Almost Periodic Solutions of One Dimensional Schrödinger Equation with the External Parameters

Geng, Jiao; Xu, Xindong

doi:10.1007/s10884-013-9302-9

Cited by 26 publications

(10 citation statements)

References 12 publications

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“…The first result on the existence of almost periodic solutions for Hamiltonian PDEs was proved by Bourgain in [16] using C-W-B method. Later, Pöschel [36] (also see [25] by Geng-Xu) constructed the almost periodic solutions for 1-dimensional NLS by the classical KAM method. The basic idea to obtain these almost periodic solutions is by perturbing the quasi-periodic ones.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Cong

Shi

et al. 2018

Journal of Differential Equations

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Cong

Shi

et al. 2018

Journal of Differential Equations

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“…The fact such potential can be used to modulate the frequencies, comes from spectral results and from the fact that the non-linearity is smoothing. Recently Geng and Xu in [GX13] proved existence of analytic almost periodic solutions for the NLS equation with external parameters given by a Fourier multipliers without assuming any smoothing assumptions on the nonlinearity. Their approach generalizes the one of [Pös02] by applying the ideas of Töplitz-Lipschitz functions which give a better control on the asymptotics of the frequencies.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Almost periodic invariant tori for the NLS on the circle

Biasco¹,

Massetti²,

Procesi³

2019

Preprint

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In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [Bou05] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find "many more" almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.

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“…The first result on the existence of almost periodic solutions for 1-dimensional NLW was given by Bourgain in [6] using C-W-B method. Later, Pöschel [25] (also see [16] by Geng-Xu) constructed the almost periodic solutions for 1-dimensional NLS by the classical KAM method. These almost periodic solutions were obtained by successive small perturbations of quasi-periodic solutions.…”

Section: Now Our Main Results Is As Followsmentioning

confidence: 99%

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

Cong¹,

Mi²,

Shi³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper we prove the existence and linear stability of full dimensional tori with subexponential decay for 1-dimensional nonlinear wave equation with external parameters, which relies on the method of KAM theory and the idea proposed by Bourgain [9].The existence and linear stability of invariant tori for Hamiltonian PDEs have drawn a lot of concerns during the last decades. There are many related works for 1-dimensional PDEs. See [1-3, 12, 14, 15, 17-20, 23, 24, 26, 27] for example. For high dimensional PDEs, Bourgain [7, 8] developed a new method initialed by Craig-Wayne [12] to prove the existence of KAM tori for d-dimensional nonlinear Schrödinger equations (NLS) and d-dimensional NLW with d ≥ 1, based on the Newton iteration, Fröhlich-Spencer techniques, Harmonic analysis and semi-algebraic set theory. This is so-called C-W-B method. Later, Eliasson-Kuksin [13] proved a classical KAM theorem which can be applied to d-dimensional NLS. It is obtained

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Almost Periodic Solutions of One Dimensional Schrödinger Equation with the External Parameters

Cited by 26 publications

References 12 publications

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

The stability of full dimensional KAM tori for nonlinear Schrödinger equation

Almost periodic invariant tori for the NLS on the circle

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

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