On the numerical solution of the sine-Gordon equation. I: Integrable discretizations and homoclinic manifolds. (English) Zbl 0866.65064

In this first of two papers on the numerical integration of the sine-Gordon equation the authors investigate the numerical behavior of a double-discrete, completely integrable discretization of this equation. The nature of potential instabilities relevant to particular initial values in the vicinity of homoclinic manifolds is clarified via analytical investigation, illustrated by numerical experiments.
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 thus opening the way to use the nonlinear spectrum as a basis for comparing different numerical schemes (in the second paper to follow).


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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