lanczos–chebyshev pseudospectral methods for wave-propagation problems

the pseudospectral approach is a well-established method for studies of the wave propagation in various settings. in this paper, we report that the implementation of the pseudospectral approach can be simplified if power-series expansions are used. there is also an added advantage of an improved computational efficiency. we demonstrate how this approach can be implemented for two-dimensional (2d) models that may include material inhomogeneities. physically relevant examples, taken from optics, are presented to show that, using collocations at chebyshev points, the power-series approximation may give very accurate 2d soliton solutions of the nonlinear schrödinger (nls) equation. to find highly accurate numerical periodic solutions in models including periodic modulations of material parameters, a real-time evolution method (rtem) is used. a variant of rtem is applied to a system involving the copropagation of two pulses with different carrier frequencies, that cannot be easily solved by other existing methods.