The Nonlinear Schrödinger Equation on the Interval

Abstract

Let $q(x,t)$ satisfy the Dirichlet initial-boundary value problem for the nonlinear Schrödinger 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 $\exp [2ikx+4i{k}^{2}t]$, 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.

References in corpus (1)

• The initial boundary value problem on the segment for the Nonlinear Schrödinger equation; the algebro-geometric approach. I

Cited by in corpus (65)

• The Unified Method: I Non-Linearizable Problems on the Half-Line
• Initial-boundary value problems for integrable evolution equations with $3\times 3$ Lax pairs
• Soliton solutions of nonlinear Schroedinger equation on simple networks
• An initial-boundary value problem for the integrable spin-1 Gross-Pitaevskii equations with a 4x4 Lax pair on the half-line
• Shock problem for MKdV equation: Long time Dynamics of the Step-like initial data
• Solitons, boundary value problems and a nonlinear method of images
• Initial-boundary value problems associated with the Ablowitz-Ladik system
• Initial-boundary value problem for an integrable spin-1 Gross-Pitaevskii system with a 4x4 Lax pair on a finite interval
• Initial-boundary value problems for discrete evolution equations: discrete linear Schrodinger and integrable discrete nonlinear Schrodinger equations
• Stability of the periodic Toda lattice under short range perturbations
• Long-time asymptotics for the Hirota equation on the half-line
• The Generalized Dirichlet to Neumann map for the KdV equation on the half-line
• A higher dispersion KdV equation on the half-line
• A Riemann-Hilbert Approach to the Complex Sharma-Tasso-Olver Equation on the Half Line
• An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons
• Transport in simple networks described by integrable discrete nonlinear SchrÄodinger equation
• $L$-functions for families of generalized Kloosterman sums and $p$-adic differential equations
• An efficient explicit full-discrete scheme for strong approximation of stochastic Allen-Cahn equation
• Initial boundary value problem for the focusing NLS equation with Robin boundary condition: half-line approach
• The derivative nonlinear Schrödinger equation on the half-line
• The nonlinear Schrödinger equation with $t$-periodic data: I. Exact results
• GHW-based Necessary and Sufficient Conditions for Support Constrained Generator Matrices
• Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data
• NLS ground states on the half-line with point interactions
• On the Riemann-Hilbert problem for the Chen-Lee-Liu derivative nonlinear Schrödinger equation
• The Kadomtsev-Petviashvili II Equation on the Half-Plane
• Multidimensional Inverse Scattering of Integrable Lattice Equations
• Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval
• A new approach to integrable evolution equations on the circle
• The Robin problem for the nonlinear Schrödinger equation on the half-line
• On the Dirichlet to Neumann Problem for the 1-dimensional Cubic NLS equation on the Half-Line; Zero Initial Data
• The coupled Fokas-Lenells equations by a Riemann-Hilbert approach
• The nonlinear steepest descent approach for long time behavior of the two-component coupled Sasa-Satsuma equation with a $5\times 5$ Lax pair
• A Riemann-Hilbert Approach to the Kundu-Eckhaus Equation on the half-Line
• Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems
• The Ablowitz-Ladik system on a finite set of integers
• The massive Thirring system in the quarter plane
• Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect
• The mKdV equation on a finite interval
• Long-time asymptotics for the derivative nonlinear Schrödinger equation on the half-line
• An initial-boundary value problem of the general three-component nonlinear Schrodinger equation with a 4x4 Lax pair on a finite interval
• On an inverse scattering transform for the nonlinear Schödinger equation on the half-line
• The Ablowitz-Ladik system on a graph
• Initial-boundary value problems for the defocusing nonlinear Schrödinger equation in the semiclassical limit
• The nonlinear Schrödinger equation with forcing involving products of eigenfunctions
• Dressing the boundary: on soliton solutions of the nonlinear Schrödinger equation on the half-line
• Well-Posedness of Initial-Boundary Value Problems for a Reaction-Diffusion Equation
• Admissible boundary values for the defocusing nonlinear Schrödinger equation with asymptotically time-periodic data
• Soliton solutions of the nonlinear Schrödinger equation with defect conditions
• The Numerical Unified Transform Method for Initial-boundary Value Problems on the Half-line
• Vector NLS solitons interacting with a boundary
• The derivative nonlinear Schrodinger equation on the interval
• Boundary value problems for the stationary axisymmetric Einstein equations: a rotating disk
• Riemann-Hilbert problem associated with the fourth-order dispersive nonlinear Schrödinger equation in optics and magnetic mechanics
• Explicit soliton asymptotics for the Korteweg-de Vries equation on the half-line
• Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations
• The initial-boundary value problems for the coupled derivative nonlinear Schrödinger equations on the half-line
• The Focusing NLS Equation on the Half-Line with Periodic Boundary Conditions
• An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times 4$ Lax pair
• The unified transform method for the Sasa-Satsuma equation on the interval
• The GLM representation of the global relation for the two-component nonlinear Schrödinger equation on the interval
• The hyperbolic Ernst--Maxwell equations in a triangular domain
• The defocusing nonlinear Schrödinger equation with $t$-periodic data: New exact solutions
• On the solutions of some boundary value problems for the general KdV equation
• The Fokas method to the Sasa-Satsuma equation on the half-line