On the numerical solution of the sine-Gordon equation

In this, the first of two papers on the numerical solution of the sine-Gordon equation, we investigate the numerical behavior of a double discrete, completely integrable discretization of the sine-Gordon equation. For certain initial values, in the vicinity of homoclinic manifolds, this discretization admits an instability in the form of grid scale oscillations. We clarify the nature of the instability through an analytical investigation supported by numerical experiments. In particular, a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold. This paves the way for the second paper where we use the non-linear spectrum as a basis for comparing different numerical schemes. 21 refs., 11 figs