Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

DOI10.1016/J.AMC.2023.128428arXiv2305.18568MaRDI QIDQ6128637

Lisandro A. Raviola, Mariano De Leo

Publication date: 16 April 2024

Published in: Applied Mathematics and Computation (Search for Journal in Brave)

Abstract: In this paper, we evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the the standard and fractional variants of the nonlinear Schr"odinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive with traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, our method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

Full work available at URL: https://arxiv.org/abs/2305.18568

zbMATH Keywords

pseudo-spectral methodsfractional nonlinear Schrödinger equationsaffine operator splittingfractional complex Ginzburg-Landau equations

Mathematics Subject Classification ID

Numerical methods for ordinary differential equations (65Lxx) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65Mxx) Partial differential equations of mathematical physics and other areas of application (35Qxx)

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