Geometric approach for evolution equations and self-similar solutions of inhomogeneous Heisenberg spin system

Abstract

In this paper, we study the integro-differential nonlinear Schr & ouml;dinger equation (INLSE) by applying a new geometric approach within the formalism of a moving space curve. Using this geometric framework, we show that the INLSE is geometrically equivalent to the inhomogeneous Heisenberg spin chain (IHC) and obtain the corresponding evolution equations through this equivalence. The self-similar solutions of the flows governing the evolution of the IHC and the geometric properties of the associated surfaces derived from the spin evolution equation are fully examined. Overall, the results provide a unified geometric interpretation linking the INLSE, the IHC dynamics, and the induced surface geometry, thereby offering new insights into the integrable structure of these systems.