Metastability phenomena in two-dimensional rectangular lattices with nearest-neighbour interaction

Abstract

We study analytically the dynamics of two-dimensional rectangular lattices with periodic boundary conditions. We consider anisotropic initial data supported on one low-frequency Fourier mode. We show that, in the continuous approximation, the resonant normal form of the system is given by integrable PDEs. We exploit the normal form in order to prove the existence of metastability phenomena for the lattices. More precisely, we show that the energy spectrum of the normal modes attains a distribution in which the energy is shared among a packet of low-frequencies modes; such distribution remains unchanged up to the time-scale of validity of the continuous approximation.

Hamiltonian field theory close to the wave equation: from Fermi-Pasta-Ulam to water waves

“…Although we decided to focus on one-dimensional systems, it is worth mentioning that the techniques presented here can be generalised to study problems in higher space dimension. In this case one can predict, for example, energy localisation for certain class of anisotropic rectangular lattices [18].…”

KP-II approximation for a scalar FPU system on a 2D square lattice

“…where A is a suitable solution to the KP-II equation ( 12) below for which derivatives of A and ∂ −1 X ∂ Y A are controlled in Sobolev spaces of sufficiently high regularity. The compatibility condition (10) rewritten in variables (X, Y, T ) is satisfied at the order of O(ε 4 ). Substitution of ( 11) into (9) rewritten in variables (X, Y, T ) results in the following KP-II equation at the order of O(ε 6 ):…”

“…In order to establish such an approximation theorem, we have to estimate the residual terms first, i.e., we have to control the terms which do not cancel after inserting the approximation (11) into system ( 9) and (10). In general, these estimates can be obtained by expanding the multipliers in Fourier space and by assuming a certain regularity of the solutions of the KP-II equation (12).…”

“…Two rigorous results were obtained only very recently. The KP-II equation was justified in the periodic domain among other integrable normal forms [10]. By using a more general setting of the vector FPU systems with particles moving in the (x, y) directions, the KP-II equation was justified in the unbounded 2D square lattice for the propagation along the axes (as well as for the diagonal propagation in the (x, y)-plane under additional constraints on the parameters of the lattice) [14].…”

Arnold diffusion in Hamiltonian systems on infinite lattices

“…On energy localization, there are several papers that apply KAM Theory techniques to prove the existence of invariant tori [30,16,59,32,33,58] which have strong decay in space and therefore have localized energy. There are also several results providing time estimates for energy localization (see for instance [57,56,2,3,31,19]).…”