This work presents a novel numerical stability analysis of a collection of pseudo-spectral methods, also known as split-step methods, for solving pulse propagation modeled by the nonlinear Schrödinger equation in the nonlinear fiber optics formalism. In order to guarantee the convergence of different pulse propagation dynamics, the numerical solutions of the pseudospectral methods (split-step Fourier method, symmetric split-step Fourier method, fourth-order Runge-Kutta in the interaction picture method, and an optimization of split-step Fourier schemes for pulse propagation over long distances) are tested by the validation of the conservation laws that govern this system. The presented numerical results are an illustrative guide to consider in the selection of an appropriate numerical method in future investigations of a wide variety of propagation problems that involve the interplay of the linear and nonlinear contribution in the nonlinear Schrödinger equation, in order to accurately reproduce a specific phenomenology using this formalism.
This work presents a numerical approach to understand the self-regeneration mechanism of the fundamental soliton propagation driven by the nonlinear Schr\"odinger equation in the nonlinear fiber formalism. This approach shows that the interplay between dispersion and nonlinearity results in a compensation effect in the phase and the instantaneous frequency representation of the pulse envelope. For a better understanding of this compensation process, 3D mapping propagation graphs are presented.
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