Integrable Kuralay equations: geometry, solutions and generalizations

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.