obtained characteristic equation (\[eq: SLpsolCE\]) determines the stability of a plane wave. Namely, for any plane wave, which is defined by the amplitude $a$, frequency $\Omega$, wave-vectors $k_{+}$ and $k_{-}$ (then also $k_{\tau}=k_{+}-\Omega\tau$ is given), the equation (\[eq: SLpsolCE\]) determines the stability with respect to the perturbation mode with the spatial perturbations $q_{+}$ and $q_{-}$. In particular, if for all $q_{+}$, $q_{-}\in[0,2\pi]$, all the solutions $\lambda$ of the characteristic equation (\[eq: SLpsolCE\]) have negative real parts, then the plane wave is asymptotically stable. The symmetry-relation $\chi^{\ast}(\lambda^{*},q_{-},-q_{+};-k_{\tau})=\chi(\lambda,q_{-},q_{+};k_{\tau})$ implies, that the stability properties of periodic solutions are invariant with respect to changing $k_{\tau}\to-k_{\tau}$. Notice that the obtained equation is a quasi-polynomial, which has generically infinitely many roots.

Although Eq. (\[eq: SLpsolCE\]) can be studied numerically for each given set of parameters, an additional useful analytical insight in the properties of its solutions is possible under the assumption of large delay $\tau$. This is performed in the next section.

### Stability of plane wave solutions: large delay approximation

### Strong spectrum {#strong-spectrum .unnumbered}

As it was discussed in Sec. \[sub:ststLong\], the strong spectrum involves only the instantaneous part of the dynamics. Therefore it does not depend on the spatial perturbation modes $\mathbf{q}$ or 