Abstract.
The goal of this work is to investigate, analytically and numerically, the dynamics of gravity water waves with the effects of the small surface tension and the bottom topography taken into account. Using a third-order perturbative approach of the Boussinesq equation, we obtain a new third-order perturbed Korteweg-de Vries (KdV) equation which includes nonlinear, dispersive, nonlocal and mixed nonlinear-dispersive terms, describing shallow water waves with a non-flat bottom and the surface tension. We show by numerical simulations, for various bottom shapes, that this new third-order perturbed KdV equation can support the propagation of solitary waves, whose profiles strongly depend on the surface tension. In particular, we show that the instability observed in the numerical simulation can be suppressed by the inclusion of small surface tension.
Similar content being viewed by others
References
D.G. Korteweg, G. de Vries, Philos. Mag. 39, 422 (1895)
J.K. Hunter, J. Scherule, Physica D 32, 253 (1988)
G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1999) reprint of the (1974) original, a Wiley-Interscience Publication
M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, in SIAM Studies in Applied Mathematics, Vol. 4 (SIAM, Philadelphia, 1981)
A.C. Newell, Solitons in Mathematics and Physics, in CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 48 (SIAM, Philadelphia, PA, 1985)
R.S. Johnson, J. Nonlinear Math. Phys. 10, 72 (2003)
T. Sakuma, Y. Kawanami, Phys. Rev. B 29, 869 (1984)
X.-F. Panga, J. Phys. B 10, 415 (1999)
H.-Y. Hao, H.J. Maris, Phys. Rev. B 64, 064302 (2001)
S. Poornakala, A. Das, P.K. Kaw, A. Sen, Z.M. Sheng, Y. Sentoku, K. Mima, K. Nishikawa, Phys. Plasmas 9, 3802 (2002)
J.R. Apel, J.R. Holbrook, A.K. Liu, J.J. Tsai, J. Phys. Oceanogr. 15, 1625 (1985)
A.K. Liu, Y.S. Chang, M.-K. Hsu, N.K. Liang, J. Geophys. Res. 103, 7995 (1998)
M.H. Orr, P.C. Mignerey, J. Geophys. Res. 108, 3064 (2003)
T.C. Kofane, B. Michaux, M. Remoissenet, J. Phys. C: Solid State Phys. 21, 1395 (1988)
W. Duan, B. Wang, R. Wei, Phys. Rev. E 55, 1773 (1997)
T.J. Pedley, The Fluid Mechanics of Large Blood Vessels (Cambridge University Press, Cambridge, 1980)
N.J. Zabusky, M.D. Krustal, Phys. Rev. Lett. 15, 6 (1965)
D.A. McDonald, Blood Flow in Arteries (Arnold, London, 1974)
G.B. Whitham, Proc. R. Soc. London A 299, 6 (1967)
A. Biswas, Appl. Math. Lett. 22, 208 (2009)
F. Chardard, Stabilité des Ondes Solitaires, PhD Thesis (2009)
T. Kawahara, J. Phys. Soc. Jpn. 33, 260 (1972)
T.B. Benjamin et al., Philos. Trans. R. Soc. London Ser. A 272, 47 (1972)
A. Jeffrey, Z. Angew. Math. Mech. 58, 38 (1978)
R.I. Joseph, R. Egri, Phys. Lett. A 61, 429 (1977)
C.C. Mei, B. Le Méhauté, J. Geophys. Res. 71, 393 (1966)
D.H. Peregrine, J. Fluid Mech. 27, 815 (1967)
O.S. Madsen, C.C. Mei, J. Fluid Mech. 39, 781 (1969)
T. Kakutani, J. Phys. Soc. Jpn. 30, 272 (1971)
R.S. Johnson, Proc. Camb. Philos. Soc. 73, 183 (1973)
R.S. Johnson, J. Fluid Mech. 54, 81 (1972)
H.-H. Dai, J. Phys. Soc. Jpn. 68, 1854 (1999)
P.J. Olver, Phys. Lett. A 126, 501 (1988)
E. Van Greosen, S.R. Pudjaprasetya, Wave Motion 18, 345 (1993)
S.R. Pudjaprasetya, E. Van Greosen, Wave Motion 23, 23 (1996)
W. Craig, P. Guyenne, D.P. Nicholls, C. Sulem, Proc. R. Soc. A 461, 839 (2005)
R. Grimshaw, J. Fluid Mech. 42, 639 (1970)
R. Grimshaw, J. Fluid Mech. 46, 611 (1971)
E. Audusse, F. Brouchut, M.O. Bristeau, R. Klein, B. Perthame, Soc. Industr. Appl. Math. 25, 2050 (2004)
N. Thürey, U. Rüde, M. Stamminger, Animation of open water phenomena with coupled shallow water and free surface simulations, in ACM SIGGRAPH Symposium on Computer Animation (Eurographics Association, 2006)
T. Gallouet, J.M. Herard, N. Seguin, Comput. Fluids 32, 479 (2003)
J.G. Zhou, D.M. Causon, D.M. Ingram, C.G. Mingham, Int. J. Numer. Methods Fluids 38, 769 (2002)
Y. **ng, C. Wang Shu, J. Comput. Phys. 208, 206 (2005)
B. Turan, K.H. Wang, J. Appl. Comput. Math. 3, 1 (2014)
R.H.J. Grimshaw, N.F. Smyth, J. Fluid Mech. 169, 429 (1986)
N.F. Smyth, Proc. R. Soc. London A 409, 79 (1987)
A.M. Kamchatnov, Y.-H. Kuo, T.-C. Lin, T.-L. Horng, S.-C. Gou, R. Clift, G.A. El, R.H.J. Grimshaw, Phys. Rev. E 86, 036605 (2012)
A.E. Green, P.M. Naghdi, J. Fluid Mech. 78, 237 (1976)
B.T. Nadiga, L.G. Margolin, P.K. Smolarkiewicz, Phys. Fluids 8, 2066 (1996)
J.W. Kim, K.J. Bai, R.C. Ertekin, W.C. Webster, J. Eng. Math. 40, 17 (2001)
A. Constantin, J. Fluid Mech. 740, 17 (2014)
H. Aspe, M.C. Depassier, Phys. Rev. A 41, 6 (1990)
H. Guo **ang, M.G. Velarde, Commun. Theor. Phys. 34, 321 (2000)
H.R. Dullin, G.A. Gottwald, D.D. Holm, Fluid Dyn. Res. 33, 73 (2003)
G.I. Burde, Commun. Nonlinear Sci. Numer. Simul. 16, 1314 (2011)
G.I. Burde, A. Sergyeyev, J. Phys. A: Math. Theor. 46, 075501 (2013)
A. Karczewska, P. Rozmej, E. Infeld, Phys. Rev. E 90, 012907 (2014)
J.K. Hunter, J. Scheurle, Physica D 32, 253 (1988)
A. Karasu-Kalkani, A. Karsau, A. Sakovich, S. Sarkovich, R. Turhan, J. Math. Phys. 49, 073516 (2008)
T.R. Marchant, N.F. Symth, J. Fluid Mech. 221, 236 (1990)
M. Fokou, T.C. Kofané, A. Mohamadou, E. Yomba, Nonlinear Dyn. 83, 2461 (2016)
T.R. Marchant, Appl. Math. 109, 1 (2002)
A. Karczewska, P. Rozmej, L. Rutkowski, Phys. Scr. 89, 054026 (2014)
C.J. Amick, J.F. Toland, Philos. Trans. R. Soc. London 303, 633 (1981)
C.J. Amick, K. Kirchgssner, Solitary water-waves in the presence of surface tension, Vol. 4 (Springer, New York, 1987)
C.J. Amick, K. Kirchgssner, A global theory of solitary water-waves in the presence of surface tension, Vol. 105 (Springer, 1989)
M. Zhao, B. Teng, L. Cheng, Ocean Eng. 31, 2047 (2004)
D.E. Mitsosakis, J. Math. Comput. Simul. 80, 860 (2009)
C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, Phys. Rev. Lett. 19, 1095 (1967)
N.J. Zabusky, Phys. Rev. 168, 124 (1968)
N.J. Zabusky, C.J. Galvin, J. Fluid Mech. 47, 811 (1971)
W.K. Melville, K.R. Helfrich, J. Fluid Mech. 178, 31 (1987)
C.Y. Lee, R.C. Beardsley, J. Geophys. Res. 7, 338 (1974)
P.G. Baines, J. Fluid Mech. 140, 127 (1984)
K.R. Helfrich, W.K. Melville, Annu. Rev. Fluid Mech. 38, 395 (2006)
A.T. Ippen, G. Kulin, MIT Hydrodynamics Laboratory Tech. Report, No. 15 (1955)
J.L. Hammack, J. Fluid Mech. 60, 769 (1973)
J.L. Hammack, H. Segur, J. Fluid Mech. 65, 289 (1974)
J.L. Hammack, H. Segur, J. Fluid Mech. 84, 337 (1978)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fokou, M., Kofané, T.C., Mohamadou, A. et al. The third-order perturbed Korteweg-de Vries equation for shallow water waves with a non-flat bottom. Eur. Phys. J. Plus 132, 410 (2017). https://doi.org/10.1140/epjp/i2017-11709-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2017-11709-0