Initial–boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method

Abstract

Boundary value problems for integrable nonlinear differential equations can be analyzed via the Fokas method. In this paper, this method is employed in order to study initial–boundary value problems of the general coupled nonlinear Schrödinger equation formulated on the finite interval with 3×3 Lax pairs. The solution can be written in terms of the solution of a 3×3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the three matrix-value spectral functions s(k), S(k), and SL(k). The associated general Dirichlet to Neumann map is also analyzed via the global relation. It is interesting that the relevant formulas can be reduced to the analogous formulas derived for boundary value problems formulated on the half-line in the limit when the length of the interval tends to infinity. It is shown that the formulas characterizing the Dirichlet to Neumann map coincide with the analogous formulas obtained via a Gelfand–Levitan–Marchenko representation.