Arnold diffusion in Hamiltonian systems on infinite lattices

Abstract

We consider a system of infinitely many penduli on an m-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying.The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasiperiodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded Z m -sequences and in spaces of decaying Z m -sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.

Sobolev norms explosion for the cubic NLS on irrational tori

“…Nowadays Bourgain's question [4] remains wide open, although results of existence of unbounded solutions to NLS equations have been provided by Hani-Pausader-Visciglia-Tzvetkov [17] on R×T 2 and by Hani [16] for a cubic almost-polynomial NLS on T 2 . For other models rather than NLS, we refer to [8], for an existence result of unbounded solutions of the Szegö equation on T, and [9], for a construction of unbounded transfers of energy orbits in an infinite pendulum lattice by means of Arnold diffusion techniques.…”