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

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

2007-03-03

arxiv.org/abs/nlin/0703002v1

SIGMA 3 (2007), 039, 19 pages

Nonlinear Sciences

Exactly Solvable and Integrable Systems

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym

Scientific paper

10.3842/SIGMA.2007.039

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 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 ${\bf B}_2$ algebra with a canonical $\mathbb{Z}_2$ reduction.

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