The Fokas Method to the Modified Camassa–Holm Equation on the Half‐Line

ABSTRACT

We investigate the initial‐boundary value problem for the modified Camassa–Holm (mCH) equation  with cubic nonlinearity on the half‐line through the unified transform method, also known as the Fokas method. By analyzing the associated Lax pair, we define spectral functions determined by the initial and boundary data, which, in turn, satisfy a global relation coupling all these values. Based on this framework, we construct a matrix Riemann‐Hilbert (RH) problem parameterized by appropriately scaled variables, from which the solution of the mCH equation can be represented. The analytic structure of the RH problem depends on the sign of a certain boundary quantity at , leading to distinct forms of the global relation. Finally, it is further shown that the global relation provides both necessary and sufficient conditions for the compatibility of the initial and boundary data.