  The stability of Marangoni roll convection in a liquid-gas system with
deformable interface is studied in the case when there is a nonlinear
interaction between two modes of Marangoni instability: long-scale surface
deformations and short-scale convection. Within the framework of a model
derived previously, it is shown that the nonlinear interaction between the two
modes substantially changes the width of the band of stable wave numbers of the
short-scale convection pattern as well as the type of the instability limiting
the band. Depending on the parameters of the system, the instability can be
either long- or short-wave, either monotonic or oscillatory. The stability
boundaries strongly differ from the standard ones and sometimes exclude the
band center. The long-wave limit of the side-band instability is studied in
detail within the framework of the phase approximation. It is shown that the
monotonic instability is always subcritical, while the long-wave oscillatory
instability can be supercritical, leading to the formation of either travelling
or standing waves modulating the pattern wavelength and the surface
deformation.
