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Riccardo Montalto

Publications and source records attributed to Riccardo Montalto.

At least 37 records · Page 2Linked to original sources
arXiv2021

Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the Benjamin-Ono equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudo-differential operator of order 0 with principal part given by a modified Fourier transform (modification by a phase factor) and (2) the pullback of the Hamiltonian of the Benjamin-Ono is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of para-differential operators. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the Benjamin-Ono equation under small, quasi-linear perturbations.

math.AP↗
arXiv2021

On the stability of periodic multi-solitons of the KdV equation

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size $\varepsilon > 0$, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least $O(\varepsilon^{-2})$. To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.

math.AP↗
arXiv2021

Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori

We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension $d\geq1$. We study the behaviour of high Sobolev norms $H^{s}$, $s\gg1$, of solutions with initial conditions in $H^{s}$ whose $H^ρ$-Sobolev norm, $1\llρ\ll s$, is smaller than $\e\ll1$. We provide a control of the $H^{s}$-norm over a time interval of order $O(\e^{-2})$. %where $\e\ll1$ is the size of the initial condition in $H^ρ$. Due to the lack of conserved quantities controlling high Sobolev norms, the key ingredient of the proof is the construction of a modified energy equivalent to the "low norm" $H^ρ$ (when $ρ$ is sufficiently high) over a nontrivial time interval $O(\e^{-2})$. This is achieved by means of normal form techniques for quasi-linear equations involving para-differential calculus. The main difficulty is to control the possible loss of derivatives due to the small divisors arising form three waves interactions. By performing "tame" energy estimates we obtain upper bounds for higher Sobolev norms $H^{s}$.

math.AP↗
arXiv2020

Almost-periodic Response Solutions for a forced quasi-linear Airy equation

We prove the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. This is the first result about the existence of this type of solutions for a quasi-linear PDE. The solutions turn out to be analytic in time and space. To prove our result we use a Craig-Wayne approach combined with a KAM reducibility scheme and pseudo-differential calculus on ${\mathbb T}^\infty$.

math.AP↗
arXiv2020

The Navier-Stokes equation with time quasi-periodic external force: existence and stability of quasi-periodic solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the $d$-dimensional torus $\T^d$, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $H^s$ (for $s$ large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $t \to + \infty$, with an exponential rate of convergence $O( e^{- αt })$ for any arbitrary $α\in (0, 1)$.

math.AP↗
arXiv2020

Quasi-periodic incompressible Euler flows in 3D

We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus $\T^3$, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.

math.AP↗
arXiv2019

Linear Schrödinger equation with an almost periodic potential

We study the reducibility of a Linear Schrödinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of the almost-periodic perturbation, we prove that such an equation is reducible to constant coefficients via an analytic almost-periodic change of variables. This implies control of both Sobolev and Analytic norms for the solution of the corresponding Schrödinger equation for all times.

math.AP↗
arXiv2019

Large KAM tori for quasi-linear perturbations of KdV

In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.

math.AP↗
arXiv2019

On the spectrum of the Schrödinger operator on $\mathbb{T}^d$: a normal form approach

In this paper we study the spectrum of the operator \begin{equation} \label{ope} H:=(-Δ)^{M/2}+\mathcal{V}\ , \quad M>0\ , \end{equation} on $L^2(\mathbb{R}^d/Γ)$, with $Γ$ a maximal dimension lattice in $\mathbb{R}^d$ and $\mathcal{V}$ a pseudodifferential operator of order strictly smaller than $M$. We prove that most of its eigenvalues admit the asymptotic expansion \begin{equation} \label{sim} λ_ξ=|ξ|^M+Z(ξ)+O(\left|ξ\right|^{-\infty})\ , \end{equation} where $Z$ is a $C^\infty(\mathbb{R}^d)$ function (symbol) and $ξ\inΓ^*$ (the dual lattice of $Γ$).

math-ph↗
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

Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.

math.DS↗
arXiv2018

Reducibility of non-resonant transport equation on $T^d$ with unbounded perturbations

We prove reducibility of a transport equation on the $d$-dimensional torus $T^d$ with a time quasi-periodic unbounded perturbation. As far as we know this is the first example of a reducibility result for an equation in more than one dimensions with unbounded perturbations. Furthermore the unperturbed problem has eigenvalues whose differences are dense on the real axis.

math-ph↗
arXiv2018

Quasi-periodic standing wave solutions of gravity-capillary water waves

We prove the existence and the linear stability of small amplitude time {\it quasi-periodic} standing wave solutions (i.e. periodic and even in the space variable $ x $) of a $ 2 $-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.

math.AP↗
arXiv2018

Growth of Sobolev norms for time dependent periodic Schrödinger equations with sublinear dispersion

In this paper we consider Schrödinger equations with sublinear dispersion relation on the one-dimensional torus $\T := \R /(2 π\Z)$. More precisely, we deal with equations of the form $\partial_t u = \ii {\cal V}(ωt)[u]$ where ${\cal V}(ωt)$ is a quasi-periodic in time, self-adjoint pseudo-differential operator of the form ${\cal V}(ωt) = V(ωt, x) |D|^M + {\cal W}(ωt)$, $0 < M \leq 1$, $|D| := \sqrt{- \partial_{xx}}$, $V$ is a smooth, quasi-periodic in time function and ${\cal W}$ is a quasi-periodic time-dependent pseudo-differential operator of order strictly smaller than $M$. Under suitable assumptions on $V$ and ${\cal W}$, we prove that if $ω$ satisfies some non-resonance conditions, the solutions of the Schrödinger equation $\partial_t u = \ii {\cal V}(ωt)[u]$ grow at most as $t^η$, $t \to + \infty$ for any $η> 0$. The proof is based on a reduction to constant coefficients up to smoothing remainders of the vector field $\ii {\cal V}(ωt)$ which uses Egorov type theorems and pseudo-differential calculus. The {\it homological equations} arising in the reduction procedure involve both time and space derivatives, since the dispersion relation is sublinear. Such equations can be solved by imposing some Melnikov non-resonance conditions on the frequency vector $ω$.

math.AP↗
arXiv2018

Quasi-periodic solutions for the forced Kirchhoff equation on $\mathbb{T}^d$

In this paper we prove the existence of small-amplitude quasi-periodic solutions with Sobolev regularity, for the $d$-dimensional forced Kirchhoff equation with periodic boundary conditions. This is the first result of this type for a quasi-linear equations in high dimension. The proof is based on a Nash-Moser scheme in Sobolev class and a regularization procedure combined with a multiscale analysis in order to solve the linearized problem at any approximate solution.

math.AP↗
arXiv2018

Reducibility of first order linear operators on tori via Moser's theorem

In this paper we prove reducibility of classes of linear first order operators on tori by applying a generalization of Moser's theorem on straightening of vector fields on a torus. We consider vector fields which are a $C^\infty$ perturbations of a constant vector field, and prove that they are conjugated --by a $C^\infty$ torus diffeomorphism-- to a constant diophantine flow, provided that the perturbation is small in some given $H^{s_1}$ norm and that the initial frequency is in some Cantor-like set. Actually in the classical results of this type the regularity of the change of coordinates which straightens the perturbed vector field coincides with the class of regularity in which the perturbation is required to be small. This improvement is achieved thanks to ideas and techniques coming from the Nash-Moser theory.

math.AP↗