On N-wave equations and their gauge equivalent

The class of nonlinear evolution equations (NLEE) – gauge equivalent to the $N$-wave equations related to the simple Lie algebra $g$ are derived and analyzed. The corresponding Lax pairs and the time evolution of the scattering data are found. The Zakharov–Shabat dressing method is appropriately modified to construct their soliton solutions. Several examples including ones describing isoparametric hypersurfaces are presented. The hierarchy of the Hamiltonian structures to the gauge equivalent systems to the $N$-wave ones is also discussed.