Integrable Kuralay equations: geometry, solutions and generalizations

TL;DR

This paper explores the integrability, geometric interpretation, and solutions of the Kuralay equations, including their nonlocal and dispersionless forms, establishing gauge equivalence and deriving soliton solutions.

Contribution

It introduces the integrable Kuralay equations, analyzes their geometric motion, and constructs explicit soliton solutions using Hirota's method, including nonlocal and dispersionless variants.

Findings

Established gauge equivalence between K-IE and K-IIE

Derived explicit soliton solutions using Hirota method

Analyzed nonlocal and dispersionless versions of the equations

Abstract

In this paper, we study the Kuralay equations, namely, the Kuralay-I equation (K-IE) and the Kuralay-II equation (K-IIE). The integrable motion of space curves induced by these equations is investigated. The gauge equivalence between these two equations is established. With the help of the Hirota bilinear method, the simplest soliton solutions are also presented. The nonlocal and dispersionless versions of the K-IE and K-IIE are considered.