$N$-Wave Equations with Orthogonal Algebras: $\mathbb Z_2$ and $\mathbb Z_2\times\mathbb Z_2$ Reductions and Soliton Solutions

We consider $N$-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first $\mathbb Z_2$-reduction is the canonical one. We impose a second $\mathbb Z_2$-reduction and consider also the combined action of both reductions. For all three types of $N$-wave equations we construct the soliton solutions by appropriately modifying the Zakharov–Shabat dressing method. We also briefly discuss the different types of one-soliton solutions. Especially rich are the types of one-soliton solutions in the case when both reductions are applied. This is due to the fact that we have two diferent configurations of eigenvalues for the Lax operator $L$: doublets, which consist of pairs of purely imaginary eigenvalues, and quadruplets. Such situation is analogous to the one encountered in the sine-Gordon case, which allows two types of solitons: kinks and breathers. A new physical system, describing Stokes-anti Stokes Raman scattering is obtained. It is represented by a $4$-wave equation related to the $\mathbf B_2$ algebra with a canonical $\mathbb Z_2$ reduction.