Initial boundary value problems for some nonlinear dispersive models on the half-line: a review and open problems

2019 ◽  
Vol 13 (2) ◽  
pp. 418-434
Author(s):  
Márcio Cavalcante
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Open Problems ◽  
Initial Boundary Value ◽  
Dispersive Models

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 .

scholarly journals Zero-dispersion limit for integrable equations on the half-line with linearisable data

2004 ◽  
Vol 2004 (5) ◽  
pp. 361-370 ◽  
Author(s):  
A. S. Fokas ◽  
S. Kamvissis
Keyword(s):  
Surface Tension ◽  
Boundary Value Problems ◽  
Kdv Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Half Line ◽  
Initial Boundary Value Problems ◽  
Nonlinear Schrödinger ◽  
Initial Boundary Value ◽  
Dispersion Limit

We study the zero-dispersion limit for certain initial boundary value problems for the defocusing nonlinear Schrödinger (NLS) equationand for the Korteweg-de Vries (KdV)equation with dominant surface tension. These problems are formulated on the half-line and they involve linearisable boundaryconditions.

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