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Series: CDSNS Colloquium

Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDEs.

Series: CDSNS Colloquium

We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifold. Furthermore, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Series: CDSNS Colloquium

The study of nonlocal transport in physically relevant systems requires
the formulation of mathematically well-posed and physically meaningful
nonlocal models in bounded spatial domains. The main problem faced by
nonlocal partial differential equations in general,
and fractional diffusion models in particular, resides in the treatment
of the boundaries. For example, the naive truncation of the
Riemann-Liouville fractional derivative in a bounded domain is in
general singular at the boundaries and, as a result, the incorporation
of generic, physically meaningful boundary conditions is not feasible.
In this presentation we discuss alternatives to address the problem of
boundaries in fractional diffusion models. Our main goal is to present
models that are both mathematically well posed
and physically meaningful. Following the formal construction of the
models we present finite-different methods to evaluate the proposed
non-local operators in bounded domains.

Series: CDSNS Colloquium

We present a general mechanism to establish the existence of diffusing
orbits in a large class of nearly integrable Hamiltonian systems. Our
approach relies on successive applications of the `outer dynamics'
along homoclinic orbits to a normally hyperbolic invariant manifold.
The information on the outer dynamics is encoded by a geometrically
defined map, referred to as the `scattering map'.
We find pseudo-orbits of the scattering map that keep moving in some
privileged direction.
Then we use the recurrence property of the `inner dynamics', restricted
to the normally hyperbolic invariant manifold, to return to those
pseudo-orbits.
Finally, we apply topological methods to show the existence of true
orbits that follow the successive applications of the two dynamics.
This method differs, in several crucial aspects, from earlier works.
Unlike the well known `two-dynamics' approach, the method relies
heavily on the outer dynamics alone.
There are virtually no assumptions on the inner dynamics, as its
invariant objects (e.g., primary and secondary tori, lower dimensional
hyperbolic tori and their
stable/unstable manifolds, Aubry-Mather sets) are not used at all.
The method applies to unperturbed Hamiltonians of arbitrary degrees of
freedom that are not necessarily convex.
In addition, this mechanism is easy to verify (analytically or
numerically) in concrete examples, as well as to establish diffusion in
generic systems.

Series: CDSNS Colloquium

We use invariant manifold
results on Banach spaces to conclude the existence of spectral
submanifolds (SSMs) in a class of nonlinear, externally forced beam
oscillations .
Reduction of the governing PDE to the SSM provides an exact
low-dimensional model which we compute explicitly. This model captures
the correct asymptotics of the full, infinite-dimensional
dynamics. Our approach is general enough to admit extensions to other
types of continuum vibrations. The model-reduction procedure we employ
also gives guidelines for a mathematically
self-consistent modeling of damping in PDEs describing structural vibrations.

Series: CDSNS Colloquium

The nonlinear Schroedinger
equation (NLS) can be derived as a formal approximating equation for the
evolution of wave packets in a wide array of nonlinear dispersive PDE’s
including the propagation of waves on the surface of an inviscid
fluid. In
this talk I will describe recent work that justifies this approximation
by exploiting analogies with the theory of normal forms for ordinary
differential equations.

Series: CDSNS Colloquium

It was shown by V.E. Zakharov that the equations for water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. This latter is joint work with C. Sulem.

Series: CDSNS Colloquium

We investigate deterministic superdiusion in nonuniformly hyperbolic system
models in terms of the convergence of rescaled distributions to the normal
distribution following the abnormal central limit theorem, which differs
from the usual requirement that the mean square displacement grow
asymptotically linearly in time. We obtain an explicit formula for the
superdiffusion constant in terms of the ne structure that originates in the
phase transitions as well as the geometry of the configuration domains of
the systems. Models that satisfy our main assumptions include chaotic
Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to
other nonuniformly hyperbolic systems with slow correlation decay rates of
order O(1/n)

Series: CDSNS Colloquium

Upon quantization, hyperbolic Hamiltonian systems generically exhibit
universal spectral properties effectively described by Random
Matrix Theory. Semiclassically this remarkable phenomenon can be
attributed to the existence of pairs of classical periodic orbits
with small action differences. So far, however, the scope of this
theory has, by and large, been restricted to single-particle systems.
I will discuss an extension of this program to hyperbolic
coupled map lattices with a large number of sites (i.e., particles).
The crucial ingredient is a two-dimensional symbolic dynamics which
allows an effective representation of periodic orbits and their
pairings. I will illustrate the theory with a specific model of
coupled cat maps, where such a symbolic dynamics can be constructed
explicitly.

Series: CDSNS Colloquium

A metric on the 2-torus T^2 is said to be "Liouville" if in some coordinate system it has the form ds^2 = (F(q_1) + G(q_2)) (dq_1^2 + dq_2^2). Let S^*T^2 be the unit cotangent bundle.A "folklore conjecture" states that if a metric is integrable (i.e. the union of invariant 2-dimensional tori form an open and dens set in S^*T^2) then it is Liouville: l will present a counterexample to this conjecture.Precisely I will show that there exists an analytic, non-separable, mechanical Hamiltonian H(p,q) which is integrable on an open subset U of the energy surface {H=1/2}. Moreover I will show that in {H=1/2}\U it is possible to find hyperbolic behavior, which in turn means that there is no analytic first integral on the whole energy surface.This is a work in progress with V. Kaloshin.