# Wave function in one-dimensional Dirac operator

1. Jan 17, 2014

### d98EVsfoW

How to understand the case in which one-dimensional Dirac operator includes a wave function?
Such as
$L= i \left( \begin{array}{ccc} 1+p & 0 \\ 0 & 1-p \end{array} \right) \frac{\partial}{\partial x} + \left( \begin{array}{ccc} 0 & \psi^{*} \\ \psi & 0 \end{array} \right)$, where $\chi= \frac{2}{1-p^2}$.
It is called one-dimensional Dirac operator in [2] and used when solving nonlinear Schrodinger equation [1]:
$i \frac{\partial\psi}{\partial t} + \frac{\partial^2\psi}{\partial x^2} + \chi|\psi^2|\psi =0$.
First equation also referred as "Zakharov-Shabat operator".
In simple cases, when operator consisting of 4x4 matrices $\gamma^\mu$ (constructed from Pauli and identity 2x2 matrices), constants and partial derivatives is applied to wave function, it yields Dirac equation:
$(i\hbar c\gamma^\mu\partial_\mu - mc^2)\psi = 0$
Is there a generalization that allows to use more complex notation?

[1] V. F . Zakharov; A. B, Shabat. "Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Wave in Nonlinear Media" Zh. Eksp. Teor. Fiz. 61, 118-134 (July, 1971)
[2] V.E. Zakharov; S.V. Manakov. "On the complete integrability of a nonlinear Schrödinger equation" Journal of Theoretical and Mathematical Physics 19 (3). Originally in: Teoreticheskaya i Matematicheskaya Fizika 19 (3): 332–343. June 1974