  The modulational instability (MI) of plane waves in nonlocal Kerr media is
studied for a general, localized, response function. It is shown that there
always exists a finite number of well-separated MI gain bands, with each of
them characterised by a unique maximal growth rate. This is a general property
and is demonstrated here for the Gaussian, exponential, and rectangular
response functions. In case of a focusing nonlinearity it is shown that
although the nonlocality tends to suppress MI, it can never remove it
completely, irrespectively of the particular shape of the response function.
For a defocusing nonlinearity the stability properties depend sensitively on
the profile of the response function. It is shown that plane waves are always
stable for response functions with a positive-definite spectrum, such as
Gaussians and exponentials. On the other hand, response functions whose spectra
change sign (e.g., rectangular) will lead to MI in the high wavenumber regime,
provided the typical length scale of the response function exceeds a certain
threshold. Finally, we address the case of generalized multi-component response
functions consisting of a weighted sum of N response functions with known
properties.
