Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 039, 19 pages      nlin.SI/0703002      http://dx.doi.org/10.3842/SIGMA.2007.039
Contribution to the Vadim Kuznetsov Memorial Issue

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

Vladimir S. Gerdjikov a, Nikolay A. Kostov a, b and Tihomir I. Valchev a
a) Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
b) Institute of Electronics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received November 21, 2006, in final form February 08, 2007; Published online March 03, 2007

Abstract
We consider N-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z2-reduction is the canonical one. We impose a second Z2-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 B2 algebra with a canonical Z2 reduction.

Key words: solitons; Hamiltonian systems.

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