Isospectral flow in loop algebras and quasiperiodic solutions of the sine-Gordon equation

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 𝔤* of a loop algebra 𝔤 , is parameterized by a finite dimensional symplectic vector space W embedded into 𝔤* 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.