### 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 language | English (US) |
---|---|

Pages (from-to) | 507-568 |

Number of pages | 62 |

Journal | Nonlinearity |

Volume | 13 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2000 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Nonlinearity*,

*13*(3), 507-568. https://doi.org/10.1088/0951-7715/13/3/303

}

*Nonlinearity*, vol. 13, no. 3, pp. 507-568. https://doi.org/10.1088/0951-7715/13/3/303

**Metastability of breather modes of time-dependent potentials.** / Miller, P. D.; Soffer, Avraham; Weinstein, M. I.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Metastability of breather modes of time-dependent potentials

AU - Miller, P. D.

AU - Soffer, Avraham

AU - Weinstein, M. I.

PY - 2000/5/1

Y1 - 2000/5/1

N2 - 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.

AB - 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.

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U2 - https://doi.org/10.1088/0951-7715/13/3/303

DO - https://doi.org/10.1088/0951-7715/13/3/303

M3 - Article

VL - 13

SP - 507

EP - 568

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 3

ER -