Geodesic
Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in $2+1$ Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions
Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 1, pp. 27-39 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{TMF_2003_137_1_a3,
     author = {M. S. Bruz\'on and M. L. Gandarias and C. Muriel and J. Ram{\'\i}res and F. R. Romero},
     title = {Traveling-Wave {Solutions} of the {Schwarz{\textendash}Korteweg{\textendash}de} {Vries} {Equation} in $2+1$ {Dimensions} and the {Ablowitz{\textendash}Kaup{\textendash}Newell{\textendash}Segur} {Equation} {Through} {Symmetry} {Reductions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {27--39},
     year = {2003},
     volume = {137},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_137_1_a3/}
}
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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. Teoretičeskaâ i matematičeskaâ fizika, Tome 137 (2003) no. 1, pp. 27-39. http://geodesic.mathdoc.fr/item/TMF_2003_137_1_a3/

[1] E. Hille, Analytic Function Theory, V. 2, Ginn, Boston, 1962 ; H. Schwerdtfeger, Geometry of Complex Numbers, Dover, New York, 1979 | MR | Zbl | MR | Zbl

[2] I. M. Krichever, S. P. Novikov, UMN, 35:6 (1980), 47 | MR | Zbl

[3] J. Weiss, J. Math. Phys., 24 (1983), 1405 ; 26, 258 | DOI | MR | Zbl | DOI | Zbl

[4] M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Stud. Appl. Math., 53 (1974), 249 | DOI | MR | Zbl

[5] B. Gambier, Acta Math., 33 (1909), 1 ; P. Painlevé, Bull. Soc. Math. France, 28 (1900), 201 ; Acta Math., 25 (1902), 1 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR

[6] M. J. Ablowitz, A. Ramani, H. Segur, Lett. Nuovo Cimento, 23 (1978), 333 | DOI | MR

[7] R. Conte (ed.), The Painlevé Property\rom: One Century Later, CRM Ser. Math. Phys., Springer, New York, 1999 | MR

[8] M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981 | MR | Zbl

[9] F. Nijhoff, “On some “Schwarzian” equations and their discrete analogues”, Algebraic Aspects of Integrable Systems, In Memory of Irene Dorfman, Prog. Nonlinear Differ. Eq. Appl., 26, eds. A. S. Fokas et al., Birkhäuser, Boston, MA, 1997, 237 | MR | Zbl

[10] K. Toda, S. Yu, J. Math. Phys., 41 (2000), 4747 | DOI | MR | Zbl

[11] J. Weiss, J. M. Tabor, G. Carnevale, J. Math Phys., 24 (1983), 522 | DOI | MR | Zbl

[12] D. David, N. Kamran, D. Levi, P. Winternitz, J. Math. Phys., 27 (1986), 1225 | DOI | MR | Zbl

[13] B. Champagne, P. Winternitz, J. Math. Phys., 29 (1988), 1 | DOI | MR | Zbl

[14] M. Senthil Velan, M. Lakshmanan, J. Nonlinear Math. Phys., 5 (1998), 190 | DOI | MR | Zbl

[15] N. Kudriashov, P. Pickering, J. Phys. A, 31 (1998), 9505 | DOI | MR

[16] J. Ramirez, M. S. Bruzón, C. Muriel, M. L. Gandarias, J. Phys. A, 36 (2003), 1467 | DOI | MR | Zbl

[17] M. L. Gandarias, M. S. Bruzón, J. Ramírez, “Classical symmetries for a Boussinesq equation with nonlinear dispersion”, Symmetry and Perturbation Theory, eds. D. Bambusi, G. Gaeta, M. Cadoni, World Scientific, River Edge, NJ, 2001 ; М. Л. Гандариас, М. С. Брузон, Х. Рамирес, ТМФ, 134:1 (2003), 74 | MR | DOI | MR | Zbl

[18] M. Wadati, J. Phys. Soc. Japan, 34 (1973), 1289 | DOI | MR | Zbl