Schrodinger-Poisson system

The Schrodinger-Poisson system

$iv_{t}^{+}+\Delta v/2=uv^{+}\,$ $iv_{t}^{-}-\Delta v/2=uv^{-}\,$ $\Delta u=-|v^{+}|^{2}+|v^{-}|^{2}\,$

arises as the non-relativistic limit of the nonlinear Klein-Gordon equation.