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.
Highlights
We study analytically the dynamics of two-dimensional rectangular lattices with periodic boundary conditions
Despite the authors believed that the approach to such an equilibrium would have occurred in a short time-scale, the outcoming Fourier spectrum was far from being flat and they observed two features of the dynamics that were in contrast with their expectations: the lack of thermalization displayed by the energy spectrum and the recurrent behaviour of the dynamics
We mention the papers [BP06] and [Bam08], in which the authors used the techniques of canonical perturbation theory for PDEs in order to show that the FPU α model can be rigorously described by a system of two uncoupled Korteweg-de Vries (KdV) equations, which are obtained as a resonant normal form of the continuous approximation of the FPU model; this result allowed to deduce a rigorous result about the energy sharing among the Fourier modes, up to the time-scales of validity of the approximation
Summary
In Theorem 2.2 we do not mention the existence of a sequence of almost-periodic functions approximating the specific energies of the modes. This is related to the construction of actionangle/Birkhoff coordinates for the KP equation, which is an open problem in the theory of integrable PDEs. 2.2. (1D NLS) the very weakly transverse regime, where the effective dynamics is described by a cubic onedimensional nonlinear Schrödinger (NLS) equation This corresponds to taking μ ≪ 1 and 1 < σ < 7. Let d ≤ 6, define ZdN1,...,Nd as in (37) and consider the d-dimensional NLKG lattice (43). These will be discussed in Remark 4.7 and Remark 4.12 respectively
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