The scattering approach for the Camassa-Holm equation

We present an approach proving the integrability of the Camassa--Holm equation for initial data of small amplitude

The solution of the global relation for the derivative nonlinear Schr\"odinger equation on the half-line

We consider initial-boundary value problems for the derivative nonlinear Schr\"odinger (DNLS) equation on the half-line $x > 0$. In a previous work, we showed that the solution $q(x,t)$ can be expressed in terms of the solution of a Riemann-Hilbert problem with jump condition specified by the initial and boundary values of $q(x,t)$. However, for a well-posed problem, only part of the boundary values can be prescribed; the remaining boundary data cannot be independently specified, but are determined by the so-called global relation. In general, an effective solution of the problem therefore requires solving the global relation. Here, we present the solution of the global relation in terms of the solution of a system of nonlinear integral equations. This also provides a construction of the Dirichlet-to-Neumann map for the DNLS equation on the half-line.Comment: 20 pages, 2 figures, minor corrections mad