Abstract
The one-dimensional sine-Gordon equation is an example of an exactly solvable nonlinear partial differential equation. When discretized on a periodic lattice in space and in time, it corresponds to a lattice of pendula coupled by linear springs. We show that the discretized system is unstable to a parametric mode when the frequency of time discretization is sufficiently small, and we obtain the instability condition and growth rate. By considering the effects of two finite amplitude modes, we also obtain the nonlinear saturation of the instability. We also examine how solutions of the exact sine-Gordon equation behave under this map, both in and away from the parametrically unstable regime.
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