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 SIGMA, 2007, Volume 3, 039, 19 pages (Mi sigma165)

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

Vladimir S. Gerdjikova, Nikolay A. Kostovab, Tihomir I. Valcheva

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

Abstract: 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 diferent 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 $\mathbf B_2$ algebra with a canonical $\mathbb Z_2$ reduction.

Keywords: solitons; Hamiltonian systems

DOI: https://doi.org/10.3842/SIGMA.2007.039

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ArXiv: nlin.SI/0703002
MSC: 37K15; 17B70; 37K10; 17B80
Received: November 21, 2006; in final form February 8, 2007; Published online March 3, 2007
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Citation: Vladimir S. Gerdjikov, Nikolay A. Kostov, Tihomir I. Valchev, “$N$-Wave Equations with Orthogonal Algebras: $\mathbb Z_2$ and $\mathbb Z_2\times\mathbb Z_2$ Reductions and Soliton Solutions”, SIGMA, 3 (2007), 039, 19 pp.

Citation in format AMSBIB
\Bibitem{GerKosVal07} \by Vladimir S.~Gerdjikov, Nikolay A.~Kostov, Tihomir I.~Valchev \paper $N$-Wave Equations with Orthogonal Algebras: $\mathbb Z_2$ and $\mathbb Z_2\times\mathbb Z_2$ Reductions and Soliton Solutions \jour SIGMA \yr 2007 \vol 3 \papernumber 039 \totalpages 19 \mathnet{http://mi.mathnet.ru/sigma165} \crossref{https://doi.org/10.3842/SIGMA.2007.039} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2299840} \zmath{https://zbmath.org/?q=an:1134.37028} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000207065200039} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84889235929} 

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This publication is cited in the following articles:
1. Gerdjikov, VS, “On classification of soliton solutions of multicomponent nonlinear evolution equations”, Journal of Physics A-Mathematical and Theoretical, 41:31 (2008), 315213
2. Gerdjikov V.S., Vilasi G., and Yanovski AB,, “The Inverse Scattering Problem for the Zakharov–Shabat System”, Lecture Notes in Physics, 748, 2008, 97–132
3. Gerdjikov V.S., Vilasi G., and Yanovski AB,, “The Lax Representation and the AKNS Approach”, Lecture Notes in Physics, 748, 2008, 37–70
4. Lombardo S., Sanders J.A., “On the classification of automorphic Lie algebras.”, Communications in Mathematical Physics, 299:3 (2010), 793-824
5. Gerdjikov V.S., Ivanov R.I., Kyuldjiev A.V., “On the N-wave equations and soliton interactions in two and three dimensions”, Wave Motion, 48:8 (2011), 791–804
6. Gerdjikov V.S., “On New Types of Integrable 4-Wave Interactions”, Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings, 1487, ed. Todorov M., Amer Inst Physics, 2012, 272–279
7. V. S. Gerdjikov, G. G. Grahovski, R. I. Ivanov, “The $N$-wave equations with $\mathcal{PT}$ symmetry”, Theoret. and Math. Phys., 188:3 (2016), 1305–1321
8. Gerdjikov V.S., “On Nonlocal Models of Kulish-Sklyanin Type and Generalized Fourier Transforms”, Advanced Computing in Industrial Mathematics, Studies in Computational Intelligence, 681, eds. Georgiev K., Todorov M., Georgiev I., Springer International Publishing Ag, 2017, 37–52
9. G. G. Grahovski, A. J. Mustafa, H. Susanto, “Nonlocal reductions of the multicomponent nonlinear Schrödinger equation on symmetric spaces”, Theoret. and Math. Phys., 197:1 (2018), 1430–1450
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