Notes on integrable motion of two interacting curves and two-layer generalized Heisenberg ferromagnet equations

Abstract

We study the integrable two-layer generalized Heisenberg ferromagnet equation (HFE). The relation between this generalized HFE and differential geometry of curves is established. Using this relation we found the geometrical equivalent counterpart of the two-layer spin system which is the two-component KdV equation. Finally, the gauge equivalence between these equations is established.