N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions
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 equation...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2007 |
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Інститут математики НАН України
2007
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| Zitieren: | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions / V.S. Gerdjikov, N.A. Kostov, T.I. Valchev // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862642386700599296 |
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| author | Gerdjikov, V.S. Kostov, N.A. Valchev, T.I. |
| author_facet | Gerdjikov, V.S. Kostov, N.A. Valchev, T.I. |
| citation_txt | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions / V.S. Gerdjikov, N.A. Kostov, T.I. Valchev // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
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| first_indexed | 2025-12-01T06:40:06Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147823 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T06:40:06Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
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| spelling | Gerdjikov, V.S. Kostov, N.A. Valchev, T.I. 2019-02-16T08:53:20Z 2019-02-16T08:53:20Z 2007 N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions / V.S. Gerdjikov, N.A. Kostov, T.I. Valchev // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 25 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K15; 17B70; 37K10; 17B80 https://nasplib.isofts.kiev.ua/handle/123456789/147823 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. This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. This work is partially supported by a contract 1410 with the National Science Foundation of Bulgaria. This work has been supported also by the programme “Nonlinear Phenomena in Physics and Biophysics”, contract 1879. We also thank the referees for the careful reading of our paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions Article published earlier |
| spellingShingle | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions Gerdjikov, V.S. Kostov, N.A. Valchev, T.I. |
| title | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions |
| title_full | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions |
| title_fullStr | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions |
| title_full_unstemmed | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions |
| title_short | N-Wave Equations with Orthogonal Algebras: Z₂ and Z₂ × Z₂ Reductions and Soliton Solutions |
| title_sort | n-wave equations with orthogonal algebras: z₂ and z₂ × z₂ reductions and soliton solutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147823 |
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