  We consider the propagation of short waves which generate waves of much
longer (infinite) wave-length. Model equations of such long wave-short wave
resonant interaction, including integrable ones, are well-known and have
received much attention because of their appearance in various physical
contexts, particularly fluid dynamics and plasma physics. Here we introduce a
new long wave-short wave integrable model which generalises those first
proposed by Yajima-Oikawa and by Newell. By means of its associated Lax pair,
we carry out the linear stability analysis of its continuous wave solutions by
introducing the stability spectrum as an algebraic curve in the complex plane.
This is done starting from the construction of the eigenfunctions of the
linearised long wave-short wave model equations. The geometrical features of
this spectrum are related to the stability/instability properties of the
solution under scrutiny. Stability spectra for the plane wave solutions are
fully classified in the parameter space together with types of modulational
instabilities.
