Long-time stability of small FPU solitary waves

Abstract: Small-amplitude waves in the Fermi-Pasta-Ulam (FPU) lattice with weakly anharmonic interaction potentials are described by the generalized Korteweg-de Vries (KdV) equation. Justification of the small-amplitude approximation is usually performed on the time scale, for which dynamics of the KdV equation is defined. We show how to extend justification analysis on longer time intervals provided dynamics of the generalized KdV equation is globally well-posed in Sobolev spaces and either the Sobolev norms are globally bounded or they grow at most polynomially. The time intervals are extended respectively by the logarithmic or double logarithmic factors in terms of the small amplitude parameter. Controlling the approximation error on longer time intervals allows us to deduce nonlinear metastability of small FPU solitary waves from orbital stability of the KdV solitary waves.