A split-step Christov method for approximating rational PDE solutions

TL;DR

This paper introduces a split-step Christov spectral method for efficiently approximating rational PDE solutions, demonstrating its effectiveness through nonlinear Schrödinger equation applications and rogue wave analysis.

Contribution

It develops a novel split-step Fourier-based approach with explicit integrators for rational PDEs, improving spectral approximation and enabling detailed rogue wave studies.

Findings

Spectral differentiation matrices are derived for rational PDEs.

Explicit fourth-order split-step integrators outperform existing methods.

Rogue wave-like structures are generated by perturbing backgrounds in nonlinear Schrödinger equations.

Abstract

Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a basis of orthogonal functions known as the Fourier series, allowing for the computation of its spectrum via the fast Fourier transform. Spectral differentiation matrices are derived. Several explicit fourth-order split-step integrators are derived and their performance compared. As an application, rogue wave solutions in a family of nonlinear Schr\"odinger equations are explored. Perturbing the constant background is found to generate rogue wave-like structures. The effects of higher-order dispersion and generalized nonlinearities are also examined.