Soliton surfaces associated with generalize symmetries of integrable equations

TL;DR

This paper characterizes the symmetries of soliton surfaces in Lie algebras related to integrable equations, providing conditions for their explicit construction and analyzing specific models like the $CP^{N-1}$ sigma model.

Contribution

It introduces a symmetry-based framework for constructing and analyzing soliton surfaces using generalized symmetries and applies it to the $CP^{N-1}$ sigma model.

Findings

Explicit conditions for the existence of soliton surfaces

Restrictions on conformal symmetries in Minkowski space

Sufficient conditions are always met in Euclidean space

Abstract

In this paper, based on the Fokas, Gel'fand et al approach [15,16], we provide a symmetry characterization of continuous deformations of soliton surfaces immersed in a Lie algebra using the formalism of generalized vector fields, their prolongation structure and links with the Fr\'echet derivatives. We express the necessary and sufficient condition for the existence of such surfaces in terms of the invariance criterion for generalized symmetries and identify additional sufficient conditions which admit an explicit integration of the immersion functions of 2D surfaces in Lie algebras. We discuss in detail the $su(N)$-valued immersion functions generated by conformal symmetries of the $CP^{N-1}$ sigma model defined on either the Minkowski or Euclidean space. We further show that the sufficient conditions for explicit integration of such immersion functions impose additional restrictions on the admissible conformal symmetries of the model defined on Minkowski space. On the other hand, the sufficient conditions are identically satisfied for arbitrary conformal symmetries of finite action solutions of the $CP^{N-1}$ sigma model defined on Euclidean space.