Assuming that the solution q(x, t) of the nonlinear Schrodinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1–39) that q(x, t) can be represented in terms of the solution of a matrix Riemann–Hilbert (RH) problem formulated in the complex k-plane. The jump matrix of this RH problem has explicit x, t dependence and it is defined in terms of the scalar functions {a(k), b(k), A(k), B(k)} referred to as spectral functions. The functions a(k) and b(k) are defined in terms of q0(x) = q(x,0), while the functions A(k) and B(k) are defined in terms of g0(t) = q(0,t) and g1(t) = qx(0,t). The spectral functions are not independent but they satisfy an algebraic global relation. Here we first prove that if there exist spectral functions satisfying this global relation, then the function q(x, t) defined in terms of the above RH problem exists globally and solves the nonlinear Schrodinger equation, and furthermore q(x, 0) = q0(x), q(0, t) = g0(t) and qx(0, t) = g1(t). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable, the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is qx(0, t) − ρq(0, t) = 0 where ρ is a real constant.

https://iopscience.iop.org/article/10.1088/0951-7715/18/4/019/pdf

The nonlinear Schrödinger equation on the half-line

Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrodinger equation on the finite interval, $0 < x < L$, with $q_{0}(x) = q(x,0)$, $g_{0}(t) = q(0,t)$, $f_{0}(t) = q(L,t)$. Let $g_{1}(t)$ and $f_{1}(t)$ denote the {\it unknown} boundary values $q_{x}(0,t)$ and $q_{x}(L,t)$, respectively. We first show that these unknown functions can be expressed in terms of the given initial and boundary conditions through the solution of a system of nonlinear ODEs. Although the question of the global existence of solution of this system remains open, it appears that this is the first time in the literature that such a characterization is explicitely described for a nonlinear evolution PDE defined on the interval; this result is the extension of the analogous result of [4] and [6] from the half-line to the interval. We then show that $q(x,t)$ can be expressed in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem formulated in the complex $k$ - plane. This problem has explicit $(x,t)$ dependence in the form $\exp[2ikx + 4ik^2t]$, and it has jumps across the real and imaginary axes. The relevant jump matrices are explicitely given in terms of the spectral data $\{a(k), b(k)\}$, $\{A(k), B(k)\}$, and $\{\A(k), \B(k)\}$, which in turn are defined in terms of $q_{0}(x)$, $\{g_{0}(t), g_{1}(t)\}$, and $\{f_{0}(t), f_{1}(t)\}$, espectively.

The Nonlinear Schr\"odinger Equation on the Interval

Boundary value problems for integrable nonlinear evolution PDEs for- mulated on the half-line can be analyzed by the unied 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 dicult step of the new method is the characteriza- tion 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 manip- ulations. Here, we present an eective 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, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated lin- earized PDE invariant, and on the computation of the large k asymptotics of the eigenfunctions dening the relevant spectral functions.

The unified method: i non-linearizable problems on the half-line

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 the 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 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, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions.

https://iopscience.iop.org/article/10.1088/1751-8113/45/19/195201/pdf

The unified method: I. Nonlinearizable problems on the half-line

We consider the one-dimensional focusing nonlinear Schr\"odinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a B\"acklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, earlier work on the problem by Holmer and Zworski.

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data

The integrable initial boundary value problem on a semi-line for the nonlinear Schrodinger equation is considered. It is shown that by means of the Backlund transformation this problem can be reduced to the well known Cauchy problem for the same equation on the line.

http://iopscience.iop.org/article/10.1088/0305-4470/24/11/017/pdf

R F Bikbaev

Papers

Initial boundary value problem for the nonlinear Schrodinger equation