Metastability of breather modes of time-dependent potentials

P. D. Miller, Avraham Soffer, M. I. Weinstein

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

We study the solutions of linear Schrödinger equations in which the potential energy is a periodic function of time and is sufficiently localized in space. We consider the potential to be close to one that is time periodic and yet explicitly solvable. A large family of such potentials has been constructed and the corresponding Schrödinger equation solved by Miller and Akhmediev. Exact bound states, or breather modes, exist in the unperturbed problem and are found to be generically metastable in the presence of small periodic perturbations. Thus, these states are long-lived but eventually decay. On a time scale of order ∈-2, where ∈ is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermi's golden rule. For times of order ∈-1 the breather modes are frequency shifted. This behaviour is derived first by classical multiple-scale expansions, and then in certain circumstances we are able to apply the rigorous theory developed by Soffer and Weinstein and extended by Kirr and Weinstein to justify the expansions and also provide longer-time asymptotics that indicate eventual dispersive decay of the bound states with behaviour that is algebraic in time. As an application, we use our techniques to study the frequency dependence of the guidance properties of certain optical waveguides. We supplement our results with numerical experiments.

Original languageEnglish (US)
Pages (from-to)507-568
Number of pages62
JournalNonlinearity
Volume13
Issue number3
DOIs
StatePublished - May 1 2000

Fingerprint

Metastability
Breathers
metastable state
Decay
Optical waveguides
Potential energy
Linear equations
Bound States
Fermi's Golden Rule
decay
Perturbation
Long-time Asymptotics
Optical Waveguides
Multiple Scales
Periodic Functions
Justify
Guidance
perturbation
periodic functions
Linear equation

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)
  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Miller, P. D. ; Soffer, Avraham ; Weinstein, M. I. / Metastability of breather modes of time-dependent potentials. In: Nonlinearity. 2000 ; Vol. 13, No. 3. pp. 507-568.
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Metastability of breather modes of time-dependent potentials. / Miller, P. D.; Soffer, Avraham; Weinstein, M. I.

In: Nonlinearity, Vol. 13, No. 3, 01.05.2000, p. 507-568.

Research output: Contribution to journalArticle

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