A Note on the Symplectic Structure on the Dressing Group in the sinh--Gordon Model

Abstract

We analyze the symplectic structure on the dressing group in the \shG\, model by calculating explicitly the Poisson bracket $\{g\x g\}$ where $g$ is the \dg\, element which creates a generic one soliton solution from the vacuum. Our result is that this bracket does not coincide with the Semenov--Tian--Shansky one. The last induces a Lie--Poisson structure on the \dg . To get the bracket obtained by us from the Semenov--Tian--Shansky bracket we apply the formalism of the constrained Hamiltonian systems. The constraints on the \dg\, appear since the element which generates one solitons from the vacuum has a specific form.

11 pages, latex, no figures, some statements corrected, the end of Sec. 3 and the whole Sec. 4 totally revised

References in corpus (3)

• Dressing Symmetries
• The Sine-Gordon Solitons as a N-Body Problem
• Affine Solitons: A Relation Between Tau Functions, Dressing and Bäcklund Transformations

Cited by in corpus (2)

• $A_n^{(1)}$ Toda solitons and the dressing symmetry
• Vertex Operator Representation of the Soliton Tau Functions in the $A_n^{(1)}$ Toda Models by Dressing Transformations