The nonlinear Schrödinger equation on the interval

Abstract

Let q(x, t) satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrodinger equation on the finite interval, 0 < x < L, with q0(x) = q(x, 0), g0(t) = q(0, t), f0(t) = q(L, t). Let g1(t) and f1(t) denote the unknown boundary values qx(0, t) and qx(L, t), respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. For the focusing case it can be shown that this system has a global solution. It appears that this is the first time in the literature that such a characterization is explicitly described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [20] from the half-line to the interval. We then show that q(x, t) can be expressed in terms of the solution of a 2 × 2 matrix Riemann–Hilbert problem formulated in the complex k-plane. This problem has explicit (x, t) dependence in the form exp[2ikx + 4ik2t], and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitly given in terms of the spectral data {a(k), b(k)}, {A(k), B(k)}, and , which in turn are defined in terms of q0(x), {g0(t), g1(t)}, and {f0(t), f1(t)}, respectively.