The Modified Korteweg-de Vries System on the Half-Line

The initial-boundary value problem (ibvp) for a coupled system of modified Korteweg-de Vries (mKdV) equations depending on a parameter a is studied on the half-line. It is shown that this system is well-posed for initial data (u(0), v(0))(x) in spatial Sobolev spaces H-s (0,infinity) x H-s (0,infinity), s > 1/4, and boundary data (g(0), h(0))(t) in the temporal Sobolev spaces suggested by the time regularity of the Cauchy problem for the corresponding linear problem. First, linear estimates in Bourgain spaces X-s,X-b for 0 < b < 1/2 are derived by utilizing the Fokas solution formula of the ibvp for the forced linear system. Then, using these and the needed trilinear estimates in X-s,X-b spaces, it is shown that the iteration map defined by the Fokas solution formula is a contraction in an appropriate solution space. Finally, via a counterexample to trilinear estimates, the criticality of s = 1/4 for well-posedness is established.