Isospectral flow in Loop Algebras and Quasiperiodic Solutions of the Sine-Gordon Equation

TL;DR

This paper explores the integrable structure of the sine-Gordon equation using loop algebra techniques, deriving quasiperiodic solutions via theta functions and Hamiltonian methods.

Contribution

It introduces a novel Hamiltonian framework for the sine-Gordon equation based on loop algebras and explicitly constructs quasiperiodic solutions using algebraic geometric methods.

Findings

Finite-dimensional phase space parametrized by symplectic vector space

Explicit quasiperiodic solutions expressed through theta functions

Hamiltonian structure linked to loop algebra and hyperelliptic curves

Abstract

The sine-Gordon equation is considered in the hamiltonian framework provided by the Adler-Kostant-Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space $\grg^*$ of a loop algebra $\grg$, is parametrized by a finite dimensional symplectic vector space $W$ embedded into $\grg^*$ by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.