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The unified Kadomtsev–Petviashvili equation for interfacial waves

Published online by Cambridge University Press:  26 April 2006

Yongze Chen  and
Philip L.-F. Liu
Yongze Chen
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA Present address: Center For Coastal Studies, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 920930209, USA.
Philip L.-F. Liu
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
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Abstract

In this paper, the propagation of interfacial waves in a two-layered fluid system is investigated. The interfacial waves are weakly nonlinear and dispersive and propagate in a slowly rotating channel with varying topography and sidewalls, and a weak steady background current field. An evolution equation for the interfacial displacement is derived for waves propagating predominantly in the longitudinal direction of the channel. This new evolution equation is called the unified Kadomtsev–Petviashvili (uKP) equation because most of the KP-type equations existing in the literature for both surface water waves and interfacial waves are special cases of the new evolution equation. The Painlevé PDE test is used to find the conditions under which the uKP equation can be solved by the inverse scattering transform. When these conditions are satisfied, elementary transformations are found to reduce the uKP equation to one of the completely integrable equations: the KP, the Korteweg–de Vries (KdV) or the cylindrical KdV equations. The integral invariants associated with the uKP equation for waves propagating in a varying channel are obtained and their relations with the conservation of mass and energy are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 383 - 408
Copyright
© 1995 Cambridge University Press

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References

Ablowitz, M. J. & Clarkson, P. A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press.
Akylas, T. R. 1994 Three-dimensional long water-wave phenomena. Ann. Rev. Fluid Mech. 26, 191–210.Google Scholar
Brugarino, T. & Greco, A. M. 1991 Painlevé analysis and reducibility to the canonical form for the generalized Kadomtsev–Petviashvili equation. J. Math. Phys. 32, 69–71.Google Scholar
Calogero, F. & Degasperis, A. 1978 Solution by the spectral-transform method of a nonlinear evolution equation including as a special case the cylindrical KdV equation. Lett. Nuovo Cim. 19, 150–154.Google Scholar
Chen, Y. & Wen, S. 1987 Traveling wave solutions to the two-dimensional Korteweg–de Vries equation. J. Maths Anal. Appl. 127, 226–236.Google Scholar
Clarkson, P. A. 1990 Painlevé analysis and the complete integrability of a generalized variablecoefficient Kadomtsev–Petviashvili equation. IMA J. Appl. Maths 44, 27–53.Google Scholar
David, D., Levi, D. & Winternitz, P. 1987 Integrable nonlinear equations for water waves in straits of varying depth and width. Stud. Appl. Maths 76, 133–168.Google Scholar
David, D., Levi, D. & Winternitz, P. 1989 Solitons in shallow seas of variable depth and in marine straits. Stud. Appl. Maths 80, 1–23.Google Scholar
Djordjevic, V. D. & Redekopp, L. G. 1978 The fission and disintegration of internal solitary waves moving over two-dimensional topography. J. Phys. Oceanogr. 8, 1016–1024.Google Scholar
Drazin, P. G. & Johnson, R. S. 1989 Solitons: An Introduction. Cambridge University Press.
Freeman, N. C. 1980 Soliton interactions in two dimensions. Adv. Appl. Mech. 20, 1–37.Google Scholar
Gardner, C. S., Greene, J. M., Kruskal, M. D. & Miura, R. M. 1967 Method for solving the KdV equation. Phys. Rev. Lett. 19, 1095–1097.Google Scholar
Grimshaw, R. 1985 Evolution equations for weakly nonlinear, long internal waves in a rotating fluid. Stud. Appl. Maths 73, 1–33.Google Scholar
Grimshaw, R. & Melville, W. K. 1989 On the Derivation of the modified Kadomtsev–Petviashvili equation. Stud. Appl. Maths 80, 183–202.Google Scholar
Grimshaw, R. & Tang, S. 1990 The rotation-modified Kadomtsev–Petviashvili equation: An analytical and numerical study. Stud. Appl. Maths 83, 223–248.Google Scholar
Hammack, J., Scheffner, N. & Segur, H. 1989 Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567–589.Google Scholar
Iizuka, T. & Wadati, M. 1992 Shallow water waves over an uneven bottom and an inhomogeneous KP equation. Chaos, Solitons and Fractals 2, 575–582.Google Scholar
Johnson, R. S. 1973 On the development of a solitary wave moving over an uneven bottom. Proc. Camb. Phil. Soc. 73, 183–203.Google Scholar
Johnson, R. S. 1980 Water waves and Korteweg–de Vries equations. J. Fluid Mech. 97, 701–719.Google Scholar
Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539–541.Google Scholar
Kakutani, T. 1971 Effect of an uneven bottom on gravity waves. J. Phys. Soc. Japan 30, 272–276.Google Scholar
Katsis, C. & Akylas, T. R. 1987 Solitary internal waves in a rotating channel: A numerical study. Phys. Fluids 30, 297–301.Google Scholar
Knickerbocker, C. J. & Newell, A. C. 1985 Reflections from solitary waves in channels of decrease depth. J. Fluid Mech. 153, 1–16.Google Scholar
Mathew, J. & Akylas, T. R. 1990 On three-dimensional long water waves in a channel with sloping sidewall. J. Fluid Mech. 215, 289–307.Google Scholar
Miles, J. W. 1979 On the Korteweg–de Vries equation for a gradually varying channel. J. Fluid Mech. 91, 181–190.Google Scholar
Weiss, J., Tabor, M. & Carnevale, G. 1983 The Painlevé property for partial differential equations. J. Math. Phys. 24, 522–526.Google Scholar