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 SIGMA, 2008, Volume 4, 029, 30 pages (Mi sigma282)

Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type

Vladimir S. Gerdjikov, Nikolay A. Kostov

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Abstract: New reductions for the multicomponent modified Korteweg–de Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived using the approach based on the reduction group introduced by A. V. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann–Hilbert problem. The minimal sets of scattering data $\mathcal T_i$, $i=1,2$ which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on $\mathcal T_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak g\simeq so(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to $\mathbb Z_2$, $\mathbb Z_3$, $\mathbb Z_4$.

Keywords: multicomponent modified Korteweg–de Vries (MMKdV) equations; reduction group; Riemann–Hilbert problem; Hamiltonian structures

DOI: https://doi.org/10.3842/SIGMA.2008.029

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ArXiv: 0803.1651
MSC: 37K20; 35Q51; 74J30; 78A60
Received: December 14, 2007; in final form February 27, 2008; Published online March 11, 2008
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Citation: Vladimir S. Gerdjikov, Nikolay A. Kostov, “Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type”, SIGMA, 4 (2008), 029, 30 pp.

Citation in format AMSBIB
\Bibitem{GerKos08} \by Vladimir S.~Gerdjikov, Nikolay A.~Kostov \paper Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type \jour SIGMA \yr 2008 \vol 4 \papernumber 029 \totalpages 30 \mathnet{http://mi.mathnet.ru/sigma282} \crossref{https://doi.org/10.3842/SIGMA.2008.029} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2393298} \zmath{https://zbmath.org/?q=an:1157.37335} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000267267800029} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84857325117} 

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This publication is cited in the following articles:
1. V. S.. Gerdjikov, “On Reductions of Soliton Solutions of Multi-component NLS Models and Spinor Bose–Einstein Condensates”, Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings, 1186, 2009, 15–27
2. V. S. Gerdjikov, G. G. Grahovski, “Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory”, SIGMA, 6 (2010), 044, 29 pp.
3. Gerdjikov V.S., Grahovski G.G., “Two soliton interactions of BD.I multicomponent NLS equations and their gauge equivalent”, Application of Mathematics in Technical and Natural Sciences, Proceedings of the 2nd International Conference, AIP Conference Proceedings, 1301, 2010, 561–572
4. Gerdjikov V.S., “On Soliton Interactions of Vector Nonlinear Schrodinger Equations”, Application of Mathematics in Technical and Natural Sciences, 3rd International Conference - Amitans'11, AIP Conference Proceedings, 1404, 2011
5. Sun X., Wang Y., “KdV Geometric Flows on Kahler Manifolds”, Internat J Math, 22:10 (2011), 1439–1500
6. Sun X.W., Wang Y.D., “Geometric Schrodinger-Airy Flows on Kahler Manifolds”, Acta. Math. Sin.-English Ser., 29:2 (2013), 209–240
7. Han, JW; Yu, J; He, JS, “A Matrix Lie Superalgebra and Its Applications”, Advances in Mathematical Physics, 2013, 416520
8. Sun XiaoWei, Wang YouDe, “New Geometric Flows on Riemannian Manifolds and Applications To Schrodinger-Airy Flows”, Sci. China-Math., 57:11 (2014), 2247–2272
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