Abstract:
In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for
establishing equivalence between nonlinear integrable equations from different areas of physics,
which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems.
In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal
nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet
equation (HFE) to investigate how nonlocality properties of one system are inherited by the other.
First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its
equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for
three basis vectors. We also show that the equation for the third of the basis vectors coincides
with the well-known integrable HFE and its generalization. Two other equations for the remaining
two vectors give new integrable spin systems. Finally, we investigated the relation between the
differential geometry of surfaces and integrable spin systems for the three basis vectors.
