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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
![$\displaystyle i \dot b_l = \epsilon \sum_{\alpha\mu\nu} { W^{l\alpha}_{\mu\nu}}...
... b_\alpha b_\mu b_\nu e^{i\omega^{l\alpha}_{\mu\nu}t} \delta^{l\alpha}_{\mu\nu}$](img62.png) |
(11) |
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
![$\displaystyle i \dot a_l = \epsilon \sum_{\alpha\mu\nu} { W^{l\alpha}_{\mu\nu}}...
...e^{i\tilde\omega^{l\alpha}_{\mu\nu}t} \delta^{l\alpha}_{\mu\nu} - \Omega_l a_l,$](img65.png) |
(12) |
where
,
, and
![$\displaystyle \Omega_l = 2 \epsilon \sum_\mu W^{l \mu}_{l \mu} A_\mu^2$](img68.png) |
(13) |
is the nonlinear frequency shift
arising from self-interactions.
Next: Setting the stage II:
Up: Setting the stage I:
Previous: Three-wave case
Dr Yuri V Lvov
2007-01-23