Inverse scattering and the symplectic form for sine-Gordon solitons

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. Laboratoire de la Direction des Sciences de la Matière du Commissariat à l’Energie Atomique