Integrable motion of two interacting curves, spin systems and the Manakov system

TL;DR

This paper explores the integrable motion of two interacting curves, establishing a connection with coupled spin systems and demonstrating their equivalence to the Manakov system, thus advancing understanding of integrable models in mathematical physics.

Contribution

It introduces a new model of two interacting curves, finds its integrable reduction related to coupled spin systems, and proves their equivalence to the Manakov system.

Findings

The model of two interacting curves is integrable.

The integrable reduction relates to a coupled spin system.

The coupled spin system is equivalent to the Manakov system.

Abstract

Integrable spin systems are an important subclass of integrable (soliton) nonlinear equations. They play important role in physics and mathematics. At present, many integrable spin systems were found and studied. They are related with the motion of 3-dimensional curves. In this paper, we consider a model of two moving interacting curves. Next, we find its integrable reduction related with some integrable coupled spin system. Then we show that this integrable coupled spin system is equivalent to the famous Manakov system.