Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval

In this paper, we study the initial-boundary value problems of the coupled modified Korteweg-de Vries equation formulated on the finite interval with Lax pairs involving \begin{document}$3× 3$\end{document} matrices via the Fokas method. We write the solution in terms of the solution of a \begin{document}$3× 3$\end{document} Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions \begin{document}$s(k)$\end{document} , \begin{document}$S(k)$\end{document} , and \begin{document}$S_{L}(k)$\end{document} , which are determined by the initial values, boundary values at \begin{document}$x = 0$\end{document} , and at \begin{document}$x = L$\end{document} , respectively. Some of the boundary values are known for a well-posed problem, however, the remaining boundary data are unknown. By using the so-called global relation, the unknown boundary values can be expressed in terms of the given initial and boundary data via a Gelfand-Levitan-Marchenko representation.