Zero-dispersion Limit for Integrable Equations on the Half-line with Linearisable Data

In recent years, there has been a series of results of Fokas and collaborators on boundary value problems for soliton equations (see [3] for a comprehensive review). The method of Fokas in [3] goes beyond existence and uniqueness. In fact, it reduces these problems to Riemann-Hilbert factorisation problems in the complex plane, thus generalising the existing theory which reduces initial value problems to Riemann-Hilbert problems via the method of inverse scattering. One of the main advantages of the Riemann-Hilbert formulation is that one can use recent powerful results on the asymptotic behaviour of solutions to these problems (as some parameter goes to infinity) to derive asymptotics for the solution of the associated soliton equation. For the study of the long-time asymptotics, such methods were pioneered by Its and then made rigorous and systematic by Deift and Zhou; the method is known as “nonlinear steepest descent” in analogy with the linear steepest descent method which is applicable to asymptotic problems for Fouriertype integrals (see, e.g., [2]). A generalisation of the steepest descent method developed in [1] is able to give rigorous results for the so-called “semiclassical” or “zero-dispersion” limit of the solution of the Cauchy problem for (1+ 1)-dimensional integrable evolution equations, in the case where the Lax operator is selfadjoint. The method has been further extended in [9] for the “nonselfadjoint” case. In a recent paper [8], Kamvissis, by making use of the nonlinear steepest descent method, has studied the “zero-dispersion” limit of the initial boundary value problem for the (1 + 1)-dimensional, integrable, defocusing, nonlinear Schrödinger (NLS) equation on the half-line, for quite general initial and boundary data. In this paper, we consider