
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
NWave Equations with Orthogonal Algebras: Z_{2} and Z_{2} × Z_{2} 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 Nwave 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
Nwave equations we construct the soliton solutions by
appropriately modifying the ZakharovShabat dressing method. We
also briefly discuss the different types of onesoliton
solutions. Especially rich are the types of onesoliton 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 sineGordon case, which
allows two types of solitons: kinks and breathers. A new physical
system, describing Stokesanti Stokes Raman scattering is
obtained. It is represented by a 4wave equation related to the
B_{2} algebra with a canonical Z_{2} reduction.
Key words:
solitons; Hamiltonian systems.
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