The Camassa-Holm equation on the half-line: A Riemann-Hilbert approach

We consider the initial-boundary value (IBV) problem for the Camassa-Holm (CH) equation u(t) -u(txx) + 2u(x) + 3uu(x) = 2u(x)u(xx) + uu(xxx) on the half-line x >= 0. In this article, we aim to provide a characterization of the solution of the IBV problem in terms of the solution of a matrix Riemann-Hilbert (RH) factorization problem in the complex plane of the spectral parameter. The data of this RH problem are determined in terms of spectral functions associated to initial and boundary values of the solution. The construction requires more boundary data than those needed for a well-posed IBV problem. Their dependence is expressed in terms of an algebraic relation to be satisfied by the spectral functions. This RH formulation gives us the long-time asymptotics of a solution of the CH-equation.