Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method

In this paper, we implement the Fokas method in order to study initial boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the half-line, with Lax pairs involving 3 x 3 matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution {p(x, t), q(x, t)} can be written in terms of the solution of a 3 x 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s(k) and S(k), which are respectively determined by the initial values and boundary values at x = 0. Finally, the associated Dirichlet to Neumann map of the equation is analyzed in detail. Some of these boundary values are unknown; however, using the fact that these specific functions satisfy a certain global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data.