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

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