Let us first obtain an asymptotic weak-nonlinearity expansion
for the generating functional
exploiting the separation of the linear and
nonlinear time scales. ^{3} To
do this, we have to calculate at the intermediate time via
substituting into it from (32)
For the amplitude and phase ``ingredients'' in we have,

and

where

(35) | |||

(36) | |||

(37) | |||

(38) |

Substituting expansions (39) and (40) into the expression for , we have

with

where

where and denote the averaging over the initial amplitudes and initial phases (which can be done independently). Note that so far our calculation for is the same for the three-wave and for the four-wave cases. Now we have to substitute expressions for and which are different for the three-wave and the four-wave cases and given by (34), (36) and (34), (36) respectively.