Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called . For the defocusing nonlinear Schr "odinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schr "odinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.
@cite_12: We are studying the semiclassical limit of the 1+1 dimensional integrable nonlinear Schrodinger equation with defocusing cubic nonlinearity on the half line. Our analysis relies on the recent theory of , which reduces boundary value problems for soliton equations to Riemann–Hilbert factorization problems. We employ the method of nonlinear steepest descent to asymptotically deform the given Riemann–Hilbert problem to an explicilty solvable one.
The semiclassical limit is, in a sense, the exact opposite to the weakly-nonlinear small-amplitude limit. Indeed, the formal semiclassical limit is given by the strongly nonlinear hyperbolic system . There is at least one other paper in the literature on the subject of semiclassical analysis of the Dirichlet initial-boundary-value problem on the half-line for the defocusing nonlinear Schr "odinger equation, namely a paper of Kamvissis @cite_12 , which directly stimulated our interest in this problem. Like we do, Kamvissis considers general nonhomogeneous Dirichlet boundary data together with homogeneous initial conditions, and he applies the steepest descent methodology for Riemann-Hilbert problems to deduce general properties of the solution in the semiclassical limit. Our Corollary is consistent with Theorem 5 of @cite_12 (the main result of that paper) albeit in the simplest case of genus @math . On the other hand, it is less clear whether the vacuum domain @math described by our Corollary is a special case of Kamvissis' Theorem 5.