Integrable spin systems with self-consistent potentials in (1+1) dimensions

Abstract

The motion of curves and surfaces in R3 leads to nonlinear evolution equations, which are often integrable. They
are also closely related to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric equivalences. In this paper it is shown that a more general situation in which the curves evolve in the presence of additional self-consistent vector potentials can lead to interesting generalized spin systems with self-consistent potentials. A general view of the main evolution equations of curves is obtained and concrete examples of generalized spin chains and soliton equations are given. These include the main chiral model and various Myrzakulov spin equations in (1 +1) dimensions and their geometric equivalents from the family of generalized nonlinear Schr¨odinger equations (NSE) in the presence of a self-consistent field potential, including the Hiroth-Maxwell-Bloch equation. The corresponding gauge equivalent Lax pairs are also presented to confirm their integrability.