Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-05T07:19:04.994Z Has data issue: false hasContentIssue false
  • English
  • Français

A Two-dimensional Boussinesq equation for water waves and some of its solutions

Published online by Cambridge University Press:  26 April 2006

R. S. Johnson
R. S. Johnson
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.

A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 323 , 25 September 1996 , pp. 65 - 78
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M. J. & Clarkson, P. A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press.
Freeman, N. C. 1979 A two-dimensional distributed soliton solution of the Korteweg-de Vries equation. Proc. R. Soc. Lond. A 366, 185–204.Google Scholar
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 Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095–1097.Google Scholar
Gibbon, J. D., Freeman, N. C. & Johnson, R. S. 1978 Correspondence between the classical λø4, double and single sine-Gordon equations for three-dimensional solitons. Phys. Lett. 65A, 380–382.Google Scholar
Hietarinta, J. 1987 A search for bilinear equations passing Hirota's three-soliton condition. I KdV-type bilinear equations. J. Math. Phys. 28, 1732–1742.Google Scholar
Hirota, R. 1971 Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194.Google Scholar
Hirota, R. 1973 Exact N-soliton of the wave equation of long waves in shallow water and in nonlinear lattices. J. Math. Phys. 14, 810–814.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
Korteweg, D. J. & De Vries, G. 1895 On the change of form of long-waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. (5) 39, 422–443.Google Scholar
Matsuno, Y. 1984 Bilinear Transformation Method. Academic.
Miles, J. W. 1977 Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.Google Scholar