Abstract
We present closed form periodic solutions of the integrable modified Korteweg-de Vries equation (mKdV). By using a Darboux transformation, we derive first-and second-order doubly-periodic lattice-like solutions. We explicitly derive first-and second-order rational solutions as limiting cases of periodic solutions. We have also found the degenerate solution which corresponds to the equal eigenvalue case. Among the second-order solutions, we single out the doubly-localized high peak solution on a constant background with an infinitely extended trough. This solution plays the role of a rogue wave of the mKdV equation.
Graphical abstract
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Scott Russell, in 14th meeting of the British Association for the Advancement of Science, 1844, Vol. 311, p. 390
Lord Rayleigh, Phil. Mag. 1, 257 (1876)
J. Boussinesq, Essai sur la Théorie des Eaux Courantes (Imprimerie Nationale, Paris, 1877)
J. Boussinesq, C. R. Acad. Sci. 72, 755 (1871)
J. Boussinesq, J. Math. Pure Appl. 17, 55 (1872)
J. Boussinesq, C. R. Acad. Sci. 73, 256 (1871)
D.J. Korteweg, G. De Vries, The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 39, 422 (1895)
N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett. 15, 240 (1965)
C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967)
H. Leblond, D. Mihalache, Phys. Rev. A 79, 063835 (2009)
H. Leblond, D. Mihalache, J. Phys. A 43, 375205 (2010)
H. Leblond, D. Mihalache, Rom. Rep. Phys. Suppl. 63, 1254 (2011)
H. Leblond, Ph. Grelu, D. Mihalache, Phys. Rev. A 90, 053816 (2014)
H. Ono, J. Phys. Soc. Jpn 61, 4336 (1992)
A.H. Khater, O.H. El-Kalaawy, D.K. Callebaut, Phys. Scr. 58, 545 (1998)
E.A. Ralph, L. Pratt, J. Nonlinear Sci. 4, 355 (1994)
T.S. Komatsu, Shin-ichi Sasa, Phys. Rev. E 52, 5574 (1995)
Z.-P. Li, Y.-C. Liu, Eur. Phys. J. B 53, 367 (2006)
H.X. Ge, S.Q. Dai, Y. Xue, L.Y. Dong, Phys. Rev. E 71, 066119 (2005)
V. Ziegler, J. Dinkel, C. Setzer, K.E. Lonngren, Chaos Solitons Fractals 12, 1719 (2001)
K.E. Lonngren, Optical and Quantum Electronics 30, 615 (1998)
M.A. Helal, Chaos Solitons Fractals 13, 1917 (2002)
A. Chowdury, D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E 91, 022919 (2015)
A. Chowdury, D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E 90, 032922 (2014)
A. Ankiewicz, N. Akhmediev, Phys. Lett. A 378, 358 (2014)
A. Ankiewicz, Y. Wang, S. Wabnitz, N. Akhmediev, Phys. Rev E 89, 012907 (2014)
Zhaqilao, Phys. Scr. 87, 065401 (2013)
L. Li, K. Porsezian, J. He, L. Wang, R. Erdelyi, Phys. Rev. E 89, 062917 (2014)
Sun Ying-Ying, Zhang Da-Jun, Commun. Theor. Phys. 57, 923 (2012)
D.-J. Zhang, S.-L. Zhao, Y.-Y. Sun, J. Zhou, Rev. Math. Phys. 26, 1430006 (2014)
R.M. Miura, J. Math. Phys. 9, 1202 (1968)
R.M. Miura, SIAM Rev. 18, 412 (1976)
N.F. Smyth, A.L. Worthy, Wave Motion 21, 263 (1995)
R. Hirota, J. Math. Phys. 14, 805 (1973)
R. Hirota, J. Phys. Soc. Jpn 33, 1456 (1972)
M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett. 31, 125 (1973)
Sh. Tanaka, Proc. Jpn Acad. 48, 466 (1972)
M. Wadati, J. Phys. Soc. Jpn 32, 1681 (1972)
P.G. Kevrekidis, A. Khare, A. Saxena, G. Herring, J. Phys. A 37, 10959 (2004)
N. Akhmediev, V.I. Korneev, N.V. Mitskevich, Zh. Eksp. Teor. Fiz. 94, 159 (1988) English translation in: [Sov. Phys. J. Exp. Theor. Phys. 67, 89 (1988)]
N. Akhmediev, V.I. Korneev, Teor. Mat. Fiz. 69, 189 (1986). English translation in: [Theor. Math. Phys. 69, 1089 (1986)]
N. Akhmediev, V. Eleonsky, N. Kulagin, Sov. Phys. J. Exp. Theor. Phys. 62, 894 (1985)
R. Grimshaw, E. Pelinovsky, T. Talipova, M. Ruderman, R. Erdelyi, Stud. Appl. Math. 114, 189, (2005)
R. Grimshaw, D. Pelinovsky, E. Pelinovsky, T. Talipova, Physica D 159, 35 (2001)
N. Akhmediev, V.M. Eleonskii, N.E. Kulagin, Theor. Math. Phys. (USSR) 72, 809 (1987)
N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80, 026601 (2009)
D.H. Peregrine, Adv. Appl. Mech. 16, 9 (1976)
B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, J.M. Dudley, Nat. Phys. 6, 790 (2010)
D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Eur. Phys. J. Special Topics 223, 4362 (2014)
N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373, 675 (2009)
N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373, 2137 (2009)
P. Dubard, P. Gaillard, C. Klein, V.B. Matveev, Eur. Phys. J. Special Topics 185, 247 (2010)
A. Ankiewicz, D.J. Kedziora, N. Akhmediev, Phys. Lett. A 375, 2782 (2011)
D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E 85, 066601 (2012)
D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phys. Rev. E 84, 056611 (2011)
T.R. Marchant, Phys. Rev. E 66, 046623 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chowdury, A., Ankiewicz, A. & Akhmediev, N. Periodic and rational solutions of modified Korteweg-de Vries equation. Eur. Phys. J. D 70, 104 (2016). https://doi.org/10.1140/epjd/e2016-70033-9
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjd/e2016-70033-9