Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

Spatial Form of a Hamiltonian Dysthe Equation for Deep-Water Gravity Waves

An overview of a Hamiltonian framework for the description of nonlinear modulation of surface water waves is presented. The main result is the derivation of a Hamiltonian version of Dysthe’s equation for two-dimensional gravity waves on deep water. The reduced problem is obtained via a Birkhoff normal form transformation which not only helps eliminate all non-resonant cubic terms but also yields a non-perturbative procedure for surface reconstruction. The free surface is reconstructed from the wave envelope by solving an inviscid Burgers’ equation with an initial condition given by the modulational Ansatz. Particular attention is paid to the spatial form of this model, which is simulated numerically and tested against laboratory experiments on periodic groups and short-wave packets. Satisfactory agreement is found in all these cases.

Chaotic-Like Transfers of Energy in Hamiltonian PDEs

We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $${\mathbb {T}}^2$$ T 2 and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered “chaotic-like” since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.

Long-time stability of the quantum hydrodynamic system on irrational tori

<abstract><p>We consider the quantum hydrodynamic system on a $ d $-dimensional irrational torus with $ d = 2, 3 $. We discuss the behaviour, over a "non-trivial" time interval, of the $ H^s $-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving form $ \varepsilon $-small initial conditions, remain bounded in $ H^s $ for a time scale of order $ O(\varepsilon^{-1-1/(d-1)+}) $, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore implement a Birkhoff normal form procedure involving small divisors arising form three waves interactions. The main difficulty is to control the loss of derivatives coming from the exchange of energy between high Fourier modes. This is due to the irrationality of the torus which prevents to have "good separation'' properties of the eigenvalues of the linearized operator at zero. The main steps of the proof are: (i) to prove precise lower bounds on small divisors; (ii) to construct a modified energy by means of a suitable high/low frequencies analysis, which gives an a priori estimate on the solutions.</p></abstract>

LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS

We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indices are stable up to arbitrary long times with respect to the size of the initial data. To prove this result, we use a normal form transformation decomposing the dynamics into low and high frequencies with weak interactions. While the low part of the dynamics can be put under classical Birkhoff normal form, the high modes evolve according to a time-dependent linear Hamiltonian system. We then control the global dynamics by using polynomial growth estimates for high modes and the preservation of Sobolev norms for the low modes. Our general strategy applies to any semilinear Hamiltonian Partial Differential Equations (PDEs) whose linear frequencies satisfy a very general nonresonance condition. The (NLW) equation on $\mathbb{T}^{d}$ is a good example since the standard Birkhoff normal form applies only when $d=1$ while our strategy applies in any dimension.

Near-Integrability of Periodic Klein-Gordon Lattices

In this paper, we study the Klein-Gordon (KG) lattice with periodic boundary conditions. It is an N degrees of freedom Hamiltonian system with linear inter-site forces and nonlinear on-site potential, which here is taken to be of the ϕ 4 form. First, we prove that the system in consideration is non-integrable in Liouville sense. The proof is based on the Morales-Ramis-Simó theory. Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is integrable. To do this we use the results of B. Rink on FPU chains. If N is odd this integrable normal form turns out to be KAM nondegenerate Hamiltonian. This implies that almost all low-energetic motions of the periodic KG lattice are quasi-periodic. We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, nondegenerate normal forth order form. Then, the KAM theorem applies as above.