Integrable Systems, Analysis and Probability Seminar
Thursday, October 23, 2003, 4:00pm, 120 Physics
Kirill Vaninsky (Michigan State University)
Poisson brackets on meromorphic functions defined on hyperelliptic Riemann surfaces and 1+1 integrable hierachies
Abstract:- The fact that equations integrable by the method of inverse spectral
transform are Hamiltonian systems was realized in the very
early days of the theory. Gardner found that the Korteveg de Vries
equation on the line with rapidly decaying initial data can be written
as a Hamiltonian system.
It was realized later that that KdV can be formulated with various
Poisson brackets and various Hamiltonians. These Poisson brackets have
a name of Lenard-Magri bracket, or induced by the formalizm of stationary
problem, or Moser--Trubowitz isomorphism. Soon after the KdV Zakharov
and Shabat integrated by the inverse spectral transform the
Nonlinear Schr\"{o}dinger equation with repulsive nonlinearity.
The NLS equation is also a Hamiltonian system.
Novikov and McKean-- Moerbecke found that the periodic problem for this
equations is connected with hyperelliptic Riemann surfaces.
At the present time we know numerous examples of integrable systems
and various Hamiltonian formulations of them. Despite the years of
development this is just a collection of examples. Untill now
it was not known how to obtain the Poisson brackets from the
corresponding Riemann surface. The goal of the present reserach is
to make a new step in this direction. Namely we relate brackets for
the KdV and NLS equations with Poisson structures on the meromorphic
functions defined on the corresponding hyperelliptic Riemann surfaces.
The Poisson (symplectic) structure on meromorphic functions was
consideredby Atiyah and Hitchin in gauge theory. They studied
a meromorphic maps of the Riemann spere into the Riemann sphere.
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