Qualitative analysis of three-wave interaction with periodic boundary   condition
  First, for 3WRI with positive wave energy, we present a regularity theorem for all spatial dimension. Second, for 3WRI with negative wave energy, we present a class of solution in general spatial dimension that will blow up in finite time. Moreover, a complete classification of spatial uniform solution is given for this particular system. 
Ze Cheng, Harvey Segur
math.AP