Integrable Tolkynay equations and related Yajima-Oikawa type equations

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.