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On the numerics of integrable discretizations. (English) Zbl 0856.65107
Levi, Decio (ed.) et al., Symmetries and integrability of difference equations. Papers from the workshop, May 22–29, 1994, Estérel, Canada. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 9, 1-11 (1996).
Today there is widespread interest in integrable discretizations of the soliton problems, both from a numerical point of view, and as interesting physical models in their own right. In this note we briefly explain some of the advantages if integrable discretizations are used as numerical schemes, using the nonlinear Schrödinger and sine-Gordon equations as prototypical examples. As might be expected, the advantages are particularly striking when situations are encountered where integrability is a key issue. One such situation occurs when initial values are chosen in the vicinity of homoclinic manifolds. For finite dimensional dynamical systems this is well known to be a region of phase space which is very sensitive to small perturbations.
For the infinite dimensional problems of soliton theory where one may encounter high dimensional homoclinic manifolds, the sensitivity is extreme. In practice one may find that standard (nonintegrable) discretizations break down completely while integrable discretizations display the expected quasi-periodic behavior. This is clearly illustrated for the nonlinear Schrödinger equation.
The extreme sensitivity in the proximity of homoclinic manifolds is illustrated by the behavior of a completely integrable discretization of the sine-Gordon equation. Because of the particular structure of the discretization, the solution decouples (for \(N\) even, see below) into two independent components. Viewed as two independent solutions, they are given initial data that differ only marginally. However, these small differences are amplified exponentially and one quickly observes a complete separation of the two solutions.
For the entire collection see [Zbl 0852.00027].

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
37-XX Dynamical systems and ergodic theory
65L12 Finite difference and finite volume methods for ordinary differential equations
35Q55 NLS equations (nonlinear Schrödinger equations)
65L05 Numerical methods for initial value problems involving ordinary differential equations
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)