Pietro Baldi: Resonant and non-resonant dynamics of the Kirchhoff equation: longer lifespan and chaotic behaviour

Abstract:

We consider the Kirchhoff equation on \(\mathbb{T}^𝑑\) (a Hamiltonian PDE with cubic quasi-linear nonlinearity) and its Cauchy problem with small initial data. After two steps of a quasi-linear normal form procedure,and taking sums over Fourier spheres, we obtain an effective system for the dynamics of the problem, with nonzero, but harmless, resonant cubic terms,resonant quintic terms giving nontrivial contributions to the energy estimates, and higher order remainders. We use the effective system to study non-resonance (𝑖) and resonance (𝑖𝑖) situations.

(𝑖) We introduce some ad hoc non-resonance conditions on the initial data of the Cauchy problem and prove a longer lifespan of the corresponding solutions. The mechanism at the base of this improvement is an averaging effect, which reduces the growth rate of the superactions of the effective equations.

(𝑖𝑖) We construct special solutions, Fourier supported on four spheres of \(\mathbb{Z}^d \) forming two “resonant triplets”. The main part of the dynamics is conjugated to a classical nearly integrable Hamiltonian system of two pendulums. From the persistence of a hyperbolic periodic orbit of that system and the transversal splitting of its invariant manifolds, using Gronwall estimates, we obtain solutions for the original Kirchhoff equation that present an interesting chaotic behaviour.

Joint works with E. Haus, M. Guardia, F. Giuliani.