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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