Analysis of the global relation for the nonlinear Schrödinger equation on the half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2 × 2 matrix Riemann-Hilbert problem. This problem is specified by the spectral functions {a(k), b(k)} which are defined in terms of the initial condition q(x, 0) = q0(x), and by the spectral functions {A(k), b(k)} which are defined in terms of the specified boundary condition q(0, t) = g0(t) and the unknown boundary value qx(0, t) = g1(t). Furthermore, it has been shown that given q0 and g0, the function g1 can be characterized through the solution of a certain 'global relation' coupling q0, g0, g1, and Φ(t, k), where Φ satisfies the t-part of the associated Lax pair evaluated at x = 0. We show here that, by using a Gelfand-Levitan-Marchenko triangular representation of φ, the global relation can be explicitly solved for g1.