2007/03/26 by Boris Gershgorin, Gershgorin, Boris, Yuri V. Lvov +3 · 2 citations
Engineering · Physics and Astronomy · #Adhesion, Friction, and Surface Interactions #Advanced Fiber Optic Sensors #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.nlin/0703051
openalex publication_date 2007/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in thermal equilibrium are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the chain in thermal equilibrium can be effectively described by a system of weakly interacting renormalized nonlinear waves that possess (i) the Rayleigh-Jeans distribution and (ii) zero correlations between waves, just as noninteracting free waves would. This renormalization is achieved through a set of canonical transformations. The renormalized linear dispersion of these renormalized waves is obtained and shown to be in excellent agreement with numerical experiments. Moreover, a dynamical interpretation of the renormalization of the dispersion relation is provided via a self-consistency, mean-field argument. It turns out that this renormalization arises mainly from the trivial resonant wave interactions, i.e., interactions with no momentum exchange. Furthermore, using a multiple time-scale, statistical averaging method, we show that the interactions of near-resonant waves give rise to the broadening of the resonance peaks in the frequency spectrum of renormalized modes. The theoretical prediction for the resonance width for the thermalized β-FPU chain is found to be in very good agreement with its numerically measured value.