Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation

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 σ. 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.