Korteweg–de Vries equation and energy sharing in Fermi–Pasta–Ulam

We show that, for long--wavelength initial conditions, the dynamics of the Fermi-Pasta-Ulam (FPU) $\alpha$-chain is described, for short times, by two uncoupled Korteweg-de Vries (KdV) equations representing the resonant Hamiltonian normal form of the system. We show that if the number of degrees of freedom of the original FPU system is large enough, the effect of the dispersive term in the KdV equations is small. As a consequence the very beginning of the energy transfer from large to small spatial scales -- the cascade -- is ruled by a pair of uncoupled inviscid Hopf-Burgers (HB) equations. The energy cascade taking place in the system is then quantitatively characterized by arguments of dimensional-singularity analysis. The form of the relaxed energy spectrum of the Fourier modes is predicted, showing that it displays an exponentially decreasing tail at high wave numbers. In particular, it is pointed out that the form of such a spectrum survives the so-called thermodynamic limit, i.e. the limit of infinite energy proportional to the infinite number of degrees of freedom.