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Emanuele Haus

Publications and source records attributed to Emanuele Haus.

At least 19 recordsLinked to original sources
arXiv2026

On the integrability of the Kirchhoff-Pohozaev equation on tori

In this paper we study the Kirchhoff-Pohozaev equation introduced in \cite{P2} and its constants of motion (see \cite{BoitiManfrin2025}, \cite{BoitiManfrin2026}) on $n$ dimensional tori. We show that these Hamiltonians are all in involution and we prove that they are generated by an infinite list of constants of motion which are all defined and in involution on a fixed phase space. Then we study the Kirchhoff-Pohozaev equation restricted to a finite Fourier support. In dimension $n=1$ we show that such finite dimensional reduction is always completely integrable and provide an analytic Brikhoff normal form in a neighborhood of the origin. We also give sufficient conditions for integrability for $n>1$. We finally show that the formal Birkhoff Normal form of the Kirchhoff-Pohozaev equation is integrable for $n=1$.

math.DS↗
arXiv2026

Quasilinear normal form for the Kirchhoff-Poho{\v z}aev equation

On the $n$-dimensional torus $\mathbb{T}^n$, we consider the Kirchhoff-Poho{\v z}aev equation, which is the only Kirchhoff-type equation known to admit global solutions for small initial data in $H^s \times H^{s-1}$, $s \geq 2$. We study this equation from a dynamical perspective by means of a quasilinear normal form approach. We perform two steps of normal form reduction and show that the resulting cubic and quintic terms do not contribute to the Sobolev energy estimates. This cancellation reveals a special algebraic structure of the equation and represents a first step towards a dynamical explanation of its exceptional global well-posedness. As a further consequence, we obtain improved bounds on the growth of Sobolev norms and improved lower bounds on the existence time for initial data in $H^s \times H^{s-1}$ with $s \in [3/2, 2)$.

math.AP↗
arXiv2025

On the reducibility of the 1d quantum harmonic oscillator with a quasi-periodic bounded potential

Using the decay along the diagonal of the matrix representing the perturbation with respect to the Hermite basis, we prove a reducibility result in $L^2(\mathbb{R})$ for the one-dimensional quantum harmonic oscillator perturbed by time quasi-periodic potential, via a KAM iteration. The potential is only bounded (no decay at infinity is required) and its derivative with respect to the spatial variable $x$ is allowed to grow at most like $|x|^δ$ when $x$ goes to infinity, where the power $δ<1$ is arbitrary.

math.DS↗
arXiv2023

Effective chaos for the Kirchhoff equation on tori

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillates in a chaotic way on certain long time scales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, that we name as effective chaos (it occurs over a long, but finite, time scale), is consequence of the existence of symbolic dynamics for an effective system. Since the first order resonant dynamics has been proved to be essentially stable, we need to perform a second order analysis to find an effective model displaying chaotic dynamics. More precisely, after some reductions, this model behaves as two weakly coupled pendulums.

math.AP↗
arXiv2022

Reducibility and nonlinear stability for a quasi-periodically forced NLS

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 $\mathbb T^2:= (\mathbb R/2π\mathbb Z)^2$, we consider a quasi-periodically forced NLS equation on $\mathbb T^2$ 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.

math.AP↗
arXiv2020

Longer lifespan for many solutions of the Kirchhoff equation

We consider the Kirchhoff equation $$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb T^d$, and its Cauchy problem with initial data $u(0,x)$, $\partial_t u(0,x)$ of size $\varepsilon$ in Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound $\varepsilon^{-6}$ for the lifespan of the corresponding solutions (the standard local theory gives $\varepsilon^{-2}$, and the normal form for the cubic terms gives $\varepsilon^{-4}$). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the "superactions" of the effective equations on large time intervals is smaller (by a factor $\varepsilon^2$) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.

math.AP↗
arXiv2020

On the normal form of the Kirchhoff equation

Consider the Kirchhoff equation $$ \partial_{tt} u - Δu \Big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb{T}^d$. In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

math.AP↗
arXiv2019

Size of data in implicit function problems and singular perturbations for nonlinear Schrödinger systems

We investigate a general question about the size and regularity of the data and the solutions in implicit function problems with loss of regularity. First, we give a heuristic explanation of the fact that the optimal data size found by Ekeland and Séré with their recent non-quadratic version of the Nash-Moser theorem can also be recovered, for a large class of nonlinear problems, with quadratic schemes. Then we prove that this heuristic observation applies to the singular perturbation Cauchy problem for the nonlinear Schrödinger system studied by Métivier, Rauch, Texier, Zumbrun, Ekeland, Séré. Using a "free flow component" decomposition and applying an abstract Nash-Moser-Hörmander theorem, we improve the existing results regarding both the size of the data and the regularity of the solutions.

math.AP↗
arXiv2019

Long time growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

We consider the semiclassical Schrödinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t ψ= \left(-\frac{\hbar^2}{2} Δ+ W_l(x) \right)ψ+ V(t,x)ψ,$$ where $W_l$ is an anharmonic trapping of the form $W_l(x)= \frac{1}{2l}\sum_{j=1}^d x_j^{2l}$, $l\geq 2$ is an integer and $\hbar$ is a semiclassical small parameter. We construct a smooth potential $V(t,x)$, bounded in time with its derivatives, and an initial datum such that the Sobolev norms of the solution grow at a logarithmic speed for all times of order $\log^{\frac12}(\hbar^{-1})$. The proof relies on two ingredients: first we construct an unbounded solution to a forced mechanical anharmonic oscillator, then we exploit semiclassical approximation with coherent states to obtain growth of Sobolev norms for the quantum system which are valid for semiclassical time scales.

math.AP↗
arXiv2018

A Nash-Moser-Hörmander implicit function theorem with applications to control and Cauchy problems for PDEs

We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by Hörmander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by Métivier). As an example of application, we apply our theorem to a control and a Cauchy problem for quasi-linear perturbations of KdV equations, improving the regularity of a previous result. With respect to other approaches to control and Cauchy problems, the application of our theorem requires lighter assumptions to be verified.

math.FA↗
arXiv2018

Controllability of quasi-linear Hamiltonian NLS equations

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-Moser-Hörmander implicit function theorem as a black box.

math.AP↗
arXiv2018

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.

math.AP↗
arXiv2018

Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional 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 $H^s(\mathbb{T}^2)$ ($0<s<1$). 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.

math.AP↗
arXiv2018

On the existence time for the Kirchhoff equation with periodic boundary conditions

We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.

math.AP↗
arXiv2017

Growth of Sobolev norms for the analytic NLS on $\mathbb T^2$

We consider the completely resonant defocusing non-linear Schrödinger equation on the two dimensional torus with any analytic gauge invariant nonlinearity. Fix $s>1$. We show the existence of solutions of this equation which achieve arbitrarily large growth of $H^s$ Sobolev norms. We also give estimates for the time required to attain this growth.

math.AP↗
arXiv2016

Exact controllability for quasi-linear perturbations of KdV

We prove that the KdV equation on the circle remains exactly controllable in arbitrary time with localized control, for sufficiently small data, also in presence of quasi-linear perturbations, namely nonlinearities containing up to three space derivatives, having a Hamiltonian structure at the highest orders. We use a procedure of reduction to constant coefficients up to order zero, classical Ingham inequality and HUM method to prove the controllability of the linearized operator. Then we prove and apply a modified version of the Nash-Moser implicit function theorems by Hörmander.

math.AP↗
arXiv2016

KAM for beating solutions of the quintic NLS

We consider the nonlinear Schrödinger equation of degree five on the circle $\mathbb{S}^1 = \mathbb{R}/2π$. We prove the existence of quasi-periodic solutions which bifurcate from "resonant" solutions (studied in [14]) of the system obtained by truncating the Hamiltonian after one step of Birkhoff normal form, exhibiting recurrent exchange of energy between some Fourier modes. The existence of these quasi-periodic solutions is a purely nonlinear effect.

math.AP↗