Abstract
We demonstrate the existence of complex solitary wave and periodic solutions of the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations. The solutions of the KdV (mKdV) equation appear in complex-conjugate pairs and are even (odd) under the simultaneous actions of parity (𝓟) and time-reversal (𝓣) operations. The corresponding localized solitons are hydrodynamic analogs of Bloch soliton in magnetic system, with asymptotically vanishing intensity. The 𝓟𝓣-odd complex soliton solution is shown to be iso-spectrally connected to the fundamental sech2 solution through supersymmetry. Physically, these complex solutions are analogous to the experimentally observed grey solitons of non-liner Schödinger equation, governing the dynamics of shallow water waves and hence may also find physical verification.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Scott-Russel, in Rep. 14th Meeting British Assoc Adv. Sci. (John Murray, London, 1844), p. 311
D.J. Korteweg, de Vries, Phil. Mag. 39, 422 (1895)
P.G. Drazin, R.S. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, 1989)
R.M. Miura, C.S. Gardner, M.D. Kruskal, J. Math. Phys. 9, 1204 (1968)
P.D. Lax, Comm. Pure Appl. Math. 21, 467 (1968)
K. Chadan, P.C. Sabatier, Inverse Problems in Quantum Scattering Theory (Springer-Verlag, New York, 1977)
I. Kay, H.E. Moses, J. Appl. Phys. 27, 1503 (1956)
V.E. Zakharov, S.V. Manakov, S.P. Novikov, L.P. Pitaeveskii, Theory of Solitons: The Inverse Scattering Method (Plenum, New York, 1984)
S.V. Manakov, Zh. Eksp. Teor. Fiz. 67, 543 (1974)
V.E. Zakharov, A.B. Shabat, Func. Anal. Appl. 8, 226 (1974)
R. Hirota, Phys. Rev. Lett. 27, 1192 (1971)
C. Rogers, W.K. Schief, Bäcklund and Darboux Transformations (Cambridge University Press, 2002)
Y. Jiang, B. Tian, W.-J. Liu, M. Li, P. Wang, K. Sun, J. Math. Phys. 51, 093519 (2010)
X.-Y. Gao, J. Math. Phys. 56, 014101 (2015)
X.-Y. Gao, Ocean Eng. 96, 245 (2015)
R.M. Miura, J. Math. Phys. 9, 1202 (1968)
K. Konno, M. Wadati, Prog. Theor. Phys. 53, 1652 (1975)
A.H. Khater, O.H. El-Kakaawy, D.K. Callebaut, Phys. Scr. 58, 545 (1998)
X.-Y. Gao, Europhys. Lett. 110, 15002 (2015)
H. Ono, J. Phys. Soc. Jpn 61, 4336 (1992)
K.R. Helfrich, W.K. Melville, J.W. Miles, J. Fluid Mech. 149, 305 (1984)
A. Chabchoub, N.P. Hoffman, N. Akhmediev, Phys. Rev. Lett. 106, 204502 (2011)
M. Akbari-Moghnjoughi, Phys. Plasmas 22, 032307 (2015)
V. Ziegler, J. Dinkel, C. Setzer, K.E. Lonngren, Chaos Solitons Fractals 12, 1719 (2001)
K.E. Lonngren, Opt. Quantum Electron. 30, 615 (1998)
S. Watanabe, J. Phys. Soc. Jpn 53, 950 (1984)
M. Tajiri, K. Nishihara, J. Phys. Soc. Jpn 54, 572 (2001)
A. Khare, A. Saxena, Phys. Lett. A 377, 2761 (2013)
A. Khare, A. Saxena, J. Math. Phys. 55, 032701 (2014)
V.G. Bar’yakhtar, M.V. Chetkin, B.A. Ivanov, S.N. Gadetskii, Dynamics of Topological Magnetic Soliton-Experiment and Theory (Springer-Verlag, Berlin, 1994)
M. Lakshmanan, Phys. Lett. A 61, 53 (1977)
A.K. Mukhopadhayay, V.M. Vyas, P.K. Panigrahi, Eur. Phys. J. B 88, 188 (2015)
A. Das, Integrable Model (World Scientific, Singapore, 1989)
C.M. Bender, D.C. Brody, H.F. Jones, Phys. Rev. Lett. 89, 270401 (2002)
Z. Ahmed, Phys. Lett. A 282, 343 (2001)
K. Abhinav, P.K. Panigrahi, Ann. Phys. 325, 6 (2011)
C.N. Kumar, J. Phys. A 20, 5397 (1987)
F. Cooper, A. Khare, U.P. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001)
Q. Wang. U.P. Sukhatme, W.-Y. Keung, T.D. Imbo, Mod. Phys. Lett. A 05, 525 (1990)
B. Bagchi, R. Roychoudhury, J. Phys. A 21, L195 (1988)
B. Birnir, SIAM J. Appl. Math. 47, 4 (1987)
A. Das, L. Greenwood, Phys. Lett. B 678, 504 (2009) and references therein
V.G. Knizhnik, A.M. Polyakov, A.B. Zamolodchikov, Mod. Phys. Lett. A 3, 819 (1988)
A.M. Polyakov, Int. J. Mod. Phys. A 5, 883 (1990)
J.-M. Lina, P.K. Panigrahi, Int. J. Mod. Phys. A 7, 27 (1992)
Q. Wang, P.K. Panigrahi, U. Sukhatme, W.-Y. Keung, Nucl. Phys. B 344, 196 (1990) and references therein
K. Pradhan, P.K. Panigrahi, J. Phys. A 39, L343 (2006)
W.-R. Sun, B. Tian, D.-Y. Liu, X.-Y. Xie, J. Phys. Soc. Jpn 84, 074003 (2015)
B. Peng, S.K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G.L. Long, S. Fan, F. Nori, C.M. Bender, L. Young, Nat. Phys. 10, 394 (2014)
C.E. Reüter, K.G. Makris, R. El-Ganainy, D.N. Christodoulides, M. Segev, D. Kip, Nat. Phys. 6, 192 (2010)
X.-Y. Xie, B. Tian, W.-R. Sun, M. Wang, Y.-P. Wang, Mod. Phys. Lett. B 29, 1550192 (2015)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Modak, S., Singh, A. & Panigrahi, P. Complex solitary waves and soliton trains in KdV and mKdV equations. Eur. Phys. J. B 89, 149 (2016). https://doi.org/10.1140/epjb/e2016-70130-7
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2016-70130-7