Simone Paleari

Families of Periodic Solutions of Resonant PDEs

Summary. We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation utt-uxx± u3=0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the nth family converge to 2π/n when the amplitude tends to zero.

Normal form and exponential stability for some nonlinear string equations

Abstract. We study small amplitude solutions of the nonlinear wave equation¶¶

Metastability and dispersive shock waves in the Fermi-Pasta-Ulam system

u_{tt}-u_{xx}=\psi(u) \qquad u(0,t)=0=u(\pi,t)

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

(0.1)¶¶ with an analytic nonlinearity of the type

Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices

\psi(u)=\pm u^{2k-1} + {\cal O}(u^{2k})

An Extensive Adiabatic Invariant for the Klein–Gordon Model in the Thermodynamic Limit

,

Breathers and Q-Breathers: Two Sides of the Same Coin

k\geq2

An extensive resonant normal form for an arbitrary large Klein–Gordon model

. For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the “simplified system”, namely a Hamiltonian system that we will prove to approximate (0.1) up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

We show the relevance of the dispersive analogue of the shock waves in the FPU dynamics. In particular we give strict numerical evidence that metastable states emerging from low frequency initial excitations are indeed constituted by dispersive shock waves travelling through the chain. Relevant characteristics of the metastable states, such as their frequency extension and their time scale of formation, are correctly obtained within this framework, using the underlying continuum model, the KdV equation.

Generalized Discrete Nonlinear Schrödinger as a Normal Form at the Thermodynamic Limit for the Klein–Gordon Chain

Small-amplitude weakly coupled oscillators of the Klein–Gordon lattices are approximated by equations of the discrete nonlinear Schrodinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss the applications of the discrete nonlinear Schrodinger equation in the context of existence and stability of breathers of the Klein–Gordon lattice.