Convergence of a Spectral Projection of the Camassa-Holm Equation

on the interval [0, 2π]. Spectral discretizations of this equation have been in use ever since the work of Camassa and Holm [2] and Camassa, Holm and Hyman [3]. However, to the knowledge of the authors, no proof that such a discretization actually converges has appeared heretofore. Therefore, this issue is taken up here. Our method of proof is related to the work of Maday and Quarteroni on the convergence of a Fourier-Galerkin and collocation method for the Korteweg-de Vries equation [22]. While they were able to treat the unfiltered collocation approximation, we resort to proving the convergence of a de-aliased collocation projection which turns out to be equivalent to a Galerkin scheme. Before we get to the heart of the subject, a few words about the range of applicability of the equation are in order. The validity of the Camassa-Holm equation as a model for water waves in a channel of uniform width and depth has been a somewhat controversial subject. The discussion seems to have finally been settled in the recent articles of Johnson [16] and Kunze and Schneider [20]. One merit of the equation is the fact that it allows wave breaking typical of hyperbolic systems. Such wave breaking is