scaled real part of the leading eigenvalues. Note that this plot is similar to the Master stability function [@Flunkert2010b]. \[pic: SL psol pcs\]](SL_lattice_acs_psol_paper){width="75.00000%"}

The trivial multiplier always denotes a critical point of the pseudo-continuous spectrum at $(\omega=0,q_{-}=0)$, where the gradient vanishes: $$\left.\nabla\gamma_{-}\right|_{\omega=0,q_{-}=0}=\left.\left(\frac{\partial}{\partial\omega},\frac{\partial}{\partial q_{-}}\right)\gamma_{-}\right|_{\omega=0,q_{-}=0}=0.$$ This can be verified by a direct calculation. Therefore the point $(\omega=0,q_{-}=0)$ is either an extremum or saddle of the pseudo-continuous spectrum. Analyzing the shape of the spectrum close to the trivial multiplier shows the appearance of the *modulational instability* [@Cross2009; @Cross1993], see also Fig. \[pic: SL psol pcs\]. For this, let us consider the second order approximation of $\gamma_{-}$ at $(\omega,q_{-})=0$, involving the Hessian matrix $$H=\left.\left(\begin{array}{cc}
\frac{\partial^{2}\gamma_{-}}{\partial\omega^{2}} & \frac{\partial^{2}\gamma_{-}}{\partial\omega\partial q_{-}}\\
\frac{\partial^{2}\gamma_{-}}{\partial\omega\partial q_{-}} & \frac{\partial^{2}\gamma_{-}}{\partial q_{-}^{2}}
\end{array}\right)\right|_{\omega=0,q_{-}=0}.$$ Direct calculation leads to the following expressions for the elements of the Hessian matrix $$\begin{aligned}
\left.\frac{\partial^{2}\gamma_{-}}{\partial\omega^{2}}\right|_{\omega=0,q_{-}=0} & =\frac{1}{R^{2}\cos^{2}{k_{\tau}}}\left(\frac{R}{a^{2}}\frac{\sin^{2}{k_{\tau}}}{\cos{k_{\tau}}}-1\right),\\
\left.\frac{\partial^{2}\gamma_{-}}{\partial q_{-}^{2}}\right|_{\omega=0,q_{-}=0} & =-1+\frac{R\tan^{2}{k_{-}}}{a^{2}\cos^{3}{k_{\tau}}}+\tan^{2}{k_{-}}\tan^{2}{k_{\tau}},\\
\left.\frac{\partial^{2}\gamma_{-}}{\partial\omega\partial q_{-}}\right|_{\omega=0,q_{-}=0} & =\frac{\tan{k_{-}}\tan{k_{\tau}}}{a^{2}\cos^{2}{k_{\tau}}}.\end{aligned}$$ The curvature of the asymptotic continuous spectrum close to the trivial multiplier is directly related to the stability of the corresponding plane wave. If the surface is locally concave close to $(\omega,q_{-})=(0,0)$, then the Hessian is negative definite and the corresponding periodic orbit is stable (at least the part of the spectrum which is 