How to understand the case in which one-dimensional Dirac operator includes a wave function?(adsbygoogle = window.adsbygoogle || []).push({});

Such as

[itex]

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)

[/itex], where [itex]

\chi=

\frac{2}{1-p^2}

[/itex].

It is called one-dimensional Dirac operator in [2] and used when solving nonlinear Schrodinger equation [1]:

[itex]

i

\frac{\partial\psi}{\partial t}

+

\frac{\partial^2\psi}{\partial x^2}

+

\chi|\psi^2|\psi

=0

[/itex].

First equation also referred as "Zakharov-Shabat operator".

In simple cases, when operator consisting of 4x4 matrices [itex]\gamma^\mu[/itex] (constructed from Pauli and identity 2x2 matrices), constants and partial derivatives is applied to wave function, it yields Dirac equation:

[itex](i\hbar c\gamma^\mu\partial_\mu - mc^2)\psi = 0[/itex]

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

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# Wave function in one-dimensional Dirac operator

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