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

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.

29 pages

References in corpus (2)

• The Unified Method: I Non-Linearizable Problems on the Half-Line
• The Generalized Dirichlet to Neumann map for the KdV equation on the half-line

Cited by in corpus (12)

• An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line
• Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval
• The nonlinear Schrödinger equation with $t$-periodic data: I. Exact results
• The nonlinear Schrödinger equation with $t$-periodic data: II. Perturbative results
• Initial-boundary value problem for integrable nonlinear evolution equations with $3\times 3$ Lax pairs on the interval
• An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval
• Perturbative and exact results on the Neumann value for the nonlinear Schrödinger equation on the half-line
• Initial-boundary value problems for the defocusing nonlinear Schrödinger equation in the semiclassical limit
• Well-Posedness of Initial-Boundary Value Problems for a Reaction-Diffusion Equation
• Riemann-Hilbert problem of the three-component coupled Sasa-Satsuma equation and its multi-soliton solutions
• On the spectral problem associated with the time-periodic nonlinear Schrödinger equation
• The Fokas method to the Sasa-Satsuma equation on the half-line