= Weighted essential spectrum of a Fisher travelling front
{title2=$\gamma=c/2,\quad\sigma_{\mathrm{ess}}=(-\infty,k-c^2/4]$}
The <linearization> at a <travelling wave> $w_c$ is $A=\partial_z^2+c\partial_z+k-2w_c$. Conjugation by $e^{\gamma z}$ changes this to $\partial_z^2+(c-2\gamma)\partial_z+\gamma^2-c\gamma+k-2w_c$. The choice $\gamma=c/2$ removes the first derivative and makes the operator <self-adjoint>. Its two limiting potentials are $-k-c^2/4$ and $k-c^2/4$, so <essential-spectrum invariance under relatively compact perturbations> gives $\sigma_{\mathrm{ess}}=(-\infty,k-c^2/4]$. This lies strictly below zero for $c>2\sqrt{k}$ but touches zero at the critical speed. Stability of this <essential spectrum> alone is not a general nonlinear stability theorem.
Back to article page
