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

TL;DR

This paper investigates the symplectic structure of the dressing group in the sinh-Gordon model, explicitly computing the Poisson brackets and revealing differences from the standard Semenov-Tian-Shansky structure.

Contribution

It provides an explicit calculation of the Poisson bracket on the dressing group for the sinh-Gordon model, highlighting deviations from known structures and applying constrained Hamiltonian formalism.

Findings

The Poisson bracket on the dressing group differs from the Semenov-Tian-Shansky bracket.

Constraints on the dressing group element are necessary due to soliton generation.

The formalism of constrained Hamiltonian systems is used to relate different brackets.

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.