 Bloch stability operator

For a spatially periodic nonlinear pattern, a linear stability disturbance is written $e^{\lambda t+iq\cdot x}v(x)$ with $v$ periodic on the pattern cell. The conjugated operator $L(q)$ acts on that fixed cell and determines the sideband eigenvalues. Translation invariance supplies zero eigenfunctions at $q=0$, namely spatial derivatives of the base pattern. Their small-$q$ eigenvalues determine long-wavelength phase stability; other parts of the spectrum can give independent instabilities.