I want to calculate the maximum growth rate from the following dispersion relation: $$ \omega(k) = \frac{1}{\bar{k}}\bigg(k^4(a+b)+k^2(a+b) \bigg) $$ where $a,b$ are constant expressions dependent on the equilibrium states of the variables being perturbed. It is my understanding that to do this you simply find the extremum of the function using basic calculus: $$ \omega'=\frac{2}{\bar{k}}(a+b)k(2k^2+1) $$ So the extremum occur at $k=0,\pm\sqrt{-\frac{1}{2}}$. Now my question is what do I make of this? The wavenumber, $k$, only takes on positive real numbers right? So what does that say about the system?
Note this dispersion relation is for a nonlinear system of PDEs that is fourth-order in space and first-order in time in one variable and second-order in space in the other variable. For now, I am just trying to carry out this analysis in a single spatial dimension, hence the single wavenumber.