I have a system of coupled PDEs which have both time and spatial dependence with the form:
\begin{align*} a_t &= D_aa_{xx}+f(a, b) \\ b_t &= D_bb_{xx}+g(a, b) \end{align*}
where $f(a, b)$ and $g(a, b)$ are linear functions in terms of $a$ and $b$ and $D_a$ and $D_b$ are diffusion coefficients.
When there is no spatial dependence I can easily calculate the fixed points, find the Jacobian and determine the stability of the fixed points.
My question is then, how can I generalise this method when I have spatial dependence too? I was told I could Fourier Transform the system and turn it into a set of ODEs so that I have
\begin{align*} \tilde{a}_t &= -k^2D_a\tilde{a}+f(\tilde{a}, \tilde{b}) \\ \tilde{b}_t &= -k^2D_b\tilde{b}+g(\tilde{a}, \tilde{b}) \end{align*}
and find the Jacobian of this. However, I struggle to see how finding the Jacobian of the Fourier transformed equations could be related to the stability of my system? I was also told that the Jacobian of my FT equations is equal to the Jacobian of my non transformed system though I can't figure out how.
Also:
$$\tilde{a}(k, t) = \int a(x, t)e^{ikx}\,\mathrm{d}x$$