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From solitons to many-body systems. (English) Zbl 1153.37031
The authors of this very interesting paper study the connection of the KP soliton equations with the Calogero-Moser many-body systems. They describe the motion of poles of meromorphic solutions of soliton equations by simple many-body systems. For this purpose they use the methods of the noncommutative algebraic geometry. It is known that the Calogero-Moser many-body systems could be interpreted as flows on spaces of spectral curves on a ruled surface. It turns out that KP Lax operators can be identified with vector bundles on quantized cotangent spaces formulated technically in terms of \(\mathcal{D}\)-modules. A geometric duality identifies the parameter space for such vector bundles with that for the spectral curves and sends the KP flows to the Calogero-Moser flows. A result being of interest in this paper is that the motion and collisions of the poles of the rational, trigonometric and elliptic solutions of the KP hierarchy, as well as of its multicomponent analogs, are governed by the Calogero-Moser systems on cuspidal, nodal and smooth genus one curves.

MSC:
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
70F10 \(n\)-body problems
81R12 Groups and algebras in quantum theory and relations with integrable systems
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