Let's assume set of equations $$ \tag 1 \frac{\partial \mathbf A}{\partial t} = \Delta \mathbf A + a [\nabla \times \mathbf A] - d\mathbf b_{k} (\mathbf b_{k} \cdot \mathbf A), \quad \mathbf A(0) = 0 $$ Here $$ \mathbf b_{k} = (sin(kz), cos(kz), 0), \quad \Delta = \nabla \cdot \nabla . $$ I only need to find how $\mathbf A$ grows with $t$.
I have tried to solve this equation by making Fourier transformation on spatial components. By using definition $F[g](\mathbf p)$ as Fourier transform of $g$ I get $$ \tag 2 \partial_{t} \mathbf A = -p^{2} \mathbf A + ia [\mathbf p \times \mathbf A] - d F\left[ \mathbf b_{k}(\mathbf A \cdot \mathbf b_{k})\right](\mathbf p). $$ The last summand can be written in a form $$ F[\mathbf b_{k} A^{j} b_{k}^{j}](\mathbf p) = \int d^{3}\mathbf l F[\mathbf b_{k}b_{k}^{j}](\mathbf p - \mathbf l) A^{j}(\mathbf l ). $$ So by rewriting $\mathbf A(p) = \int d^{3}\mathbf l \delta (\mathbf l - \mathbf p) \mathbf A(\mathbf l)$ etc, I get something like $$ \tag 3 \delta (\mathbf p - \mathbf l)\sigma \partial_{0}\mathbf A (\mathbf l) = -\delta (\mathbf l -\mathbf p) \left( l^{2} \mathbf A (\mathbf l) - ik[\mathbf l \times \mathbf A(l)] \right) - F[\mathbf b_{k}b_{k}^{j}](\mathbf p - \mathbf l) A^{j}(\mathbf l). $$ $(3)$ is only other form of $(2)$.
Then I've tried to analyze $(3)$. By using explicit form of $b_{k}^{i}$, I can conclude that $F[b^{i}_{k}b^{j}_{k}](\mathbf p - \mathbf l)$ contains only Dirac delta-functions. For example, $$ F[b^{1}_{k}b^{1}_{k}](\mathbf p - \mathbf l) = F[sin^{2}(kz)](\mathbf p -\mathbf l) $$ takes the form $$ -\frac{1}{4}\delta (p_{x} - l_{x})\delta (p_{y} - l_{y})\left[\delta (p_{z} - l_{z} - 2k) + \delta (p_{z} - l_{z} + 2k) - \delta(p_{z} - l_{z})\right], $$ etc. From here I have concluded that when (for example) $p_{z} - l_{z} = 2k$, coefficient near corresponding delta-function must be equal to zero. This statement leads to the equalities like $A_{x} + iA_{y} = A_{x} - iA_{y} = 0$, from which I've got $A_{x} = A_{y} = A_{z} = 0$.
Is this analysis correct?