Initial-Boundary Value Problems for the Coupled Nonlinear Schrödinger Equation on the Half-Line

2015 ◽  
Vol 135 (3) ◽  
pp. 310-346 ◽  
Author(s):  
Xianguo Geng ◽  
Huan Liu ◽  
Junyi Zhu
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problems ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Schrodinger Equation ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Coupled Nonlinear Schrödinger Equation ◽  
Initial Boundary Value

scholarly journals Asymptotics of linear initial boundary value problems with periodic boundary data on the half-line and finite intervals

2009 ◽  
Vol 465 (2111) ◽  
pp. 3341-3360 ◽  
Author(s):  
Guillaume Michel Dujardin
Keyword(s):  
Schrödinger Equation ◽  
Asymptotic Behaviour ◽  
Boundary Value Problems ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using Fokas’ transformation method, we show that, for the linear Schrödinger equation, the linear heat equation and the linearized KdV equation on the half-line, the solutions indeed become periodic for large time. However, for the same linear Schrödinger equation on a finite interval, we show that the solution, in general, is not asymptotically periodic; actually, the asymptotic behaviour of the solution depends on the commensurability of the time period T of the boundary data with the square of the length of the interval over π .

Initial Boundary-Value Problems for Derivative Nonlinear Schroedinger Equation. Justification of Two-Step Algorithm

2002 ◽  
Vol 7 (2) ◽  
pp. 69-104 ◽  
Author(s):  
T. Meškauskas ◽  
F. Ivanauskas
Keyword(s):  
Difference Scheme ◽  
Boundary Value Problems ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Schrodinger Equation ◽  
Initial Boundary Value Problems ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Step Algorithm ◽  
Initial Boundary Value

We investigate two different initial boundary-value problems for derivative nonlinear Schrödinger equation. The boundary conditions are Dirichlet or generalized periodic ones. We propose a two-step algorithm for numerical solving of this problem. The method consists of Bäcklund type transformations and difference scheme. We prove the convergence and stability in C and H1 norms of Crank–Nicolson finite difference scheme for the transformed problem. There are no restrictions between space and time grid steps. For the derivative nonlinear Schrödinger equation, the proposed numerical algorithm converges and is stable in C1 norm.

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