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

Abstract

We consider the initial-boundary value (IBV) problem for the modified Camassa--Holm (mCH) equation $ \tilde m_t+\left((\tilde u^2-\tilde u_x^2+2\tilde u)\tilde m\right)_x = 0$, $\tilde m:=\tilde u-\tilde u_{xx}+1$ on the half line $x \ge 0$. We 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 with the initial and boundary values of the solution, whose compatibility is characterized in spectral terms.