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 TMF, 2003, Volume 137, Number 1, Pages 27–39 (Mi tmf242)

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in $2+1$ Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

M. S. Bruzóna, M. L. Gandariasa, C. Muriela, J. Ramíresa, F. R. Romerob

b University of Seville

Abstract: One of the more interesting solutions of the $(2+1)$-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in $2+1$ dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the $(2+1)$-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in $2+1$ dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in $2+1$ dimensions. It is interesting that neither of the $(2+1)$-dimensional integrable systems considered admit Virasoro-type subalgebras.

Keywords: partial differential equations, Lie symmetries

DOI: https://doi.org/10.4213/tmf242

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English version:
Theoretical and Mathematical Physics, 2003, 137:1, 1378–1389

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Citation: M. S. Bruzón, M. L. Gandarias, C. Muriel, J. Ramíres, F. R. Romero, “Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in $2+1$ Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions”, TMF, 137:1 (2003), 27–39; Theoret. and Math. Phys., 137:1 (2003), 1378–1389

Citation in format AMSBIB
\Bibitem{BruGanMur03} \by M.~S.~Bruz\'on, M.~L.~Gandarias, C.~Muriel, J.~Ram{\'\i}res, F.~R.~Romero \paper Traveling-Wave Solutions of the Schwarz--Korteweg--de Vries Equation in $2+1$ Dimensions and the Ablowitz--Kaup--Newell--Segur Equation Through Symmetry Reductions \jour TMF \yr 2003 \vol 137 \issue 1 \pages 27--39 \mathnet{http://mi.mathnet.ru/tmf242} \crossref{https://doi.org/10.4213/tmf242} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2048086} \elib{http://elibrary.ru/item.asp?id=13974300} \transl \jour Theoret. and Math. Phys. \yr 2003 \vol 137 \issue 1 \pages 1378--1389 \crossref{https://doi.org/10.1023/A:1026092304047} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000186557700003} 

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• https://doi.org/10.4213/tmf242
• http://mi.mathnet.ru/eng/tmf/v137/i1/p27

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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. Ramirez, J, “Multiple solutions for the Schwarzian Korteweg-de Vries equation in (2+1) dimensions”, Chaos Solitons & Fractals, 32:2 (2007), 682
2. Ozer, T, “New traveling wave solutions to AKNS and SKdV equations”, Chaos Solitons & Fractals, 42:1 (2009), 577
3. Wazwaz A.-M., “N-soliton solutions for shallow water waves equations in (1+1) and (2+1) dimensions”, Applied Mathematics and Computation, 217:21 (2011), 8840–8845
4. Liu Na, Liu Xi-Qiang, “Application of the Binary Bell Polynomials Method to the Dissipative (2+1)-Dimensional AKNS Equation”, Chin. Phys. Lett., 29:12 (2012), 120201
5. Guner O., Bekir A., Karaca F., “Optical Soliton Solutions of Nonlinear Evolution Equations Using Ansatz Method”, Optik, 127:1 (2016), 131–134
6. Wang H., Wang Yu.-H., “Cre Solvability and Soliton-Cnoidal Wave Interaction Solutions of the Dissipative (2+1)-Dimensional AKNS Equation”, Appl. Math. Lett., 69 (2017), 161–167
7. Jadaun V., Kumar S., “Lie Symmetry Analysis and Invariant Solutions of -Dimensional Calogero-Bogoyavlenskii-Schiff Equation”, Nonlinear Dyn., 93:2 (2018), 349–360
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