• Corpus ID: 221090661

Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation

@article{Anco2020IntegrableSG,
  title={Integrable spinor/quaternion generalizations of the nonlinear Schrodinger equation},
  author={Stephen C. Anco and Ahmed M. G. Ahmed and Esmaeel Asadi},
  journal={arXiv: Mathematical Physics},
  year={2020}
}

• S. Anco, Ahmed M. G. Ahmed, E. Asadi
• Published 7 August 2020
• Mathematics
• arXiv: Mathematical Physics

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.