On immersion formulas for soliton surfaces

TL;DR

This paper explores the relationships between different analytic descriptions of soliton surface immersions, focusing on gauge, conformal, and generalized symmetries, and provides conditions linking these descriptions through gauge transformations, exemplified by the sigma model.

Contribution

It establishes necessary and sufficient conditions for connecting immersion formulas via gauge transformations for various symmetries in soliton surfaces.

Findings

Derived conditions linking immersion formulas through gauge transformations.

Analyzed the role of symmetries in soliton surface theory.

Illustrated results with examples from the sigma model.

Abstract

This paper is devoted to a study of the connections between three different analytic descriptions for the immersion functions of 2D-surfaces corresponding to the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations in the spectral parameter and generalized symmetries of the associated integrable system. After a brief exposition of the theory of soliton surfaces and of the main tool used to study classical and generalized Lie symmetries, we derive the necessary and sufficient conditions under which the immersion formulas associated with these symmetries are linked by gauge transformations. We illustrate the theoretical results by examples involving the sigma model.