Reduced systems for Intrinsic Localized Modes on an infinite oscillator array

We reduce the complex mechanics of an Intrinsic Localized Mode (ILM) on an infinite monodirectional nonlinear coupled oscillator array. By using Hamiltonian Mechanics, we derive two systems – a simple case when only the position of the ILM is allowed to vary, and the more complex case where both position and amplitude are allowed to change with time. We arrive at a systems of nonlinear equations of motion. For the position-only system, the dynamics are determined solely by the level sets of the Hamiltonian. The four-variable system has a richer structure. We also analyze the effect of a defect, or an impurity, on the motion of the ILM within our framework.