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 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 for given . For the general fCH equation, we demonstrate spectral accuracy in spatial discretization with optimal error estimates for initial data with and exponential convergence for smooth solutions. Numerical experiments validate stability of solitary waves achieving optimal convergence, confirming theoretical findings.