Consider a weakly nonlinear wavefield dominated by the 4-wave interactions, e.g.
the water-surface gravity waves [1,6,29,28],
Langmuir waves in plasmas [1,3] and the waves described by the
nonlinear Schroedinger equation [7].
The a Hamiltonian is given by (in the appropriately chosen variables)
as

As in three-wave case the most convenient form of equation of motion is obtained in interaction representation, , so that

where is an interaction coefficient, . We are going expand in and consider the long-time behaviour of a wave field, but it will turn out that to do the perturbative expansion in a self-consistent manner we have to renormalise the frequency of (14) as

where , , and

is the nonlinear frequency shift arising from self-interactions.