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AKNS Lax pair for the nonlinear Schrodinger equation

For complex fields

q, r

, the standard two-by-two AKNS pair has zero-curvature equations

i r_{t} + r_{xx} + 2 q r^{2} = 0, i q_{t} - q_{xx} - 2 r q^{2} = 0.

The reductions

q = \overline{r}

and

q = - \overline{r}

give the focusing and defocusing cubic nonlinear Schrödinger equations, respectively.

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