The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method

Abstract

In this paper, we implement the Fokas method to study initial-boundary value problems of the mixed coupled nonlinear Schrödinger equation formulated on the half-line with Lax pairs involving 3×3 matrices. The solution can be written in terms of the solution to a 3×3 Riemann–Hilbert problem. The relevant jump matrices are explicitly expressed in terms of the matrix-value spectral functions s ( k ) and S ( k ), which are determined by the initial values and boundary values at x =0, respectively. 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.