ABSTRACT:

We consider a long chain of particles. Each one is subjected to a nonlinear restoring force and is coupled to its first neighbor through a linear spring (Klein Gordon lattice). We consider initial data with the energy concentrated in the first Fourier modes. We prove two normal form theorems showing that up to higher order corrections the dynamics is well described by the Nonlinear Schroedinger equation in the small dispersion limit, namely $$ idotpsi=-frac{1}{(N+1)^2}psi_{xx}+psi|psi|^2 . $$ We use the first theorem, together with an analysis of the NLS in the zero disperion limit, to prove that if the total energy ${E}$ is small enough, then the energy flows to modes with wavelength of order ${E}^{-1/4}$ (mesoscale), and not to modes with wavelength shorter than ${E^{-1/2}}$ (microscale). The result is uniform in the size of the lattice. We use the second theorem to discuss a situation with small specific energy. Assuming that the solution of the NLS fulfills a suitable estimate, we prove that if the specific energy $mathcal{E}:=E/(N+1)$ is small enough, then the energy flows to modes with wavelength of order $mathcal{E}^{-1/2}$, and not to modes with shorter wavelength. The proofs of the normal form theorems are based on techniques that could be interesting in themselves. In particular the solution of the homological equation is based on the use of Neumann series. The case of finite specific energy requires a first step which consists in putting the linearized system in a suitable exact normal form. In order to prove the results on the NLS in the zero dispersion limit we use a technique that, as far as we know, is new.