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

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.