Generalizations of the Kardar-Parisi-Zhang equation

We generalize the Kardar-Parisi-Zhang (KPZ) equation to an O(3)N=2j+1 component model. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty} we show that the mode coupling equations become exact. Solving these approximately we find that the dynamic exponent z increases from 3/2 for d=1 to 2 at the dimension d\ensuremath{\approxeq}3.6. For d=1 it can be shown analytically that z=3/2 for all j. The case j=2 for d=2 is investigated by numerical integration of the KPZ equation.