scholarly journals Initial-Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

2015 ◽  
Vol 134 (3) ◽  
pp. 276-362 ◽  
Author(s):  
Peter D. Miller ◽  
 Zhenyun Qin
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problems ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Semiclassical Limit ◽  
Boundary Value ◽  
Initial Boundary ◽  
Nonlinear Schrodinger Equation ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

scholarly journals On an initial-boundary value problem for the nonlinear Schrödinger equation

1979 ◽  
Vol 2 (3) ◽  
pp. 503-522 ◽  
Author(s):  
Herbert Gajewski
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Initial Boundary Value

We study an initial-boundary value problem for the nonlinear Schrödinger equation, a simple mathematical model for the interaction between electromagnetic waves and a plasma layer. We prove a global existence and uniqueness theorem and establish a Galerkin method for solving numerically the problem.

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 .

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