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 3 x 3 matrices via the Fokas method. We write the solution in terms of the solution of a 3 x 3 Riemann-Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s(k), S(k), and S-L(k), which are determined by the initial values, boundary values at x = 0, and at x = L, 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.