A convergent Fourier spectral Galerkin method for the fractional Camassa-Holm equation

TL;DR

This paper develops and analyzes a Fourier spectral Galerkin method for the fractional Camassa-Holm equation, demonstrating spectral accuracy, stability, and convergence properties through rigorous proofs and numerical experiments.

Contribution

It introduces a mass- and energy-preserving spectral Galerkin scheme for the fractional Camassa-Holm equation with proven convergence and stability.

Findings

Spectral accuracy with optimal error estimates for initial data.

Exponential convergence for smooth solutions.

Numerical validation of orbital stability of solitary waves.

Abstract

We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $\alpha \in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and energy invariants of the fCH equation. For the fractional Benjamin-Bona-Mahony reduction, we establish existence and uniqueness of semi-discrete solutions and prove strong convergence to the unique solution in $ C^1([0, T];H^{\alpha}_{\mathrm{per}}(I))$ for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates $\mathcal{O}(N^{-r})$ for initial data $u_0 \in H^r(I)$ with $r \geq \alpha + 2$ and exponential convergence $\mathcal{O}(e^{-cN})$ for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings.