This is similar; in this case it is a concatenation of 1st order operators
[$]\frac{\partial }{\partial z} - i\frac{\partial }{\partial \overline{z}}[$] and [$]\frac{\partial u}{\partial z} + i\frac{\partial u}{\partial \overline{z}}[$]. This is is the elliptic case.
The hyperbolic case is similar but in real space.
[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$] = [$](\frac{\partial }{\partial x} - \frac{\partial }{\partial {y}})[$] [$](\frac{\partial u}{\partial x} + \frac{\partial u}{\partial {y}})[$].
Then [$]x = (\xi + \eta), y = (\xi - \eta)[$] leads to [$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$] .
Putting in a convection/drift term seems to cause meltdown fubar.
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used \overline for complex conjugation, better than \bar?