Аннотации:
We consider some nonlinear models describing resonance interactions of long
waves and short-waves (shortly, the LS waves models). Such LS models were derived and
proposed due to various motivations, which mainly come from the different branches of
modern physics, especially, from the fluid and plasma physics. In this paper, we study
some of integrable LS models, namely, the Yajima-Oikawa equation, the Newell equation,
the Ma equation, the Geng-Li equation and their different modifications and extensions.
In particular, the gauge equivalent counterparts of these integrable LS models (equations),
namely, different integrable spin systems are constructed. In fact, these gauge equivalent
counterparts of these LS equations are integrable generalized Heisenberg ferromagnet
type models (equations) (HFE) with self-consistent potentials (HFESCP). The associated
Lax representations of these HFESCP are presented. Using these Lax representations
of these HFESCP, they can be studied by the inverse scattering method. For instance,
the equivalence established using the Lax representation also makes it possible to find a
connection between the solutions of the corresponding integrable equations.
