The nonlinear Schrödinger equation on the half-line

Abstract

Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the\nnonlinear Schr\\"odinger equation on the finite interval, $0 < x < L$, with\n$q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$\nand $f_{1}(t)$ denote the {\\it unknown} boundary values $q_{x}(0,t)$ and\n$q_{x}(L,t)$, respectively. We first show that these unknown functions can be\nexpressed in terms of the given initial and boundary conditions through the\nsolution of a system of nonlinear ODEs. Although the question of the global\nexistence of solution of this system remains open, it appears that this is the\nfirst time in the literature that such a characterization is explicitely\ndescribed for a nonlinear evolution PDE defined on the interval; this result is\nthe extension of the analogous result of [4] and [6] from the half-line to the\ninterval. We then show that $q(x,t)$ can be expressed in terms of the solution\nof a $2\\times 2$ matrix Riemann-Hilbert problem formulated in the complex $k$ -\nplane. This problem has explicit $(x,t)$ dependence in the form $\\exp[2ikx +\n4ik^2t]$, and it has jumps across the real and imaginary axes. The relevant\njump matrices are explicitely given in terms of the spectral data $\\{a(k),\nb(k)\\}$, $\\{A(k), B(k)\\}$, and $\\{\\A(k), \\B(k)\\}$, which in turn are defined in\nterms of $q_{0}(x)$, $\\{g_{0}(t), g_{1}(t)\\}$, and $\\{f_{0}(t), f_{1}(t)\\}$,\nespectively.\n