F$ are given as: $$\begin{aligned}
    \frac{\partial}{\partial \omega }
    [F(\alpha, \omega,c)]_k 
    =& \;
    i k ( \alpha^{-1}  -    e^{ - i \omega k }) c_k 
    -i   \sum_{j=1}^{k-1} j e^{-i \omega j } c_j c_{k-j}  \nonumber \\ &
    \;- i  \sum_{j=1}^{\infty} \left( (j+k)e^{-i \omega (j+k) }  -j e^{i \omega j } \right) c_j^* c_{j+k}   . \\
    \frac{\partial}{\partial a_n} [F(\alpha,\omega,c)]_k 
    =& \;
        ( i \tfrac{\omega}{\alpha}  k + e^{ - i \omega k })+
    \begin{cases}
    \gamma_1 c_{n+k} + \gamma_2  c_{k-n}  & \mbox{if } 1\leq n < k \\
    \gamma_1 c_{n+k} + \gamma_2  c_{n-k}^* 
    & \mbox{if } k \leq n . 
    \end{cases} \\
                \frac{1}{i} \frac{\partial}{\partial b_n} [F(\alpha,\omega,c)]_k 
    =& \;
    ( i \tfrac{\omega}{\alpha}  k + e^{ - i \omega k })+
    \begin{cases}
    -\gamma_1  c_{n+k} + \gamma_2  c_{k-n}  & \mbox{if } 1 \leq n < k \\
    -\gamma_1 c_{n+k} + \gamma_2  c_{n-k}^* 
    & \mbox{if } k \leq  n. 
    \end{cases} 
    \end{aligned}$$

Decomposition of Phase Space
----------------------------

By working in a space of rapidly decaying Fourier coefficients, we are able to closely approximate the value of $F$ using a Galerkin projection. Since $F: {\mathbb{R}}^2 \times  {\Omega}^{s} \to {\Omega}^{s-1}$ has distinct domain and range, we need to define two sets of projection maps. We define projection maps $ \pi_\alpha ,\pi_{\omega} 