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We consider $N$-wave type equations related to the orthogonal algebrasobtained 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 bothreductions. For all three types of $N$-wave equations we construct the solitonsolutions by appropriately modifying the Zakharov-Shabat dressing method. Wealso briefly discuss the different types of one-soliton solutions. Especiallyrich are the types of one-soliton solutions in the case when both reductionsare applied. This is due to the fact that we have two different configurationsof eigenvalues for the Lax operator $L$: doublets, which consist of pairs ofpurely imaginary eigenvalues, and quadruplets. Such situation is analogous tothe one encountered in the sine-Gordon case, which allows two types ofsolitons: kinks and breathers. A new physical system, describing Stokes-antiStokes Raman scattering is obtained. It is represented by a 4-wave equationrelated to the ${\bf B}_2$ algebra with a canonical $\mathbb{Z}_2$ reduction.

N-Wave Equations with Orthogonal Algebras: Z2 and Z2 × Z2 Reductions and Soliton Solutions

N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions