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 SIGMA, 2012, Volume 8, 087, 23 pages (Mi sigma764)

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

Alexandar B. Yanovskia, Gaetano Vilasib

a Department of Mathematics & Applied Mathematics, University of Cape Town, Rondebosch 7700, Cape Town, South Africa
b Dipartimento di Fisica, Universitè degli Studi di Salerno, INFN, Sezione di Napoli-GC Salerno, Via Ponte Don Melillo, 84084, Fisciano (Salerno), Italy

Abstract: We consider the recursion operator approach to the soliton equations related to the generalized Zakharov–Shabat system on the algebra $\mathrm{sl}(n,\mathbb C)$ in pole gauge both in the general position and in the presence of reductions. We present the recursion operators and discuss their geometric meaning as conjugate to Nijenhuis tensors for a Poisson–Nijenhuis structure defined on the manifold of potentials.

Keywords: Lax representation; recursion operators; Nijenhuis tensors.

DOI: https://doi.org/10.3842/SIGMA.2012.087

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Bibliographic databases:

ArXiv: 1211.3803
Document Type: Article
MSC: 35Q51; 37K05; 37K10
Received: May 17, 2012; in final form November 5, 2012; Published online November 16, 2012
Language: English

Citation: 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.

Citation in format AMSBIB
\Bibitem{YanVil12}
\by Alexandar~B.~Yanovski, Gaetano~Vilasi
\paper 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
\jour SIGMA
\yr 2012
\vol 8
\papernumber 087
\totalpages 23
\mathnet{http://mi.mathnet.ru/sigma764}
\crossref{https://doi.org/10.3842/SIGMA.2012.087}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3007272}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84870040901}

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This publication is cited in the following articles:
1. Gerdjikov, V.S., Yanovski, A.B., “On soliton equations with Zh and Dh reductions: Conservation laws and generating operators”, Journal of Geometry and Symmetry in Physics, 31 (2013), 57–92
2. 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
3. Gerdjikov V.S., Mladenov D.M., Stefanov A.A., Varbev S.K., “Integrable Equations and Recursion Operators Related To the Affine Lie Algebras a(R)((1))”, J. Math. Phys., 56:5 (2015), 052702
4. Gerdjikov, V. S., “On Kaup-Kupershchmidt-type equations and their soliton solutions”, Il Nuovo Cimento C - Colloquia and communications in physics, 38:5 (2015), 161
5. Yanovski, A.; Vilasi, G., “Recursion Operators for CBC system with reductions. Geometric theory”, NUOVO CIMENTO C-COLLOQUIA AND COMMUNICATIONS IN PHYSICS, 38:5 (2015), 172
6. Gerdjikov, V. S.; Mladenov, D. M.; Stefanov, A. A.; Varbev, S. K., “MKdV-type of equations related to $B^{(1)}_2$ and $A^{(2)}_4$”, Springer Proceedings in Physics, 163 (2015), 59-69
7. 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
8. 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
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