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Linearizing flows and a cohomological interpretation of Lax equations. (English) Zbl 0585.58028

The author considers a Lax equation of the type \(dA(\xi,t)/dt=[B(\xi,t),A(\xi,t)]\) (\(\xi\) is a rational parameter) and associates to it an algebraic curve (its spectral curve) together with a dynamical system \(L_ t\) on its Jacobian variety J(C). He gives necessary and sufficient conditions on B for the flow \(t\to L_ t\) to be linear on J(C). These conditions are cohomological: the Lax equations turn out to have a very natural cohomological interpretation.
Reviewer: Yu.E.Gliklikh

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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.)
37C10 Dynamics induced by flows and semiflows
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