On the Numerical Solution of the Sine–Gordon Equation: I. Integrable Discretizations and Homoclinic Manifolds

✍ Scribed by M.J. Ablowitz; B.M. Herbst; Constance Schober

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 614 KB
• Volume: 126
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.0139

✦ Synopsis

completely integrable through the inverse scattering transform (see, for example, [1,11]) this allows us to study the In this, the first of two papers on the numerical solution of the sine-Gordon equation, we investigate the numerical behavior of a subsequent nonlinear evolution of the instabilities. There double discrete, completely integrable discretization of the sineare two interesting boundary value problems which can Gordon equation. For certain initial values, in the vicinity of homobe considered, the ''infinite line'' case with decaying initial clinic manifolds, this discretization admits an instability in the form values, and the periodic boundary value problem. In the of grid scale oscillations. We clarify the nature of the instability through an analytical investigation supported by numerical experi-infinite line case the instability results in solitons-the ments. In particular, a perturbation analysis of the associated linear number depending on the amplitude of the initial values. spectral problem shows that the initial values used for the numerical On the other hand, with periodic boundary values, the experiments lie exponentially close to a homoclinic manifold. instabilities are a manifestation of a ''homoclinic'' manifold This paves the way for the second paper where we use the nonassociated with the NLS equation. The dimension or comlinear spectrum as a basis for comparing different numerical schemes.