A simple formula for the conserved charges of soliton theories

TL;DR

This paper introduces a straightforward formula to compute all conserved charges in soliton theories, demonstrated explicitly on the sine-Gordon and mKdV models, revealing that energy and momentum are boundary terms for a broad class of solutions.

Contribution

It provides a simple, explicit method to evaluate conserved charges for solutions in soliton theories, including solitons, breathers, and their combinations.

Findings

Energy and momentum are boundary terms for solutions on the vacuum orbit.

The formula applies to a wide class of physically relevant solutions.

Explicit calculations are shown for sine-Gordon and mKdV models.

Abstract

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.