
SIGMA 3 (2007), 039, 19 pages nlin.SI/0703002
https://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.
pdf (277 kb)
ps (201 kb)
tex (20 kb)
References
 Ackerhalt J.R., Milonni P.W., Solitons and fourwave
mixing, Phys. Rev. A 33 (1986), 31853198.
 Armstrong J., Bloembergen N., Ducuing J., Persham P., Interactions
between light waves in a nonlinear dielectric, Phys. Rev.
127 (1962), 19181939.
 Degasperis A., Lombardo S., Multicomponent
integrable wave equations: I. Darbouxdressing transformation,
J. Phys. A: Math. Theor. 40 (2007), 961977,
nlin.SI/0610061.
 Ferapontov E., Isoparametric hypersurfaces in spheres, integrable
nondiagonalizable systems of hydrodynamic type and Nwave
systems, Differential Geom. Appl. 5 (1995), 335369.
 Gerdjikov V.,
On the spectral theory of the integrodifferential operator
L, generating nonlinear evolution equations, Lett.
Math. Phys. 6 (1982), 315324.
 Gerdjikov V.,
Generalised Fourier transforms for the soliton equations. Gauge
covariant formulation, Inverse Problems 2 (1986),
5174.
 Gerdjikov V., Algebraic and analytic aspects of soliton type
equations, Contemp. Math. 301 (2002), 3567,
nlin.SI/0206014.
 Gerdjikov V., Grahovski G., Ivanov R., Kostov N.,
Nwave interactions related to simple Lie algebras.
Z_{2}reductions and soliton solutions, Inverse Problems
17 (2001), 9991015,
nlin.SI/0009034.
 Gerdjikov V., Grahovski G., Kostov N.,
Reductions of Nwave interactions related to lowrank simple Lie
algebras, J. Phys A: Math. Gen. 34 (2001), 94259461,
nlin.SI/0006001.
 Gerdjikov V., Kaup D.,
How many types of soliton solutions do we know? In Proceedings of
Seventh International Conference on Geometry, Integrability and
Quantization (June 210, 2005, Varna, Bulgaria), Editors
I. Mladenov and M. de Leon, Softex, Sofia, 2006, 1134.
 Gerdjikov V., Kostov N.,
Inverse scattering transform analysis of Stokesanti Stokes
stimulated Raman scattering, Phys. Rev. A 54 (1996),
43394350,
pattsol/9502001.
 Gerdjikov V., Valchev T., Breather solutions of Nwave
equations, in Proceedings of Eighth International Conference on
Geometry, Integrability and Quantization (June 914, 2006, Varna,
Bulgaria), Editors I. Mladenov and M. de Leon, to appear.
 Goto M., Grosshans F., Semisimple Lie algebras, Marcel Dekker, New
York, 1978.
 Ivanov R., On the dressing method for the
generalised ZakharovShabat system, Nuclear Phys. B
694 (2004), 509524,
mathph/0402031.
 Matveev V., Salle M., Darboux transformations and solitons,
Springer Verlag, Berlin, 1991.
 Mikhailov A., The Reduction problem and the inverse scattering
method, Phys. D 3 (1981), 73117.
 Rogers C., Schief W.K., Bäcklund and Darboux
transformations, Cambridge University Press, London, 2002.
 Rogers C., Shadwick W., Bäcklund transformations and their
applications, Academic Press, New York, 1982.
 Shabat A., The inverse scattering problem for a system of
differential equations, Funktsional. Anal. i Prilozhen.
9, (1975), no. 3, 7578 (in Russian).
 Shabat A., The inverse scattering problem, Differ. Uravn. 15 (1979), 18241834
(in Russian).
 Takhtadjan L., Faddeev L., The Hamiltonian approach to soliton
theory, Springer Verlag, Berlin, 1987.
 Zakharov V., Mikhailov A., On the integrability of classical
spinor models in twodimensional spacetime, Comm. Math.
Phys. 74 (1980), 2140.
 Zakharov V., Manakov S., On the theory of resonant interactions of
wave packets in nonlinear media, JETP 69 (1975),
no. 5, 16541673.
 Zakharov V., Manakov S., Novikov S., Pitaevskii L., Theory of
solitons: the inverse scattering method, Plenum, New York, 1984.
 Zakharov V., Shabat A., Integration of the nonlinear equations of
mathematical physics by the inverse scattering method,
Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 1322
(in Russian).

