Four-wave case

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

$${\cal H}={\cal H}_2 + {\cal H}_4 = \sum_{n=1}^\infty \omega_n\ver... ...^2 \sum_{m,n,\mu,\nu=1}^\infty W^{lm}_{\mu\nu} \bar c_{l} \bar c_m c_\mu c_\nu.$$

As in three-wave case the most convenient form of equation of motion is obtained in interaction representation, $c_l=b_l e^{-i \omega_l t}$, so that

$$i \dot b_l = \epsilon \sum_{\alpha\mu\nu} { W^{l\alpha}_{\mu\nu}}... ... b_\alpha b_\mu b_\nu e^{i\omega^{l\alpha}_{\mu\nu}t} \delta^{l\alpha}_{\mu\nu}$$ (11)

where $W^{l \alpha} _{\mu \nu} \sim 1$ is an interaction coefficient, $\omega^{l\alpha}_{\mu\nu} = \omega_l+\omega_{\alpha}-\omega_{\mu}-\omega_{\nu}$. We are going expand in $\epsilon$ 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

$$i \dot a_l = \epsilon \sum_{\alpha\mu\nu} { W^{l\alpha}_{\mu\nu}}... ...e^{i\tilde\omega^{l\alpha}_{\mu\nu}t} \delta^{l\alpha}_{\mu\nu} - \Omega_l a_l,$$ (12)

where $a_l = b_l e^{i \Omega_l t} \;\;$, $\tilde \omega^{l\alpha}_{\mu\nu} = \omega^{l\alpha}_{\mu\nu} +\Omega_l+\Omega_\alpha-\Omega_\mu-\Omega_\nu$, and

$$\Omega_l = 2 \epsilon \sum_\mu W^{l \mu}_{l \mu} A_\mu^2$$ (13)

is the nonlinear frequency shift arising from self-interactions.