Chebyshev spectral method for numerical simulations of counter-propagating optical waves interaction in nonlinear media

Chebyshev spectral methods for two-point boundary value problems describing the processes of counter interaction of optical waves in media with cubic nonlinearity and linear media with periodic modulation of the refractive index are considered. On the example of a linear problem, it is shown that the spectral method for achieving a given accuracy requires of two-three orders less time in comparison with the spline collocation method of the $5^{th}$ accuracy order. Moreover, Chebyshev mesh has natural adaptive properties for the considered problems of the nonlinear interaction of optical waves. A conservative iterative algorithm for implementation of the nonlinear spectral model is proposed. The proposed method has a lower sensitivity to the choice of an appropriate initial guess and provides a higher rate of convergence in comparison with Newton’s method under conditions of strong coupling of interacting waves.