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A. Khan and D. Pelinovsky

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Long-time stability of small FPU solitary waves

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Discrete and Continuous Dynamical Systems Series A 37 (2017) 2065-2075

**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.

**Keywords:**

FPU lattice, generalized KdV equation, stability of nonlinear waves, justification of amplitude equations.