FPU models, in dimension one, are perturbations either of the linear model or of the Toda model; perturbations of the linear model include the usual β-model, perturbations of Toda include the usual α + β model. In this paper we explore and compare two families, or hierarchies, of FPU models, closer and closer to either the linear or the Toda model, by computing numerically, for each model, the maximal Lyapunov exponent χ. We study the asymptotics of χ for large N (the number of particles) and small ε (the specific energy E/N ), and find, for all models, asymptotic power laws χ ≃ Cε a , C and a depending on the model. The asymptotics turns out to be, in general, rather slow, and producing accurate results requires a great computational effort. We also revisit and extend the analytic computation of χ introduced by Casetti, Livi and Pettini, originally formulated for the β-model. With great evidence the theory extends successfully to all models of the linear hierarchy, but not to models close to Toda.

We consider the dispersive Degasperis-Procesi equation ut−uxxt−cuxxx+4cux−uuxxx−3uxuxx+4uux = 0 with c ∈ R \ {0}. In [12] the authors proved that this equation possesses infinitely many conserved quantities. We prove that there are infinitely many of such constants of motion which control the Sobolev norms and which are analytic in a neighborhood of the origin of the Sobolev space H s with s ≥ 2, both on R and T. By the analysis of these conserved quantities we deduce a result of global well-posedness for solutions with small initial data and we show that, on the circle, the formal Birkhoff normal form of the Degasperis-Procesi at any order is action-preserving. * This research was supported by PRIN 2015 "Variational methods, with applications to problems in mathematical physics and geometry".

We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider an order-r normalized approximation of NLKG (which corresponds to the NLS at order r = 1), and prove that when M = R d , d ≥ 2, small radiation solutions of the order-r normalized equation approximate solutions of the NLKG up to times of order O(c 2(r−1) ).

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.