The Fermi-Pasta-Ulam problem in the thermodynamic limit: scaling laws of the energy cascade

In the present contribution we justify and discuss the scaling laws characterizing the first phase of the energy transfer from large to small spatial scales in a chain of nonlinear oscillators (the so-called Fermi- Pasta-Ulam α-model). By means of qualitative estimates, we show that large scale initial excitations (long wavelength Fourier modes) produce injection of energy into smaller scales on times t > τ_c = ε^{−3/4} and up to a cutoff spatial scale ell_c = ε^{−1/4} , where ε is the energy per degree of freedom of the system.