Analytical estimate of stochasticity thresholds in Fermi-Pasta-Ulam and ${\phi }^4$ models

We consider an infinitely extended Fermi-Pasta-Ulam model. We show that the slowly modulating amplitude of a narrow wave packet asymptotically satisfies the nonlinear Schrödinger equation (NLS) on the real axis. Using well known results from inverse scattering theory, we then show that there exists a threshold of the energy of the central normal mode of the packet, with the following properties. Below threshold the NLS equation presents a quasilinear regime with no solitons in the solution of the equation, and the wave packet width remains narrow. Above threshold generation of solitons is possible instead and the packet of normal modes can spread out. Analogous results are obtained for the ${\phi }^4$ model. We also give an analytical estimate for such thresholds. Finally, we make a comparison with the numerical results known to us and show that, they are in remarkable agreement with our estimates.