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Toward a topological classification of integrable PDE’s. (English) Zbl 0743.58020
The geometry of Hamiltonian systems, Proc. Workshop, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 22, 111-129 (1991).
[For the entire collection see Zbl 0733.00016.]
Recently A. T. Fomenko presents a nice classification of integrable Hamiltonian systems with two degrees of freedom. This classification is based on studying the critical points lying on the isoenergetic threefold in the phase space of the second integral (which is supposed to be Bottean).
In the paper under review the authors develop a parallel theory capable to produce detailed stratifications of infinite solution systems and illustrate it via nonlinear Schrödinger equation. The basic object of their study is the Floquet discriminant and its critical points. The paper ends with discussion about the impact which the topological classification may have on studies of perturbed and truncated systems.
Reviewer: I.Mladenov (Sofia)

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q58 Other completely integrable PDE (MSC2000)
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.