Inverse scattering and the symplectic form for sine-Gordon solitons

TL;DR

This paper explicitly computes the canonical symplectic form for sine-Gordon solitons using inverse scattering, deriving a simple boundary-term expression and confirming its consistency with previous results, while also exploring conserved charges.

Contribution

It provides an explicit evaluation of the symplectic form for sine-Gordon solitons within the inverse scattering framework, connecting boundary terms to soliton parameters.

Findings

Derived a simple explicit expression for the symplectic form in boundary terms.

Confirmed agreement with previous results by Babelon and Bernard.

Analyzed higher conserved charges and their Poisson commutation.

Abstract

We consider the canonical symplectic form for sine-Gordon evaluated explicitly on the solitons of the model. The integral over space in the form, which arises because the canonical argument uses the Lagrangian density, is done explicitly in terms of functions arising in the group doublecrossproduct formulation of the inverse scattering procedure, and we are left with a simple expression given by two boundary terms. The expression is then evaluated explicitly in terms of the changes in the positions and momenta of the solitons, and we find agreement with a result of Babelon and Bernard who have evaluated the form using a different argument, where it is diagonal in terms of `in' or `out' co-ordinates. Using the result, we also investigate the higher conserved charges within the inverse scattering framework, check that they Poisson commute and evaluate them on the soliton solutions.