Asymptotic expansion of the generating functional.

Let us first obtain an asymptotic weak-nonlinearity expansion for the generating functional $Z\{\lambda, \mu\}$ exploiting the separation of the linear and nonlinear time scales. ³ To do this, we have to calculate $Z$ at the intermediate time $t=T$ via substituting into it $a_j(T)$ from (32) For the amplitude and phase ``ingredients'' in $Z$ we have,

$$e^{\lambda_j \vert a_j\vert^2}= e^{\lambda_j \vert a_j^{(0)}+{\ep... ...ambda_j A_j^{(0) 2}} ( 1+{\epsilon}{\alpha_{1j}} + {\epsilon}^2 {\alpha_{2j}}),$$ (33)

and

$$\psi_j^{\mu_j}=\left(\frac{a_j^{(0)}+{\epsilon}a_j^{(1)}+{\epsilo... ...}}= = \psi_j^{(0)\mu_j} (1+{\epsilon}{\beta_{1j}} +{\epsilon}^2 {\beta_{2j}} ),$$ (34)

where

$${\alpha_{1j}} = \lambda_j(a_j^{(1)}\bar a_j^{(0)}+ \bar a_j^{(1)} a_j^{(0)}),$$ (35)

$${\alpha_{2j}} = {(\lambda_j +\lambda_j^2 A_j^{(0)2}\vert a_j^{(1)}\vert^2 +\lambd... ...frac{\lambda_j^2}{2}(a_j^{(1)}\bar a_j^{(0)})^2 + (\bar a_j^{(1)}a_j^{(0)})^2},$$ (36)

$${\beta_{1j}} = \frac{\mu_j}{2A_j^{(0)2}}(a_j^{(1)}\bar a_j^{(0)}-\bar a_j^{(1)} a_j^{(0)}),$$ (37)

$${\beta_{2j}} = \frac{\mu_j}{2A_j^{(0)2}}(a_j^{(2)}\bar a_j^{(0)}-\bar a_j^{(2)}a... ... a_j^{(0)}}\right)^2 \right]-\frac{\mu_j^2\vert a_j^{(1)}\vert^2}{4A_j^{(0)2}}.$$ (38)

Substituting expansions (39) and (40) into the expression for $Z$, we have

$$Z\{\lambda, \mu, T\} = X\{\lambda, \mu,T\} + \bar X \{\lambda, - \mu,T\}$$ (39)

with

$$X\{\lambda, \mu,T\} = X\{\lambda, \mu,0\} + (2 \pi)^{2N} \left<\p... ...^2}[{\epsilon}J_1 +{\epsilon}^2(J_2 +J_3+J_4+J_5)] \right>_A + O({\epsilon}^4),$$ (40)

where

$$J_1 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j +\frac{\mu_j}{2\vert a_j^{(0)}\vert^2})a_j^{(1)}\bar a_j^{(0)}\right>_\psi,$$ (41)

$$J_2 = {1 \over 2} \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j+ \l... ...^2-\frac{\mu_j^2}{2\vert a_j^{(0)}\vert^2})\vert a_j^{(1)}\vert^2 \right>_\psi,$$ (42)

$$J_3 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j (\lambda_j + \frac{\mu_j}{2\vert a_j^{(0)}\vert^2})a_j^{(2)}\bar a_j^{(0)} \right>_\psi,$$ (43)

$$J_4 = \left<\prod_l \psi_l^{(0)\mu_l} \sum_j \left[\frac{\lambda_j^2}{2... ...mu_j}{2\vert a_j^{(0)}\vert^2} \right](a_j^{(1)}\bar a_j^{(0)})^2 \right>_\psi,$$ (44)

$$J_5 = {1 \over 2} \left<\prod_l \psi_l^{(0)\mu_l} \sum_{j \ne k}\lambda... ...)}\bar a_k^{(0)} -\bar a_k^{(1)}a_k^{(0)})a_j^{(1)}\bar a_j^{(0)} \right>_\psi,$$ (45)

where $\left< \cdot \right>_A$ and $\left< \cdot \right>_\psi$ denote the averaging over the initial amplitudes and initial phases (which can be done independently). Note that so far our calculation for $Z(T)$ is the same for the three-wave and for the four-wave cases. Now we have to substitute expressions for $a^{(1)}$ and $a^{(2)}$ which are different for the three-wave and the four-wave cases and given by (34), (36) and (34), (36) respectively.