2024/07/04 by Justin T. Cole, Cole, Justin T., Troy I. Johnson +1
Earth and Planetary Sciences · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Numerical Analysis (math.NA) #Ocean Waves and Remote Sensing #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2407.04013
openalex publication_date 2024/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Schrödinger 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.