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Boundary value problems for integrable nonlinear evolution PDEs formulated onthe half-line can be analyzed by the unified method introduced by one of theauthors and used extensively in the literature. The implementation of thisgeneral method to this particular class of problems yields the solution interms of the unique solution of a matrix Riemann-Hilbert problem formulated inthe complex $k$-plane (the Fourier plane), which has a jump matrix withexplicit $(x,t)$-dependence involving four scalar functions of $k$, calledspectral functions. Two of these functions depend on the initial data, whereasthe other two depend on all boundary values. The most difficult step of the newmethod is the characterization of the latter two spectral functions in terms ofthe given initial and boundary data, i.e. the elimination of the unknownboundary values. For certain boundary conditions, called linearizable, this canbe achieved simply using algebraic manipulations. Here, we first present aneffective characterization of the spectral functions in terms of the giveninitial and boundary data for the general case of non-linearizable boundaryconditions. This characterization is based on the analysis of the so-calledglobal relation and on the introduction of the so-calledGelfand-Levitan-Marchenko representations of the eigenfunctions defining thespectral functions. We then concentrate on the physically significant case of$t$-periodic Dirichlet boundary data. After presenting certain heuristicarguments 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 asDirichlet data, the asymptotics of the Neumann boundary values can be computedexplicitly at least up to third order in a perturbative expansion and indeed atleast up to this order are asymptotically periodic.

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