The $\epsilon^0$ order -

As in all WKB based theories we first derive a linear dispersion relationship from the lowest order terms. At zeroth order in ${\bf \epsilon}$, the spatial derivative of a Gabor transform is $\nabla\hat{a} = i \vec{k}\hat{a}$ which is similar to the corresponding rule in Fourier calculus. Then, at the lowest order, equations () and () become

$$\partial_{t}\hat{a} - k^{2} \hat{b} =0,$$ (41)

$$\partial_{t}\hat{b} + k^{2} \hat{a} + 2\varrho\hat{a} =0.$$ (42)

These two linear coupled equations make up an eigenvalue problem. Diagonalizing these equations we obtain

$$\partial_{t}\lambda = +i\omega\lambda, \partial_{t}\mu = -i \omega\mu.$$

Correspondingly, we find the eigenvectors

$$\lambda = \frac{1}{2}\left(\hat{a} - \frac{i k^{2}}{\omega}\hat{b}\right), \mu = \frac{1}{2}\left(\hat{a} + \frac{i k^{2}}{\omega}\hat{b}\right),$$ (43)

or, re-arranging for $\hat{a}$ and $\hat{b}$

$$\hat{a} = \lambda + \mu, \hat{b} = \frac{i\omega}{k^{2}}(\lambda-\mu).$$ (44)

The eigenvalues are given by the dispersion relationship,

$$\omega^{2} = k^{2}(k^{2}+2\varrho),$$ (45)

which is identical to the famous Bogoliubov form [21] which was also obtained for waves on a homogeneous condensate in the weak turbulence context in [13].

Therefore, at the zeroth order, we see that $\lambda$ rotates with frequency $-\omega$ and $\mu$ rotates at $+\omega$. Note that the $\lambda$ and $\mu$ are related via

$$\lambda^{*}(\vec{k}) = \mu (-\vec{k}) .$$ (46)