The Dressing Method as Non Linear Superposition in Sigma Models

TL;DR

This paper demonstrates that the dressing method for the Non Linear Sigma Model allows for a non-linear superposition principle, enabling the construction of solutions without solving differential equations, by relating solutions with the same Pohlmeyer counterpart.

Contribution

It introduces a novel application of the dressing method that expresses solutions as a non-linear superposition, simplifying the solution process for the NLSM on 2\times S^2.

Findings

Dressed solutions can be obtained without solving differential equations.

The method establishes a superposition principle for NLSM solutions.

Solutions are expressed in terms of seed solutions with the same Pohlmeyer counterpart.

Abstract

We apply the dressing method on the Non Linear Sigma Model (NLSM), which describes the propagation of strings on $\mathbb{R}\times \mathrm{S}^2$, for an arbitrary seed. We obtain a formal solution of the corresponding auxiliary system, which is expressed in terms of the solutions of the NLSM that have the same Pohlmeyer counterpart as the seed. Accordingly, we show that the dressing method can be applied without solving any differential equations. In this context a superposition principle emerges: The dressed solution is expressed as a non-linear superposition of the seed with solutions of the NLSM with the same Pohlmeyer counterpart as the seed.