Equations of Camassa--Holm type and Jacobi elliptic coordinates

Details Equations of Camassa--Holm type and Jacobi elliptic coordinates Equations of Camassa--Holm type and Jacobi elliptic coordinates

2003-03-26

arxiv.org/abs/math-ph/0303063v4

Comm. Pure Appl. Math. 58 (2005), no. 9, 1149-1187

Physics

Mathematical Physics

42 pages, 3 figures, final version

Scientific paper

We consider the integrable Camassa--Holm equation on the line with positive initial data rapidly decaying at infinity. On such phase space we construct a one parameter family of integrable hierarchies which preserves the mixed spectrum of the associated string spectral problem. This family includes the CH hierarchy. We demonstrate that the constructed flows can be interpreted as Hamiltonian flows on the space of Weyl functions of the associated string spectral problem. The corresponding Poisson bracket is the Atiyah--Hitchin bracket. Using an infinite dimensional version of the Jacobi ellipsoidal coordinates we obtain a one parameter family of canonical coordinates linearizing the flows.

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