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SIGMA, 2011, Volume 7, 096, 48 pages (Mi sigma654)  

This article is cited in 20 scientific papers (total in 20 papers)

Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

Vladimir S. Gerdjikova, Georgi G. Grahovskiab, Alexander V. Mikhailovc, Tihomir I. Valcheva

a Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784, Bulgaria
b School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
c Applied Math. Department, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK

Abstract: A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the ‘squared solutions’ (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.

Keywords: reduction group, Riemann–Hilbert problem, spectral decompositions, integrals of motion.

DOI: https://doi.org/10.3842/SIGMA.2011.096

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Full text: http://emis.mi.ras.ru/.../096
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Bibliographic databases:

ArXiv: 1108.3990
Document Type: Article
MSC: 37K20; 35Q51; 74J30; 78A60
Received: May 26, 2011; in final form October 4, 2011; Published online October 20, 2011
Language: English

Citation: Vladimir S. Gerdjikov, Georgi G. Grahovski, Alexander V. Mikhailov, Tihomir I. Valchev, “Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces”, SIGMA, 7 (2011), 096, 48 pp.

Citation in format AMSBIB
\Bibitem{GerGraMik11}
\by Vladimir S. Gerdjikov, Georgi G. Grahovski, Alexander V. Mikhailov, Tihomir I. Valchev
\paper Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on {\bf A.III}-type Symmetric Spaces
\jour SIGMA
\yr 2011
\vol 7
\papernumber 096
\totalpages 48
\mathnet{http://mi.mathnet.ru/sigma654}
\crossref{https://doi.org/10.3842/SIGMA.2011.096}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2861180}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296010300001}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Alexandar B. Yanovski, Gaetano Vilasi, “Geometric Theory of the Recursion Operators for the Generalized Zakharov–Shabat System in Pole Gauge on the Algebra $\mathrm{sl}(n,\mathbb C)$ with and without Reductions”, SIGMA, 8 (2012), 087, 23 pp.  mathnet  crossref  mathscinet
    2. Grahovski G.G., “The Generalised Zakharov-Shabat System and the Gauge Group Action”, J. Math. Phys., 53:7 (2012), 073512  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Gerdjikov, V., Grahovski, G., Mikhailov, A., Valchev, T., “On soliton interactions for the hierarchy of a generalised Heisenberg ferromagnetic model on Su(3)/S(U(1)+U(2)) symmetric space”, Journal of Geometry and Symmetry in Physics, 25 (2012), 23–55  crossref  mathscinet  zmath
    4. Yanovski, A.B., “Geometry of the recursion operators for Caudrey-Beals-Coifman system in the presence of Mikhailov type ?p reductions”, Journal of Geometry and Symmetry in Physics, 25 (2012), 77–97  crossref  mathscinet  zmath  isi
    5. F. Khanizadeh, A. V. Mikhailov, Jing Ping Wang, “Darboux transformations and recursion operators for differential–difference equations”, Theoret. and Math. Phys., 177:3 (2013), 1606–1654  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Valchev, T.I., “On the quadratic bundles related to Hermitian symmetric spaces”, Journal of Geometry and Symmetry in Physics, 29 (2013), 83–110  crossref  mathscinet  zmath  scopus
    7. Yanovski, A., “Recursion Operators and expansions over adjoint solutions for the Caudrey–Beals–Coifman system with ZP reductions of Mikhailov type”, Journal of Geometry and Symmetry in Physics, 30 (2013), 105–120  crossref  mathscinet  zmath  scopus
    8. Valchev T., “Remarks on Quadratic Bundles Related to Hermitian Symmetric Spaces”, Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP 2013), Journal of Physics Conference Series, 482, IOP Publishing Ltd, 2014, 012044  crossref  isi  scopus
    9. Yanovski A., “Locality of the Conservation Laws For the Soliton Equations Related to Caudrey-Beals-Coifman System Via the Theory of Recursion Operators”, Proceedings of the Fifteenth International Conference on Geometry, Integrability and Quantization, eds. Mladenov I., Ludu A., Yoshioka A., Inst Biophysics & Biomedical Engineering Bulgarian Acad Sciences, 2014, 292–308  mathscinet  isi
    10. Gerdjikov V.S., Yanovski A.B., “Cbc Systems With Mikhailov Reductions By Coxeter Automorphism: i. Spectral Theory of the Recursion Operators”, Stud. Appl. Math., 134:2 (2015), 145–180  crossref  mathscinet  zmath  isi  elib  scopus
    11. Valchev T., “on Mikhailov'S Reduction Group”, Phys. Lett. A, 379:34-35 (2015), 1877–1880  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    12. Yanovski A., “Recursion Operators For Rational Bundle on Sl (3; C) With Z(2) X Z(2) X Z(2) Reduction of Mikhailov Type”, Proceedings of the Sixteenth International Conference on Geometry, Integrability and Quantization, eds. Mladenov I., Ludu A., Yoshioka A., Inst Biophysics & Biomedical Engineering Bulgarian Acad Sciences, 2015, 301–311  crossref  mathscinet  isi
    13. Valchev T.I., “Dressing Method and Quadratic Bundles Related To Symmetric Spaces. Vanishing Boundary Conditions”, J. Math. Phys., 57:2 (2016), 021508  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Habibullin I.T., Khakimova A.R., Poptsova M.N., “On a method for constructing the Lax pairs for nonlinear integrable equations”, J. Phys. A-Math. Theor., 49:3 (2016), 035202  crossref  mathscinet  zmath  isi  scopus
    15. Yanovski A., “Some Aspects of the Spectral Theory For Sl (3, C) System With Z(2) X Z(2) X Z(2) Reduction of Mikhailov Type With General Position Boundary Conditions”, Proceedings of the Seventeenth International Conference on Geometry, Integrability and Quantization, eds. Mladenov I., Meng G., Yoshioka A., Inst Biophysics & Biomedical Engineering Bulgarian Acad Sciences, 2016, 379–391  crossref  mathscinet  isi
    16. Yanovski A.B., Valchev T.I., “Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. i. Auxiliary System and Fundamental Properties”, J. Nonlinear Math. Phys., 25:2 (2018), 324–350  crossref  mathscinet  isi
    17. Valchev I T., Yanovski A.B., “Pseudo-Hermitian Reduction of a Generalized Heisenberg Ferromagnet Equation. II. Special Solutions”, J. Nonlinear Math. Phys., 25:3 (2018), 442–461  crossref  mathscinet  zmath  isi  scopus
    18. G. G. Grahovski, A. Mohammed, H. Susanto, “Nonlocal reductions of the Ablowitz–Ladik equation”, Theoret. and Math. Phys., 197:1 (2018), 1412–1429  mathnet  crossref  crossref  adsnasa  isi  elib
    19. 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  mathnet  crossref  crossref  adsnasa  isi  elib
    20. Yanovski A.B., Valchev T.I., “Hermitian and Pseudo-Hermitian Reduction of the Gmv Auxiliary System. Spectral Properties of the Recursion Operators”, Advanced Computing in Industrial Mathematics (Bgsiam 2017), Studies in Computational Intelligence, 793, eds. Georgiev K., Todorov M., Georgiev I., Springer International Publishing Ag, 2019, 433–446  crossref  mathscinet  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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