Tame majorant analyticity for the Birkhoff map of the defocusing nonlinear Schrödinger equation on the circle

Maspero, Alberto

doi:10.1088/1361-6544/aaa7ba

Cited by 7 publications

(11 citation statements)

References 33 publications

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“…Kuksin-Perelman's Theorem. In this section we recall the Vey type theorem obtained by Kuksin and Perelman in [KP10] (see also [BM16,Mas18]) and prove that it can be obtained as a corollary of Theorem 2.1. We come to the assumptions of the Kuksin-Perelman's Theorem.…”

Section: 4mentioning

confidence: 92%

“…In the present paper we show that Kuksin-Perelman's result can also be deduced from our Theorem 2.1, in the sense that the assumptions of Kuksin-Perelman's Theorem imply the assumptions of Theorem 2.1. Thus, in particular our main result applies to all the systems for which the assumptions of Kuksin-Perelman's Theorem hold ([KP10, BM16,Mas18]).…”

Section: Introductionmentioning

confidence: 95%

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Convergence to Normal Forms of Integrable PDEs

Bambusi

Stolovitch

2020

Commun. Math. Phys.

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In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincaré normal form and of the transformation to it, through a rapid convergence algorithm.

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Section: 4mentioning

confidence: 92%

Section: Introductionmentioning

confidence: 95%

Convergence to Normal Forms of Integrable PDEs

Bambusi

Stolovitch

2020

Commun. Math. Phys.

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“…Remark that, for autonomous system, the phenomenon of having all solutions bounded is very interesting and often associated to some sort of integrability, for example as it happens in the defocusing cubic NLS on T or the Toda lattice (see e.g. [29,5]).…”

Section: Unbounded Orbits For Classical Anharmonic Oscillatormentioning

confidence: 99%

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

Haus

Maspero

2020

Journal of Functional Analysis

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“…In particular, we need that it "behaves well" in 1 . This is done in the paper [Mas18b] and summarized in Section 3. In Birkhoff coordinates, the finite gap solutions are supported in a finite set of variables.…”

Section: Comments and Remarks On Theorem 11mentioning

confidence: 99%

A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

Guàrdia

Hani

Haus

et al. 2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the twodimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces H s (T 2 ) (0 < s < 1). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H s topology and whose H s norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.1.1. The dynamical system and its quasiperiodic objects. We start by describing the dynamical system and its quasiperiodic invariant objects at the center of our analysis. Consider the periodic cubicNow, we describe the invariant objects around which we will study and construct our long-time nonlinear dynamics. Of course, such a task requires a very precise understanding of the linearized dynamics around such objects. For this reason, we take the simplest non-trivial family of invariant quasiperiodic tori admitted by (2D-NLS), namely those inherited from its completely integrable 1D counterpartThis is a subsystem of (2D-NLS) if we consider solutions that depend only on the first spatial variable. It is well known that equation (1D-NLS) is integrable and its phase space is foliated by tori of finite or infinite dimension with periodic, quasiperiodic, or almost periodic dynamics. The quasiperiodic orbits are usually called finite-gap solutions. Such tori are Lyapunov stable (for all time!) as solutions of (1D-NLS) (as will be clear once we exhibit its integrable structure) and some of them are linearly stable as solutions of (2D-NLS), but we will be interested in their long-time nonlinear stability (or lack of it) as invariant objects for the 2D equation (2D-NLS). In fact, we shall show that they are nonlinearly unstable as solutions of (2D-NLS), and in a strong sense, in certain topologies and after very long times. Such instability is transversal in the sense that one drifts along the purely 2-dimensional directions: solutions which are initially very close to 1-dimensional become strongly 2-dimensional after some long time scales 2 .1.2. Energy Cascade, Sobolev norm growth, and Lyapunov instability. In addition to studying long-time dynamics close to invariant objects for NLS, another purpose of this work is to make progress on a fundamental problem in nonlinear wave theory, which is the transfer of energy between characteristically different scales for a nonlinear dispersive PDE. This is called the energy cascade phenomenon....

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Tame majorant analyticity for the Birkhoff map of the defocusing nonlinear Schrödinger equation on the circle

Cited by 7 publications

References 33 publications

Convergence to Normal Forms of Integrable PDEs

Convergence to Normal Forms of Integrable PDEs

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

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