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 TMF, 1994, Volume 99, Number 2, Pages 292–299 (Mi tmf1589)

The generalized Zakharov–Shabat system and the soliton perturbations

V. S. Gerdjikov

Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences

Abstract: The nonlinear evolution equations and their inhomogeneous versions related through the inverse scattering method to the generalized Zakharov–Shabat system $L=i d/dx + q(x) -\lambda J$ are studied. Here we assume that the potential $q(x)=[J,Q(x)]$ takes values in the simple Lie algebra $\mathfrak {g}$ and that $J$ is a nonregular element of the Cartan subalgebra $\mathfrak {h}$. The corresponding systems of equations for the scattering data of $L$ are derived. These can be applied to the study of soliton perturbations of such equations as the matrix nonlinear Schrödinger equation, the matrix $n$–wave equations etc.

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English version:
Theoretical and Mathematical Physics, 1994, 99:2, 593–598

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Language: English

Citation: V. S. Gerdjikov, “The generalized Zakharov–Shabat system and the soliton perturbations”, TMF, 99:2 (1994), 292–299; Theoret. and Math. Phys., 99:2 (1994), 593–598

Citation in format AMSBIB
\Bibitem{Ger94} \by V.~S.~Gerdjikov \paper The generalized Zakharov--Shabat system and the soliton perturbations \jour TMF \yr 1994 \vol 99 \issue 2 \pages 292--299 \mathnet{http://mi.mathnet.ru/tmf1589} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1308791} \zmath{https://zbmath.org/?q=an:0850.35108} \transl \jour Theoret. and Math. Phys. \yr 1994 \vol 99 \issue 2 \pages 593--598 \crossref{https://doi.org/10.1007/BF01016144} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PV07100016} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. S. Gerdjikov, G. G. Grahovski, N. A. Kostov, “Multicomponent NLS-Type Equations on Symmetric Spaces and Their Reductions”, Theoret. and Math. Phys., 144:2 (2005), 1147–1156
2. Kostov N.A., Atanasov V.A., Gerdjikov V.S., Grahovski G.G., “On the soliton solutions of the spinor Bose–Einstein condensate”, 14th International School on Quantum Electronics: Laser Physics and Applications, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), 6604, 2007, T6041–T6041
3. Grahovski, G, “Generalised Fourier Transform and Perturbations to Soliton Equations”, Discrete and Continuous Dynamical Systems-Series B, 12:3 (2009), 579
4. V. S. Gerdjikov, G. G. Grahovski, “Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory”, SIGMA, 6 (2010), 044, 29 pp.
5. Grecu D., Visinescu A., Fedele R., De Nicola S., “Periodic and Stationary Wave Solutions of Coupled NLS Equations”, Romanian J Phys, 55:5–6 (2010), 585–600
6. 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, AIP Conference Proceedings, 1301, 2010, 561–572
7. 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.
8. Gerdjikov V.S., “On Soliton Interactions of Vector Nonlinear Schrodinger Equations”, Application of Mathematics in Technical and Natural Sciences, AIP Conference Proceedings, 1404, 2011
9. Grahovski G.G., “The Generalised Zakharov-Shabat System and the Gauge Group Action”, J. Math. Phys., 53:7 (2012), 073512
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