Size and formation of a long wavelength mode packet in the FPU oscillator chain at low energy

The formation of a mode packet in the Fermi–Pasta–Ulam (FPU) oscillator chain, starting from a low frequency mode or modes, at low energy, is reexamined. Both the chain with cubic nonlinearity ( F P U – α ) and quartic nonlinearity ( F P U – β ) are examined for fixed boundary conditions. Some of the basic equations are reformulated to bring out the important scalings, which are compared with existing numerical data. It is shown, that for a single linear mode initial condition or for a small set of modes with random phases the resulting size of the mode packet scales as n e f = ϵ x where n e f is the fraction of occupied modes, ϵ is the energy density, and x = 1 / 4 , 1 / 2 for the α , β chains, respectively. For other initial conditions, the results can depend on the initial number of modes and their phases. Some of the scaling relations are shown to have counterparts in exact nonlinear periodic solutions on the chains ( q -breathers) and the similarities and differences to the results arising from the chaotic dynamics are pointed out.