Dario Bambusi, Universita di Milano
An averaging theorem for FPU in the thermodynamic limit
Coauthors: A.~Maiocchi, A.~Carati

Consider an FPU chain composed of $N\gg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $\beta^{-1}$. Given a fixed $K<N$, we construct $K$ packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order $\beta^{1-a}$, $a>0$) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order $\beta$. The restrictions on the shape of the packets are very mild. All estimates are uniform in the number $N$ of particles and thus hold in the thermodynamic limit $N\to\infty$, $\beta>0$.

This is a joint work with Andrea Carati and Alberto Maiocchi.

Claude Bardos, Universite Paris 7
The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in Fluid Mechanics and Semi-classical Limits (joint work with Nicolas Besse) (slides)

Thiscontributionconcernsaone-dimensionalversionoftheVlasov equation dubbed the Vlasov-Dirac-Benney equation (in short V-D-B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with the so called Benney equation leads to new stability results. Eventually the V-D-B appears to be at the “cross road” of several problems of mathematical physics which have as far as stability is concerned very similar properties.

Massimiliano Berti, Universita di Napoli
KAM for quasi-linear KdV (slides)

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear autonomous Hamiltonian and reversible perturbations of KdV. We underline that this equation do not depend on parameters. The role of parameters is played by the initial conditions.

The discussion will center on wave propagation in unbounded domains. The motion is presumed to be long-crested, so propagating mainly in one direction. Variations in the long-crested direction are also permitted, however, so a three-dimensional model is needed. The wave motion is not assumed to evanesce in all directions as the spatial variables becomes unbounded. An interesting feature of the analysis is that the lateral boundaries have dynamics of their own that are not imposed at the outset.

Thanks to Walter Criag, random matrix theory has found an application in inflationary cosmology: it can be used to study the effects of noise on the parametric resonance instability at the end of the inflationary phase of early universe cosmology. I will mention some possible future applications.

Constantine Dafermos, Brown University
Long Time Behaviour of Periodic Solutions to Scalar Conservation Laws in Several Space Dimensions (slides)

It will be shown that spatially periodic solutions of scalar conservation laws in several space dimensions decay to their time - invariant mean, as time tends to infinity, provided that the flux function is linearly nondegenerate just in the vicinity of the mean, in a countable family of directions, depending on the period.

Bernard Deconinck,Washington University
High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs (slides)

Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. The entire theory with the exception of a Krein signature calculation can be phrazed in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian PDE is changed. Two important cases are worked out and different examples are presented.

Carlo Fazioli, Drexel University
Overlapping Patches for Dynamic Problems (slides)

The method of overlapping patches has been employed with considerable success to static problems of potential theory and scattering. It is desirable then, to consider developing an overlapping patches method for free surface problems. We discuss the benefits of such a method. We introduce a framework for covering a moving surface with overlapping patches and evolving them in time, including some preliminary proofs-of-concept. We conclude by considering a physical problem of vortex sheet motion, and present some new analytical results needed to obtain accurate error bounds.

Niky Kamran, McGill University
A Singular initial-boundary value problem for non-linear wave equations and holography in asymptotically anti-de Sitter manifolds

We will present a well-posedness result for a singular initial-boundary value problem for non-linear wave equations in asymptotically anti-de Sitter manifolds. Time permitting, the case of asymptotically anti-de Sitter Einstein metrics with prescribed conformal boundary will be discussed. This is joint work with Alberto Enciso (ICMAT, Madrid, Spain).

Sergei Kuksin, Universite Paris 7
Weakly non-linear completely resonant hamiltonian PDEs and the problem of weak turbulence (slides)

I will discuss long-time behaviour of small oscillations in a non-linear Shroedinger equation on a torus, perturbed by a random force and linear dissipation. The equation is scaled in such a way that its solutions are small, but their limiting dynamics is non-trivial. The limiting behaviour turns out to be described by another damped/driven Hamiltonian PDE, where the new Hamiltonian is constructed out of the resonant terms of the original one.Next I will discuss behaviour of the new system under the limit "space-period goes to infinity". Using heuristic approximation, commonly used in the weak turbulence, I will derive for the second limit a KZ type kinetic equation which leads to KZ energy spectra. This is a joint work with Alberto Maiocchi.

The Zakharov-Craig-Sulem formulation of the (irrotational) water waves equations has been extensively used to study theoretical and practical aspects of water waves. In this joint work with Angel Castro we propose an extension of this formulation in presence of vorticity. We prove local well-posedness and stability in the shallow water limit of this new formulation, and show how to derive and justify shallow water models in presence of vorticity.

In this talk we describe a class of Integral Equations to compute Dirichlet-Neumann operators for the Helmholtz equation on periodic domains inspired by the recent work of Fokas and collaborators on novel solution formulas for boundary value problems. These Integral Equations have a number of advantages over standard alternatives including: (i.) ease of implementation (high-order spectral accuracy is realized without sophisticated quadrature rules), (ii.) seamless enforcement of the quasiperiodic boundary conditions (no periodization of the fundamental solution, e.g. via Ewald summation, is required), and (iii.) reduced regularity requirements on the interface proles (derivatives of the deformations do not appear explicitly in the formulation). We show how these can be efficiently discretized and utilized in the simulation of scattering of linear acoustic waves by families of periodic layered media which arise in geoscience applications.

We will review results, obtained with C. Klein, F. Linares and D. Pilod on weakly dispersive perturbations of the Burgers equation which can be seen as a tool model for the general question of weak dispersive perturbations of quasilinear hyperbolic systems. We will first show how dispersion enlarge the space of the resolution of the local Cauchy problem, and then address global issues. Numerical simulations will lead to conjectures on the nature of the blow-up in the $L^2$ supercritical or energy supercritical cases and on the long time asymptotics of solutions in the $L^2$ subcritical case.

Harvey Segur, University of Colorado
Three-wave resonant interactions

A resonant interaction of three wavetrains is the simplest nonlinear and non-trivial interaction of dispersive waves, propagating in a medium without dissipation. Coppi, Rosenbluth & Sudan (1969) noted that this kind of interaction takes two forms: the "decay case", where energy is conserved, and the "explosive case", in which almost all solutions blow up in finite time. Either way, the partial differential equations that describe these processes were shown to be completely integrable by Zakharov & Manakov (1973), and solutions of the problem for spatially localized wave packets were given by Zakharov & Manakov (1976), Kaup (1976) and others. Numerical simulations of the process usually impose periodic boundary conditions, and the known methods of analytical solution fail with these boundary conditions.
We present an alternative way to study this problem, in terms of convergent Laurent series (in "time"), which contain five, real-valued functions (in "space"). These functions must obey some differentiability constraints, but are otherwise arbitrary - they can be periodic, or almost periodic, or localized in space. A general solution of the problem would involve six such functions, so our current work stops short of a general solution. For simplicity, we work in one spatial dimension, and we analyze only the explosive case.

Peter J. Sternberg, Indiana University
Kinematic vortices in a thin film driven by an applied current (slides)

Using a Ginzburg-Landau model, we study the vortex behavior of a rectangular thin film superconductor subjected to an applied current fed into a portion of the sides and an applied magnetic field directed orthogonal to the film. Through a center manifold reduction we develop a rigorous bifurcation theory for the appearance of periodic solutions in certain parameter regimes near the normal state. The leading order dynamics, based on the behavior of the first eigenfunction to a PT-symmetric operator taking the form of a purely imaginary perturbation of the magnetic Schrodinger operator, yield in particular a motion law for kinematic vortices moving up and down the center line of the sample. We also present computations that reveal the co-existence and periodic evolution of kinematic and magnetic vortices.

This is joint work with Jacob Rubinstein and Lydia Peres Hari

Olga A. Trichtchenko, University of Washington
Stability of near-resonant gravity-capillary waves (slides)

I will present results on the computation and stability of periodic surface gravity-capillary waves that are in a near-resonant regime. In the zero amplitude limit, the parameters defining these solutions almost satisfy the resonance condition that leads to Wilton ripples. This manifests itself as a small divisor problem in the Stokes expansion for these solutions. I will compute such solutions and investigate their stability using Hill’s method.

Konstantina Trivisa, University of Maryland
On a nonlinear model for tumor growth: Global in time weak solutions (slides)

We investigate the dynamics of a class of tumor growth models known as mixed models. The key characteristic of these type of tumor growth models is that the different populations of cells are continuously present everywhere in the tumor at all times. In this work we focus on the evolution of tumor growth in the presence of proliferating, quiescent and dead cells as well as a nutrient.
The system is given by a multi-phase flow model and the tumor is described as a growing continuum $\Omega$ with boundary $\partial \Omega$ both of which evolve in time. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion and viscosity in the weak formulation.

This is joint work with D. Donatelli.

Gene Wayne, Boston University
Justification of the nonlinear Schödinger equation for two-dimensional gravity driven water waves (slides)

In 1968 V.E. Zakharov derived the Nonlinear Schödinger equation for the 2D water wave problem in the absence of surface tension in order to describe slow temporal and spatial modulations of a spatially and temporarily oscillating wave packet. I will describe a recent proof that the wave packets in the two-dimensional water wave problem in a canal of finite depth can be accurately approximated by solutions of the Nonlinear Schödinger equation. This is joint work with Guido Schneider and Wolf-Patrick Düll.


Vladimir Zakharov, University of Arizona
Hasselmann Equations Revisited (slides)

The Hasselmann kinetic equation is the main tool for modeling of wind-driven sea. We reexamined the derivation of this equation and determined the conditions of its applicability. We found more compact form for the kernel of this equation and studied its asymptotic in the following limit: two wave vectors are much shorter than two others. It makes possible to simplify essentially the description of the short-long wave interaction. In this case the Hasselmann equation can be reduced to the linear diffusion equation similar to the equation used in financial mathematics. We studied numerically isotopic powerlike solutions of the stationary Hasselmann equation and found the values of Kolmogorov constants. By special choice variables we transformed the Hasselmann equation to the special "multiquadruplete form" that is very convenient for numerical simulation.

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