Integrable geometric flows of interacting curves/surfaces, multilayer spin systems and the vector nonlinear Schr\"odinger equation

In this paper, we study integrable multilayer spin systems, namely, the multilayer M-LIII equation. We investigate their relation with the geometric flows of interacting curves and surfaces in some space $R^{n}$. Then we present their the Lakshmanan equivalent counterparts. We show that these equivalent counterparts are, in fact, the vector nonlinear Schr\"odinger equation (NLSE). It is well-known that the vector NLSE is equivalent to the $\Gamma$-spin system. Also, we have presented the transformations which give the relation between solutions of the $\Gamma$-spin system and the multilayer M-LIII equation. It is interesting to note that the integrable multilayer M-LIII equation contains constant magnetic field ${\bf H}$. It seems that this constant magnetic vector plays an important role in theory of "integrable multilayer spin system" and in nonlinear dynamics of magnetic systems. Finally, we present some classes of integrable models of interacting vortices.