Solving the Initial Value Problem for the 3-Wave Interaction Equations in Multidimensions

Starting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space dimensions and one time dimension), we complexify the independent variables, thus doubling the number of real variables, and hence we work in 4+2 dimensions: $$x_1$$ x 1 , $$x_2$$ x 2 , $$y_1$$ y 1 , $$y_2$$ y 2 and $$t_1$$ t 1 , $$t_2$$ t 2 . In this paper we solve the initial value problem of the 3-wave interaction equations in 4+2 dimensions.