Dynamics of Oscillator Chains from High Frequency Initial Conditions: Comparison of Phi4 and FPU-beta Models.

The dynamics of oscillator chains are studied, starting from high frequency initial conditions (h.f.i.c.). In particular, the formation and evolution of chaotic breathers (CB’s) of the Klein–Gordon chain with quartic nonlinearity in the Hamiltonian (the ϕ4 model) are compared to the results of the previously studied Fermi–Pasta–Ulam (FPU-β) chain. We find an important difference for h.f.i.c. is that the quartic nonlinearity, which drives the high frequency phenomena, being a self-force on each individual oscillator in the ϕ4 model is significantly weaker than the quartic term in the FPU-β model, which acts between neighboring oscillators that are nearly out-of-phase. The addition of a self-force breaks the translational invariance and adds a parameter. We compare theoretical results, using the envelope approximation to reduce the discrete coupled equations to a partial differential equation for each chain, indicating that various scalings can be used to predict the relative energies at which the basic phenomena of parametric instability, breather formation and coalescence, and ultimately breather decay to energy equipartition, will occur. Detailed numerical results, comparing the two chains, are presented to verify the scalings.