Isospectral flow in loop algebras and quasiperiodic solutions of the sine-Gordon equation. (English) Zbl 0780.35100
Summary: 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 \({\mathfrak g}^*\) of a loop algebra \({\mathfrak g}\), is parameterized by a finite dimensional symplectic vector space \(W\) embedded into \({\mathfrak g}^*\) 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.
MSC:
35Q53 | KdV equations (Korteweg-de Vries equations) |
58J70 | Invariance and symmetry properties for PDEs on manifolds |
14H42 | Theta functions and curves; Schottky problem |
53C80 | Applications of global differential geometry to the sciences |
17B66 | Lie algebras of vector fields and related (super) algebras |
Keywords:
Liouville-Arnold integration method; sine-Gordon equation; Hamiltonian framework; Adler-Kostant-Symes theorem; loop algebraReferences:
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