Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation

TL;DR

This paper introduces new integrable generalizations of the nonlinear Schrödinger equation involving spinor and quaternion variables, providing Lax pairs, bi-Hamiltonian structures, and demonstrating their equivalence to geometric flows and Schrödinger map equations.

Contribution

It presents novel integrable multi-component NLS equations with nonlocal terms, derived from symmetric Lie algebras, and establishes their geometric and algebraic equivalences.

Findings

Derived Lax pairs and bi-Hamiltonian structures for the new systems

Established equivalence to bi-normal flow and Schrödinger map equations

Described spinor/quaternion NLS-type equations with integrable properties

Abstract

An integrable generalization of the NLS equation is presented, in which the dynamical complex variable $u(t,x)$ is replaced by a pair of dynamical complex variables $(u_1(t,x),u_2(t,x))$, and $i$ is replaced by a Pauli matrix $J$. Integrability is retained by the addition of a nonlocal term in the resulting 2-component system. A further integrable generalization is obtained which involves a dynamical scalar variable and an additional nonlocal term. For each system, a Lax pair and a bi-Hamiltonian formulation are derived from a zero-curvature framework that is based on symmetric Lie algebras and that uses Hasimoto variables. The systems are each shown to be equivalent to a bi-normal flow and a Schrodinger map equation, generalizing the well-known equivalence of the NLS equation to the bi-normal flow in $R^3$ and the Schrodinger map equation in $S^2$. Furthermore, both of the integrable systems describe spinor/quaternion NLS-type equations with the pair $(u_1(t,x),u_2(t,x))$ being viewed as a spinor variable or equivalently a quaternion variable.