The present work addresses the study and characterization of the integrability of some generalized Heisenberg ferromagnet equations (GHFE) in 1+1 dimensions. Lax representations for these GHFE are successfully obtained. The gauge equivalent counterparts of these integrable GHFE are presented.

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 Kuralay equations are considered. Some integrable generalizations and other related nonlinear differential equations are presented.

In this paper the generalized (3+1)-dimensional Landau-Lifshitz equation with potential is investigated. Its exact localized and topological solutions are constructed by generalizing Hirota's method.

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