Dario Bambusi

Birkhoff normal form for partial differential equations with tame modulus

We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call of Tame Modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions and we use it to study some concrete equations (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrodinger equation on the

Time Quasi-Periodic Unbounded Perturbations of Schrödinger Operators and KAM Methods

d

Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators

-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular we get lower bounds on the existence time of solutions.

Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems

Abstract: We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ2 subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H0+εP(ωt) for ε small. Here H0 is the one-dimensional Schrödinger operator p2+V, V(x)∼|x|α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksins estimate of solutions of homological equations with non-constant coefficients.

Families of Periodic Solutions of Resonant PDEs

We prove existence and practical stability of breathers in chains of weakly coupled anharmonic oscillators. Precisely, for a large class of chains, we prove that there exist periodic solutions exponentially localized in space, with the property that, given an initial datum (with ) close to the phase space trajectory of the breather, then the corresponding solution remains at a distance from the above trajectory, up to times growing exponentially with the inverse of , being a parameter measuring the size of the interaction among the particles. This result is deduced from a general normal form theorem for abstract Hamiltonian systems in Banach spaces, which we think could be interesting in itself.

On long time stability in Hamiltonian perturbations of non-resonant linear PDEs

We develop canonical perturbation theory for a physically interesting class of infinite-dimensional systems. We prove stability up to exponentially large times for dynamical situations characterized by a finite number of frequencies. An application to two model problems is also made. For an arbitrarily large FPU-like system with alternate light and heavy masses we prove that the exchange of energy between the optical and the acoustical modes is frozen up to exponentially large times, provided the total energy is small enough. For an infinite chain of weakly coupled rotators we prove exponential stability for two kinds of initial data: (a) states with a finite number of excited rotators, and (b) states with the left part of the chain uniformly excited and the right part at rest.

On dispersion of small energy solutions of the nonlinear Klein Gordon equation with a potential

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.

Korteweg–de Vries equation and energy sharing in Fermi–Pasta–Ulam

We consider small Hamiltonian perturbations of a system of infinitely many harmonic oscillators. We assume that the frequencies j, j 1, fulfil j ~ jd with d 1, and a suitable non-resonance condition of Diophantine-type. We prove a Nekhoroshev-type theorem for solutions corresponding to initial data such that the energy on the jth linear oscillator is bounded by Kexp(-ja) with a given a > 1 and positive , K. We show, precisely that up to times of order exp|ln|1+b, where is the size of the perturbation and b a positive parameter, the solution remains close to a quasi-periodic motion. Closedness is measured in a weighted 2 norm with an exponential weight. For our Diophantine-type condition we show that if the frequencies depend on a real parameter, and a suitable non-degeneracy condition is fulfilled, it is satisfied for almost all values of the parameter. Finally, we apply the general result to some nonlinear Klein-Gordon equations.

Normal forms, symmetry and linearization of dynamical systems

In this paper we study small amplitude solutions of nonlinear Klein Gordon equations with a potential. Under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small energy solutions are asymptotically free. In cases where the linear system has at most one bound state the result was already proved by Soffer and Weinstein: we obtain here a result valid in the case of an arbitrary number of possibly degenerate bound states. The proof is based on a combination of Birkhoff normal form techniques and dispersive estimates.

Normal form and exponential stability for some nonlinear string equations

We address the problem of equipartition in a long Fermi-Pasta-Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg-de Vries we use the latter equation to show that, corresponding to initial data a la Fermi, the time average of the energy on the kth mode decreases exponentially with kN. The result persists in the thermodynamic limit.