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. © 2009 The Royal Society.
|Original language||English (US)|
|Number of pages||20|
|Journal||Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|
|State||Published - Aug 12 2009|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: The author thanks A.S. Fokas and P.A. Markowich for their ideas and comments on this work. This publication is based on work supported by Award No. KUK-I1-007-43, made by the King Abdullah University of Science and Technology.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.