Chaotic breather formation, coalescence, and evolution to energy equipartition in an oscillatory chain

We study the formation and evolution of chaotic breathers (CBs) on the Fermi–Pasta–Ulam oscillator chain with quartic nonlinearity (FPU-β system). Starting with most of the energy in a single high-frequency mode, the mode is found to breakup on a fast time scale into a number of spatially localized structures (CBs) which, on a slower time scale, coalesce into a single CB. On a usually longer time scale, depending strongly on the energy, the CB gives up its energy to lower frequency modes, approaching energy equipartition among modes. We analyze the behavior, theoretically, using an envelope approximation to the discrete chain of oscillators. For fixed boundaries, periodic nonlinear solutions are found. The numerical structures formed after the fast breakup are found to approximate the underlying equilibrium. These structures are shown, theoretically, to undergo slow translational motions, and an estimated time for them to coalesce into a single chaotic breather are found to agree with the numerically determined scaling τB∝E−1. A previously developed theory of the decay of the CB amplitude to approach equipartition is modified to explicitly consider the interaction of the breather with background modes. The scaling to equipartition of Teq∝E−2 agrees with the numerical scaling and gives the correct order of magnitude of Teq.