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Asymptotic expansion of the generating functional.

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,

 (33)

and

 (34)

where
 (35) (36) (37) (38)

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

 (39)

with

 (40)

where
 (41) (42) (43) (44) (45)

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.

Next: Evolution of statistics of Up: Wave Turbulence "Recent developments Previous: Weak nonlinearity expansion: four-wave
Dr Yuri V Lvov 2007-01-23