Soliton surfaces and generalized symmetries of integrable systems

TL;DR

This paper explores the symmetries of integrable systems to construct soliton surfaces using the Fokas-Gel'fand formula, establishing conditions for its applicability based on symmetry group relations.

Contribution

It introduces a criterion for selecting symmetries suitable for the Fokas-Gel'fand formula in the context of integrable systems.

Findings

Established a sufficient condition for the formula's applicability

Derived explicit relations for symmetry group-related vector fields

Provided examples demonstrating the method's application

Abstract

In this paper, we discuss some specific features of symmetries of integrable systems which can be used to contruct the Fokas-Gel'fand formula for the immersion of 2D-soliton surfaces, associated with such systems, in Lie algebras. We establish the sufficient condition for the applicability of this formula. This condition requires the existence of two vector fields which generate a common symmetry of the initial system and its corresponding linear spectral problem. This means that these two fields have to be group-related and we determine an explicit form of this relation. It provides a criterion for the selection of symmetries suitable for the use of the Fokas-Gel'fand formula. We include some examples illustrating its application.