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 $3\times 3$ matrices. This equation can be considered as a generalization of the modified KdV equation. We show that the solution $\left\{p\right(x,t),q(x,t\left)\right\}$ can be written in terms of the solution of a $3\times 3$ Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions $s\left(k\right)$ and $S\left(k\right)$, 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. (paper)