New connections between dynamical systems and Hamiltonian PDEs
NAPOLI, April 1- June 6, 2009
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Indam intensive period New connections between dynamical systems and Hamiltonian PDEs NAPOLI, April 1- June 6, 2009 |
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| Home | Speakers April-May | Speakers workshop | Timetable | Locations |
| Dario Bambusi
(Università di Milano) Dispersion of small amplitude solutions of the Klein Gordon equation in R3 |
Joint work with Scipio Cuccagna.
We study small amplitude solutions of the nonlinear Klein Gordon equation
in R3, where V is a bounded potential, m a parameter and β an analytic nonlinearity. We prove that, for generic values of the parameters (namely m and the Taylor coefficients of β), all small amplitude solution with finite energy give rise to dispersive solutions. More precisely the solution decouples into the sum of a part localized in space, which tends to zero as t → ∞, and a dispersive part which is asymptotically free.
The result is a generalization to the case of general potentials of previous results obtained by Soffer and Weinstein in the case of potentials with a single eigenvalue which is close to the continuous spectrum.
The proof is based on the combination of normal form techniques for Hamiltonian PDEs and dispersive Strichartz estimates.
| Vieri Benci
(Università di Pisa) Solitary waves, solitons and vortices |
Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. The first part of this talk is an introduction to the study of solitary waves relative to the Shroedinger equation, to the Nonlinear Wave Equation and to the Gauge Theories (GT). GT's consist of a class of field equations obtained by coupling in a suitable way the nonlinear wave equation with the Maxwell equations (in the Abelian case) or the Yang-Mills equation (in the SU(2) case). A particular type of solitary waves which might occur in GT's are the vortices. A three dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid.
In the last part of the talk, we will present some new results realative to the dynamics of solitons when an external potential is present.
| F. Calogero
(Dipartimento di Fisica, Università di Roma "La Sapienza"
and Istituto Nazionale di Fisica Nucleare, Sezione di Roma) Isochronous dynamical systems and the arrow of time |
A vector-valued time-dependent function is called isochronous if all its components are periodic in time with the same fixed period T. A dynamical system is called isochronous if its generic solution is isochronous: periodic in all its degrees of freedom with a fixed period T independent of the initial data. It will be shown how essentially any (autonomous) dynamical system can be extended or modified into another (also autonomous) dynamical systems which is isochronous with an (arbitrarily !) assigned period T, and which moreover behaves, over time periods very short with respect to T, essentially as the original (unmodified) system---up to a constant time rescaling. This can also be done for a large class of Hamiltonian systems (both the unmodified and the modified one), including the Hamiltonian describing the most general (classical, nonrelativistic) many-body problem (provided it is, overall, translation-invariant). Some implications of this fact for statistical mechanics and thermodynamics will be mentioned, and for the distinction among integrable and nonintegrable dynamical systems (all isochronous systems are integrable, in fact maximally superintegrable). It will also be shown how the most general Hamiltonian system can be embedded inside a superintegrable Hamiltonian system.
These findings have all been obtained together with F. Leyvraz: some of them are reported in my monograph entitled Isochronous systems (Oxford University Press, 2008), others are more recent.
| Cheng Chong-Qing Non-existence of KAM torus |
Given an integrable Hamiltonian h0 with n-degrees of freedom and a Diophantine frequency ω, then, arbitrarily close to h0 in the Cr topology with r < 2n, there exists an analytical Hamiltonian hε with no KAM torus of rotation vector ω.
| Luigi Chierchia The Kolmogorov's set for the planetary n-body problem |
We present some new results on the structure of the Kolmogorov's set for the planetary, spatial (1+N)-body problem. In particular, measure estimates for the Kolmogorov's set of (3N-1)-dimensional tori, as well as related tori of dimension (3N-2) in a fully reduced system, are discussed. The results are based on the PhD thesis by Gabriella Pinzari.
| James Colliander Recent Progress on NLS-type Equations |
This talk will survey some recent results on nonlinear Schrödinger and closely related equations. In particular, I will report on:
| Walter Craig Normal forms for free surface water waves |
It has been known since Zakarov (1968) that the problem of water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point (in the sense of dynamical systems). Taking the analogy with Hamiltonian mechanics a step further, we develop a normal form for the Hamltonian in a neighborhood of u=0 in an appropriate Sobolev space setting, eliminating all but essential nonlinear terms in the evolution equations. This development will give rise to several technical advances: a long time existence theorem for the water waves equations, a justification of the NLS modulational scaling regime, and a new view of the KdV long wave scaling limit.
This is work in collaboration with B. Alvarez-Samaniego and C. Sulem.
| Scipio Cuccagna On asymptotic stability of ground states and on orbital instability of standing waves with nodes of the Nonlinear Schroedinger Equation |
We show how under generic conditions, and for short range nonlinearities, a coupling assumption (the Nonlinear Fermi Golden Rule), allows to conclude that orbitally stble ground states are asymptotically stable (in the sense that the difference between ground state and solution scatters like a solution of the linear Schroedinger equation). By a similar argument, and using a similar assumption, one can also show that standing waves with nodes of the NLS cannot be orbitally stable, even in the case when they are linearly stable (in the traditional sense).
| Norman Dancer Finite Morse Index Solutions of Nonlinear Elliptic Equations |
We show how the study of stable and finite Morse index solutions on Rn can be used to stable or not too unstable solutions of nonlinear elliptic equations on bounded equations when the diffusion is small. In particular, we can frequently completely determine the stable solutions.
| R. de la Llave Periodic and almost periodic whiskered solutions in lattice systems |
We consider lattices of coupled systems. We assume that each system contains a hyperbolic periodic orbit and a positive measure set of KAM tori. We also assume that the interactions decay fast enough, that the system satisfy some non-degeneracy conditions and that it is differentiable enough.
Then, we show that there are quasi-periodic and almost-periodic solutions. These can be described as solutions in which there are clumps of systems close to moving in a KAM torus, but the rest of systems are near hyperbolic orbits. These solutions are at the border of propagation and can be used to construct even more complicated solutions.
This is joint work with E. Fontich, P. Martin and Y. Sire.
| Dmitry Dolgopyat Dynamics of bouncing balls |
We consider a ball bouncing of infinitely heavy periodically moving plate in a presence of a potential force. Assuming that the potential equals to the power of ball's height we present conditions guaranteeing recurrence in the sense that the total energy of almost every orbit does not go to infinity.
| Jiansheng Geng Quasi-periodic solutions for two dimensional cubic Schrodinger equation without the outer parameter |
In this talk, we give an infinite dimensional KAM theorem. Applying it to the two dimensional nonlinear Schröedinger equation iut-Δ u +¦u¦2u=0, t ∈ R, x ∈ T2 with periodic boundary conditions, we prove that the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. This is a joint work with X.Xu and J.You.
| Benoit Grébert
(Université de Nantes) Normal forms and geometric numerical integration of semi linear Hamiltonian PDEs |
We consider a class Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part.
First we show how standard numerical approximation methods cannot avoid numerical resonances issues except if the linear part is truncated in the high frequencies.
We then consider abstract splitting methods where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size and under a condition of zero momentum on the nonlinearity. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for numerical schemes controlling the round-off error at each step to avoid possible high frequency energy drift. We apply these results to nonlinear Schrödinger equation. (Joint work with E. Faou and E. Paturel)
| V. Y. Kaloshin Examples of Arnold diffusion |
We construct a nearly integrable Hamiltonian system of 3 degrees of freedom which has an orbit dense in a set of maximal Hausdorff dimension. This is a joint work with M. Levi and M. Saprykina.
| A. Kiselev Nonlocal maximum principles for active scalar equations |
I will discuss nonlocal maximum principles for active scalar equations. This is a relatively new technique that gives proofs of global regularity for the critical surface quasi-geostrophic equation, critical Burgers equation, and some other related models in fluid dynamics. This talk is based on works joint with Fedya Nazarov, Roman Shterenberg and Sasha Volberg.
| Michela Procesi
(Università Napoli Federico II) Nonlinear wave and Schrödinger equations on compact Lie groups and homogeneous spaces |
We prove the existence of Cantor families of small amplitude periodic solutions for wave and Schrodinger equations on compact Lie groups and homogeneous spaces with differentiable nonlinearities. The highly degenerate eigenvalues of the Laplace Beltrami operator give rise to huge clusters of "small divisors"; to solve this problem we apply a Lyapunov-Schmidt decomposition and an abstract Nash-Moser implicit function theorem. We provide an algebraic framework to prove the key tame estimates for the inverse linearized operators along Banach scales of Sobolev functions. A main ingredient are the properties of the eigenvalues and eigenfunctions of the Laplacian on Lie groups.
| Susanna Terracini
(Università Milano Bicocca) Symbolic dynamics and parabolic trajectories in the n-centre and n-body problems |
We consider both the planar N-centre problem and the classical three-dimensional n-body problem with Coulombic potentials. We discuss the existence of noncollision periodic and parabolic trajectories featuring chaotic behaviour and the scattering problem. The proofs are based on variational methods.
| C. Viterbo Symplectic Homogenization |
Let H(q,p) be a Hamiltonian on T*Tn. We show that the sequence Hk(q,p) = H(kq,p) converges for some adequate topology defined by the author, to K(p). This is extended to the case where only some of the variables are homogenized, that is we study the sequence H(kx,y,q,p) where the limit is of the type K(y,q,p) and thus yields an $quot;effective Hamiltonian".
We shall give some examples and applications of this construction including to some concrete physical problems.
| S. Volberg (Michigan
State University) Jacobi matrices generated by polynomial dynamics, Toda lattice, and estimates of Hilbert transform |
This talk is based on a joint work with P. Yuditskii and Franz Peherstorfer, in it we are going to show a Lipschitz property of Jacobi matrices built by orthogonalizing polynomials with respect to measures in the orbit of classical Perron-Frobenius-Ruelle operator associated to hyperbolic polynomial dynamics. This Lipschitz estimate will not depend on the dimension of the Jacobi matrix. In the second part of the talk for all polynomials with sufficiently big hyperbolicity we can prove that Lipschitz estimate becomes exponentially better when dimension of Jacobi matrix grows. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. We suggest the scheme how to approach Bellissard's problem for all hyperbolic dynamics by uniting the methods of the first and the second part. On the other hand, the nearness of Jacobi matrices under consideration in operator norm implies a certain nearness of their canonical spectral measures. One can notice that this last claim just gives us the classical commutative Perron-Frobenius-Ruelle theorem (it is concerned exactly with the nearness of such measures). In particular, in many situations we can see that the classical Perron-Frobenius-Ruelle theorem is a corollary of a certain non-commutative observation concerning the quantitative nearness of pertinent Jacobi matrices in operator norm.
| Wei-Min Wang Resonant perturbations of Hamiltonian systems in infinite dimensions |
We develop a resonant perturbation theory for Hamiltonian PDE's. In this talk, we focus on the linear theory, where we prove eigenfunction localization for the 2D periodic Schrödinger operator on the square torus, solving an old problem in spectral theory. Time permitting, we will also touch upon the nonlinear theory.
| Jiangong You Almost Reducibility and Non-perturbative Reducibility of Quasi-Periodic Linear Systems |
We will talk about the almost reducibility of the quasi-periodic linear differential equation in sl(2, R) with two frequencies (1, α) assuming that the coefficient is analytic and close to a constant. In case that α is Diophantine we get a non-perturbative reducibility result. When applying to the quasi-periodic Schrödinger operators with two frequencies and small potentials, we proved that (i) the Liapunov exponents are always vanish in the spectrum; (ii) the spectrum is absolutely continuous for a.e. phase; (iii) the spectrum is purely absolutely continuous for a.e. α. (joint work with Xuanji Hou)
