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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
E. Hille, Analytic Function Theory, Vol. 2, Ginn, Boston (1962); H. Schwerdtfeger, Geometry of Complex Numbers, Dover, New York (1979).
I. M. Krichever and S. P. Novikov, Russ. Math. Surv., 35, 53 (1980).
J. Weiss, J. Math Phys., 24, 1405 (1983); 26, 258 (1983).
M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Stud. Appl. Math., 53, 249 (1974).
B. Gambier, Acta Math., 33, 1 (1909); P. Painlevé, Bull. Soc. Math. France, 28, 201 (1900); Acta Math., 25, 1 (1902).
M. J. Ablowitz, A. Ramani, and H. Segur, Lett. Nuovo Cimento, 23, 333 (1978).
R. Conte, ed., The Painlevé Property: One Century Later (CRM Series in Mathematical Physics), Springer, New York (1999); M. Musette, “Painlevé analysis for nonlinear partial differential equations,” in: The Painlevé Property: One Century Later (CRM Series in Mathematical Physics, R. Conte, ed.), Springer, New York (1999), p. 517.
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia (1981).
F. Nijhoff, “On some 'Schwarzian' equations and their discrete analogues,” in: Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman (Prog. Nonlinear Differ. Equ. Appl., Vol. 26, A. S. Fokas et al., eds.), Birkhäuser, Boston, Mass. (1997), p. 237.
K. Toda and S. Yu, J. Math. Phys., 41, 4747 (2000).
J. Weiss, J. M. Tabor, and G. Carnevale, J. Math. Phys., 24, 522 (1983).
D. David, N. Kamran, D. Levi, and P. Winternitz, J. Math. Phys., 27, 1225 (1986).
B. Champagne and P. Winternitz, J. Math. Phys., 29, 1 (1988).
M. Senthil Velan and M. Lakshmanan, J. Nonlinear Math. Phys., 5, 190 (1998).
K. Kudriashov and P. Pickering, J. Phys. A, 31, 9505 (1998).
J. Ramirez, M. S. Bruzón, C. Muriel, and M. L. Gandarias, J. Phys. A, 36, 1467 (2003).
M. L. Gandarias, M. S. Bruzón, and J. Ramírez, “Classical symmetries for a Boussinesq equation with nonlinear dispersion,” in: Symmetry and Perturbation Theory (D. Bambusi, G. Gaeta, and M. Cadoni, eds.), World Scientific, River Edge, NJ (2001); M. L. Gandarias, M. S. Bruzón, and J. Ramírez, Theor. Math. Phys., 134, 62 (2003).
M. Wadati, J. Phys. Soc. Japan, 34, 1289 (1973).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bruzón, M.S., Gandarias, M.L., Muriel, C. et al. Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions. Theoretical and Mathematical Physics 137, 1378–1389 (2003). https://doi.org/10.1023/A:1026092304047
Issue Date:
DOI: https://doi.org/10.1023/A:1026092304047