquadratic and cubic nonlinearities in one dimension (1D) and 2D.