Stability of an oscillator chain from high-frequency initial conditions

We study the stability of the Fermi–Pasta–Ulam chain of coupled oscillators with a quartic coupling nonlinearity (FPU- β system), for initial conditions that give rise primarily to high-frequency oscillations. We consider initial conditions that are either close to a high-frequency mode of the linearized system or to the nonlinear equilibria that can be constructed from an envelope approximation for the high-frequency oscillations. eory indicates that broad mode-like initial conditions are unstable and the instability structure can often be approximately calculated from the stability analysis of the π -mode equilibrium in which there is exact alternation of the phases of the individual oscillators. In contrast, examination of the stability of breather-like structures indicates that they are probably stable to the fast parametric instability. These theoretical observations are generally confirmed by numerical integration of the exact equations of motion. The study clarifies previous observations of the formation of nonlinear structures, and the relative stability of those structures on time scales that are long compared to the growth times of parametric instabilities calculated within the envelope approximation.