The Unified Method: II NLS on the Half-Line with $t$-Periodic Boundary Conditions

TL;DR

This paper extends the unified method for solving boundary value problems of integrable PDEs on the half-line, providing a new characterization of spectral functions for non-linearizable boundary conditions and analyzing the asymptotic periodicity of boundary data in the NLS case.

Contribution

It introduces an effective spectral function characterization for non-linearizable boundary conditions and analyzes the asymptotic periodicity of boundary data in the NLS with periodic Dirichlet conditions.

Findings

Spectral functions can be characterized using the global relation and Gelfand-Levitan-Marchenko representations.

For sine-wave Dirichlet data, Neumann boundary values become asymptotically periodic as t→∞.

Explicit asymptotic expansion of Neumann data up to third order in perturbation theory.

Abstract

Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem formulated in the complex $k$-plane (the Fourier plane), which has a jump matrix with explicit $(x,t)$-dependence involving four scalar functions of $k$, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand-Levitan-Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of $t$-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as $t\to\infty$, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.