On the Stability of Periodic Multi-Solitons of the KdV Equation

Kappeler, Thomas; Montalto, Riccardo

doi:10.1007/s00220-021-04089-9

Cited by 5 publications

(2 citation statements)

References 35 publications

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“…The next studied invariant objects have been the so-called finite gap solutions, which are solutions for the integrable 1d cubic NLS. In particular, they are solutions of (1.1) depending only on 1 variable, say x 1 , but they are quasi-periodic in time and fill invariant tori of finite dimension d ∈ N. Their long time stability has been studied in [35] (see also [29]) whereas their instability, again in H s with s ∈ (0, 1) in [24]. As the reader might guess, again a fundamental role is played by the linearized frequencies at finite gap solutions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Haus,

Langella,

Maspero

et al. 2024

Pure and Applied Mathematics Quarterly

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Motivated by the problem of long time stability vs. instability of KAM tori of the Nonlinear cubic Schrödinger equation (NLS) on the two dimensional torus T 2 := (R/2πZ) 2 , we consider a quasi-periodically forced NLS equation on T 2 arising from the linearization of the NLS at a KAM torus. We prove a reducibility result as well as long time stability of the origin. The main novelty is to obtain the precise asymptotic expansion of the frequencies which allows us to impose Melnikov conditions at arbitrary order.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Haus,

Langella,

Maspero

et al. 2024

Pure and Applied Mathematics Quarterly

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show abstract

“…An interesting research direction in the study of Hamiltonian PDEs concerns the orbital stability of invariant objects rather than fixed points, such as plane waves and quasi-periodic tori. About that, we mention [31,45,47,56]. We also quote the stability result [18] for traveling waves of the Burger-Hilbert equation.…”

Section: Introductionmentioning

confidence: 99%

Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle

Feola,

Giuliani

2024

J Dyn Diff Equat

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We consider a class of Hamiltonian Klein–Gordon equations with a quasilinear, quadratic nonlinearity under periodic boundary conditions. For a large set of masses, we provide a precise description of the dynamics for an open set of small initial data of size $$\varepsilon $$ ε showing that the corresponding solutions remain close to oscillatory motions over a time scale $$\varepsilon ^{{-\frac{9}{4}+\delta }}$$ ε - 9 4 + δ for any $$\delta >0$$ δ > 0 . The key ingredients of the proof are normal form methods, para-differential calculus and a modified energy approach.

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A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

Bambusi,

Gérard

2024

Math. Z.

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We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map from the energy space to itself. Let $$\epsilon $$ ϵ be the size of the perturbation. We prove that for initial data close in energy norm to an N-gap state of the unperturbed equation all the actions of the Benjamin Ono equation remain $${\mathcal {O}}(\epsilon ^{\frac{1}{2(N+1)}})$$ O ( ϵ 1 2 ( N + 1 ) ) close to their initial value for times exponentially long with $$\epsilon ^{-\frac{1}{2(N+1)}}$$ ϵ - 1 2 ( N + 1 ) .

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On the Stability of Periodic Multi-Solitons of the KdV Equation

Cited by 5 publications

References 35 publications

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Reducibility and nonlinear stability for a quasi-periodically forced NLS

Long Time Dynamics of Quasi-linear Hamiltonian Klein–Gordon Equations on the Circle

A Nekhoroshev theorem for some perturbations of the Benjamin-Ono equation with initial data close to finite gap tori

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