Weak nonlinearity expansion: four-wave case

Substituting (32) in (15) we get in the zeroth order $a_l^{(0)}(T)=a_l(0)$, and the first iteration of (15) gives

$$a_l^{(1)}(T) = - i \sum_{\alpha\mu\nu} W^{l\alpha}_{\mu\nu} \bar ... ... a_\nu \delta^{l\alpha}_{\mu\nu} \Delta^{l \alpha}_{\mu\nu} + i \Omega_l a_l T.$$ (31)

Iterating one more time we get

$$a_l^{(2)} (T) = \sum_{\alpha\mu\nu v u}\left( W^{\mu\nu}_{\alpha u} W^{lu}_{v\bet... ... E(\tilde\omega^{l\mu\nu}_{\alpha v \beta}, \tilde\omega^{lu}_{v\beta}) \right.$$

$$\left. - 2 W^{\alpha v}_{\mu\nu} W^{l u}_{v \beta} \delta^{\alpha... ...\mu\nu\beta}, \tilde \omega^{lu}_{v\beta})\right) -\Omega_l^2 a_l \frac{T^2}{2}$$

$$+ \sum_{\alpha\mu\nu}\left( \Omega_l W^{l\alpha}_{\mu\nu} \delta^... ...nu) \int \limits_0^T \tau e^{i\Omega^{l \alpha}_{\mu\nu } \tau} d \tau \right).$$ (32)