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

We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent α∈ [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 C¹([0, T];H^αₚₑᵣ(I)) for given $T>0$. For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates O(N⁻ʳ) for initial data u₀ ∈ Hʳ(I) with r ≥ α+ 2 and exponential convergence O(e-cN) for smooth solutions. Numerical experiments validate orbital stability of solitary waves achieving optimal convergence, confirming theoretical findings.