Thursday, June 13 |
07:30 - 09:00 |
Breakfast (Restaurant at your assigned hotel) |
09:15 - 10:00 |
Marcel Guardia: On the breakdown of small amplitude breathers for the reversible Klein-Gordon equation ↓ Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied. This is a joint work with O. Gomide, T. Seara and C. Zeng. (Conference Room San Felipe) |
10:00 - 10:45 |
Renato Calleja: Construction of quasi-periodic response solutions for forced systems with strong damping ↓ I will present a method for constructing quasi-periodic response solutions (i.e. quasi-periodic solutions with the same frequency as the forcing) for over-damped systems. Our method applies to non-linear wave equations subject to very strong damping and quasi-periodic external forcing and to the varactor equation in electronic engineering. The strong damping leads to very few small divisors which allows to prove the existence by using a contraction mapping argument requiring very weak non-resonance conditions on the frequency. This is joint work with A. Celletti, L. Corsi, and R. de la Llave. (Conference Room San Felipe) |
10:45 - 11:30 |
Coffee Break (Conference Room San Felipe) |
11:30 - 12:15 |
Jessica Elisa Massetti: Almost-periodic tori for the nonlinear Schrödinger equation ↓ The problem of persistence of invariant tori in infinite dimension is a challenging problem in the study of PDEs. There is a rather well established literature on the persistence of n-dimensional invariant tori carrying a quasi-periodic Diophantine flow (for one-dimensional system) but very few on the persistence of infinite-dimensional ones.
Inspired by the classical "twisted conjugacy theorem" of M. Herman for perturbations of degenerate Hamiltonians possessing a Diophantine invariant torus, we intend to present a compact and unified frame in which recover the results of Bourgain and Pöschel on the existence of almost-periodic solutions for the Nonlinear Schrödinger equation. We shall discuss the main advantages of our approach as well as new perspectives. This is a joint work with L. Biasco and M. Procesi. (Conference Room San Felipe) |
12:15 - 13:00 |
Raffaele Carlone: Microscopic derivation of ionization models ↓ The so called "time-dependent point interactions” are solvable models with singular potentials whose “strength” changes in time. They are typically useful to investigate the ionization of a bound state by the action of a time-dependent localized interaction. We prove that time-dependent point interactions can be derived from the microscopic dynamics of a quantum particle – a Frohlich polaron – interacting with a bosonic scalar quantum field – the lattice field –, in suitable field’s configurations. (Conference Room San Felipe) |
13:30 - 15:00 |
Lunch (Restaurant Hotel Hacienda Los Laureles) |
16:00 - 16:30 |
Coffee Break (Conference Room San Felipe) |
16:30 - 17:15 |
Zaher Hani: On the kinetic description of the long-time behavior of dispersive PDE. ↓ Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation". This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah). (Conference Room San Felipe) |
17:15 - 18:00 |
Dario Valdebenito: Partially localized solutions of elliptic equations in RN+1. ↓ Using spatial dynamics and results from the KAM theory, we develop a framework to find solutions of semilinear elliptic equations on the entire space which are quasiperiodic in one variable, decaying in the other variables. These results apply to a wide class of nonhomogeneous (and some homogeneous) problems. A careful application of Birkhoff normal form allows us to obtain a nondegeneracy condition for KAM that works even for some purely quadratic nonlinearities. (Conference Room San Felipe) |
18:00 - 18:15 |
Victor Arnaiz Solórzano: Renormalization of semiclassical KAM systems. ↓ I will present a renormalization problem about the
convergence of normal forms in the presence of counterterms for some
semiclassical systems that are close to be completely integrable. I
will also explain some applications of these normal-form constructions
to the study of quantum limits and semiclassical measures. (Conference Room San Felipe) |
18:15 - 18:30 |
Rosa Vargas-Magana: Dispersive Shock waves in Hamiltonian Bidirectional Whitham systems. ↓ The research is concerned with the study of dispersive shock waves (DSW), also termed undular bores, on the surface of fluids, DSWs arise due to the dispersive resolution of step, or near-step, initial conditions and are a common waveform in nature. They consist of a modulated dispersive wavetrain linking distinct levels ahead and behind it. In this talk, we will present some preliminary results related to the accuracy of DSW solutions of the Hamiltonian Bidirectional Whitham compared with fully nonlinear results. All research on DSWs to date has been based on weakly nonlinear approximations of full systems of equations, for instance, the water wave equations. The research of this project is the first attempt for the study of fully nonlinear DSWs. We will use the Whitham-Boussinesq (W-B) model that I introduced in my doctoral thesis as the bridge between the water wave equations (the free-surface Euler equations) and in which the "Shock fitting method” can be applied. This "Shock fitting method", derived by G. El and his collaborators, is a general method for determining the leading and trailing edges of DSWs, based on the dispersion relation for the governing nonlinear, dispersive wave equation. (Conference Room San Felipe) |
19:00 - 21:00 |
Dinner (Restaurant Hotel Hacienda Los Laureles) |