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 sinh-Gordon model by calculating explicitly the Poisson bracket { g ⊗, g} where g is the dressing group 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 latter induces a Lie-Poisson structure on the dressing group. 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 dressing group appear since the element which generates one-soliton solutions from the vacuum has a specific form.