On Longtime Dynamics of Perturbed KdV Equations

Guan Huang

doi:10.1007/s10883-014-9238-3

On Longtime Dynamics of Perturbed KdV Equations

Guan Huang^*

^*Corresponding author for this work

Ecole Polytechnique

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Consider a perturbed KdV equation:(Formula presented.), (Formula presented.), (Formula presented.) where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ϵ-quasi-invariant measure μ and satisfies some additional mild assumptions. Let u^ϵ(t) be a solution. Then on time intervals of order ϵ⁻¹, as ϵ→0, its actions I(u^ϵ(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

Original language	English
Pages (from-to)	379-400
Number of pages	22
Journal	Journal of Dynamical and Control Systems
Volume	21
Issue number	3
DOIs	https://doi.org/10.1007/s10883-014-9238-3
Publication status	Published - 1 Jul 2015
Externally published	Yes

Keywords

Averaging
Gibbs measure
KdV
Longtime behaviour
Perturbations

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10.1007/s10883-014-9238-3

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title = "On Longtime Dynamics of Perturbed KdV Equations",

abstract = "Consider a perturbed KdV equation:(Formula presented.), (Formula presented.), (Formula presented.) where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ϵ-quasi-invariant measure μ and satisfies some additional mild assumptions. Let uϵ(t) be a solution. Then on time intervals of order ϵ−1, as ϵ→0, its actions I(uϵ(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.",

keywords = "Averaging, Gibbs measure, KdV, Longtime behaviour, Perturbations",

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N2 - Consider a perturbed KdV equation:(Formula presented.), (Formula presented.), (Formula presented.) where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ϵ-quasi-invariant measure μ and satisfies some additional mild assumptions. Let uϵ(t) be a solution. Then on time intervals of order ϵ−1, as ϵ→0, its actions I(uϵ(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

AB - Consider a perturbed KdV equation:(Formula presented.), (Formula presented.), (Formula presented.) where the nonlinear perturbation defines analytic operators u(⋅)↦f(u(⋅)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ϵ-quasi-invariant measure μ and satisfies some additional mild assumptions. Let uϵ(t) be a solution. Then on time intervals of order ϵ−1, as ϵ→0, its actions I(uϵ(t,⋅)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

KW - Averaging

KW - Gibbs measure

KW - KdV

KW - Longtime behaviour

KW - Perturbations

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SP - 379

EP - 400

JO - Journal of Dynamical and Control Systems

JF - Journal of Dynamical and Control Systems

IS - 3

ER -

On Longtime Dynamics of Perturbed KdV Equations

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Keywords

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