Georgia Institute of Technology College of Sciences

In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behavior of  ''typical'' solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are indeed typical in the integrable case. Up to now almost all results in the literature deal with very regular solutions for model PDEs with external parameters giving a large modulation. In this talk I shall discuss a new result constructing Gevrey solutions for models with a weak parameter modulation. 

This is a joint work with G.Gentile and M.Procesi.

 We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.

We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators.

This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.

 I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.

 We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the  two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast  variable.

The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408