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Displaying similar documents to “Algebro-geometric Integration in Classical and Statistical Mechanics”

Algebro-geometric approach to the Ernst equation I. Mathematical Preliminaries

O. Richter, C. Klein (1997)

Banach Center Publications

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1. Introduction. It is well known that methods of algebraic geometry and, in particular, Riemann surface techniques are well suited for the solution of nonlinear integrable equations. For instance, for nonlinear evolution equations, so called 'finite gap' solutions have been found by the help of these methods. In 1989 Korotkin [9] succeeded in applying these techniques to the Ernst equation, which is equivalent to Einstein's vacuum equation for axisymmetric stationary fields. But, the...

On the integrability of the generalized Yang-Mills system

A. Lesfari, A. Elachab (2004)

Applicationes Mathematicae

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We consider a hamiltonian system which, in a special case and under the gauge group SU(2), can be considered as a reduction of the Yang-Mills field equations. We prove explicitly, using the Lax spectral curve technique and the van Moerbeke-Mumford method, that the flows generated by the constants of motion are straight lines on the Jacobi variety of a genus two Riemann surface.

Geodesic flow on , Kac-Moody Lie algebra and singularities in the complex t-plane.

Ahmed Lesfari (1999)

Publicacions Matemàtiques

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The article studies geometrically the Euler-Arnold equations associatedto geodesic flow on for a left invariant diagonal metric. Such metric were first introduced by Manakov [17] and extensively studied by Mishchenko-Fomenko [18] and Dikii [6]. An essential contribution into the integrability of this problem was also made by Adler-van Moerbeke [4] and Haine [8]. In this problem there are four invariants of the motion defining in C = Lie( ⊗ C) an affine Abelian surface as complete intersection...

The complex geometry of an integrable system

Ahmed Lesfari (2003)

Archivum Mathematicum

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In this paper, a finite dimensional algebraic completely integrable system is considered. We show that the intersection of levels of integrals completes into an abelian surface (a two dimensional complex algebraic torus) of polarization 2 , 8 and that the flow of the system can be linearized on it.

Generalised elliptic functions

Matthew England, Chris Athorne (2012)

Open Mathematics

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We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ℘-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study. The first approach...