Exponential averaging for Hamiltonian evolution equations

Karsten Matthies, Arnd Scheel

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.
Original languageEnglish
Pages (from-to)747-773
JournalTransactions of the American Mathematical Society
Volume355
Issue number2
DOIs
Publication statusPublished - 2003

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Hamiltonians
Ordinary differential equations
Partial differential equations
Evolution Equation
Averaging
Forcing
Ordinary differential equation
Initial conditions
Gevrey Spaces
Partial differential equation
Infinite Dimensional Hamiltonian Systems
Nonlinear equations
Strong Resonance
Estimate
Refining
Coupled Nonlinear Schrödinger Equations
Anharmonic Oscillator
Nonlinear Oscillator
Analyticity
Single Mode

Cite this

Exponential averaging for Hamiltonian evolution equations. / Matthies, Karsten; Scheel, Arnd.

In: Transactions of the American Mathematical Society, Vol. 355, No. 2, 2003, p. 747-773.

Research output: Contribution to journalArticle

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