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Substituting the expansion (32) in (12) we
get in the zeroth order
i.e. the zeroth order term is time independent. This corresponds to
the fact that the interaction representation wave amplitudes are
constant in the linear approximation. For simplicity, we will write
, understanding that a quantity is taken at
if its time argument is not mentioned explicitly.
Here we have taken into account that
and
.
The first order is
given by







(30) 
Here we have taken into account that
and
. Perform the second iteration, and integrate over
time to obtain
To calculate the second iterate, write
Substitute (34) into (35) and integrate over time to obtain
where we used
. Hereafter, we drop superscript in
expressions like (36) for brevity of notations.
Next: Weak nonlinearity expansion: fourwave
Up: Wave Turbulence "Recent developments
Previous: Separation of timescales: general
Dr Yuri V Lvov
20070123