Dispersionless Toda and Toeplitz operators. (English) Zbl 1024.37047

Bloch, A.; Golse, F.; Paul, T.; Uribe, A.

Duke Math. J. 117, No. 1, 157-196 (2003).

The goal of this paper is to clarify, in a specific example the link between large systems of ordinary differential equations and certain limiting nonlinear partial differential equations (PDEs). The authors present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. They consider both periodic and nonperiodic boundary conditions. They show that the Toda equations, although they are nonlinear, propagate a Toeplitz operator into an operator arbitrary close to a Toeplitz operator as long as the Toda PDE (dispersionless limit) admits smooth solutions. The authors believe that their methods should be valid in other situations since they only use the Lax pair structure of the lattice and not its complete integrability.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37K60	Lattice dynamics; integrable lattice equations
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators

Keywords:

dispersionless limit; Toda lattice equations; Toeplitz operators; Lax pair

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