ON THE NUMERICAL-SOLUTION OF THE SINE-GORDON EQUATION .1. INTEGRABLE DISCRETIZATIONS AND HOMOCLINIC MANIFOLDS

In this, the first of two papers on the numerical solution of the sine -Gordon equation, we investigate the numerical behavior of a double di screte, completely integrable discretization of the sine-Gordon equati on. For certain initial values, in the vicinity of homoclinic manifold s, this discretization admits an instability in the form of grid scale oscillations. We clarify the nature of the instability through an ana lytical investigation supported by numerical experiments. In particula r, a perturbation analysis of the associated linear spectral problem s hows that the initial values used for the numerical experiments lie ex ponentially close to a homoclinic manifold. This paves the way for the second paper where we use the nonlinear spectrum as a basis for compa ring different numerical schemes. (C) 1996 Academic Press. Inc.