In this section we are going to calculate the four-wave analog of
PBP equation. These calculations are similar in spirit to
those presented in the orevious section, but with the four-wave equation
of motion (15). The calculation will be slightly
more
involved due to the frequency renormalisation, but it will be significantly
simplified by our knowledge that
(for the same reason as for three-wave case) all
dependent terms will not contribute to the final result. Indeed,
any term with nonzero would have less summations and therefore
will be of lower order. On the diagrammatic language that would mean
that all diagrams with non-zero valence can be discarded, as it was explained
in the previous section. Below, we will only keep the leading order terms
which are and wich correspond to the zero-valence diagrams.
Thus, to calculate we start with the terms
(47),(48),(49),(50),(51) in which we put and
substutute into them the values of and
from (37) and (38).
For the terms proportional to we have

where we have used the fact that . We see that our choice of the frequency renormalisation (16) makes the contribution of equal to zero, and this is the main reason for making this choice. Also, this choice of the frequency correction simplifies the order and, most importantly, eliminates from them the cumulative growth, e.g.

Finally we get,

(65) |

By putting everything together in (45) and (46) and explointing the symmeteries introduced by the summation variables, we finally obtain in limit followed by

By applying to this equation the inverse Laplace transform, we get a four-wave analog of the PBP equation for the -mode PDF:

where

This equation can be easily written in the continuity equation form (65) with the flux given in this case by