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.