N-Wave Equations with Orthogonal Algebras: Z_2 and Z_2 × Z_2 Reductions and Soliton Solutions

Abstract

We consider N-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z_2-reduction is the canonical one. We impose a second 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 different 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 B_2 algebra with a canonical Z_2 reduction