A simple formula for the conserved charges of soliton theories

We present a simple formula for all the conserved charges of soliton theories, evaluated on the solutions belonging to the orbit of the vacuum under the group of dressing transformations. For pedagogical reasons we perform the explicit calculations for the case of the sine-Gordon model, taken as a prototype of soliton theories. We show that the energy and momentum are boundary terms for all the solutions on the orbit of the vacuum. That orbit includes practically all the solutions of physical interest, namely solitons, multi-solitons, breathers, and combinations of solitons and breathers. The example of the mKdV equation is also given explicitly.

💡 Research Summary

The paper introduces a compact and universal formula for computing the entire hierarchy of conserved charges in soliton theories, specifically for solutions that lie on the orbit of the vacuum under dressing transformations. The authors begin by recalling that integrable field theories possess an infinite set of conserved quantities, traditionally obtained through inverse scattering, recursion operators, or direct integration of the Noether currents. However, these methods become cumbersome when applied to explicit multi‑soliton, breather, or mixed configurations.

To overcome this, the authors exploit the dressing group – a gauge‑like symmetry that maps the trivial vacuum solution of a zero‑curvature (Lax) representation into non‑trivial field configurations while preserving the flatness condition. If (A_\mu) denotes the Lax connection satisfying (\partial_\mu A_\nu-\partial_\nu A_\mu+