The Benjamin–Bona–Mahony equation (or BBM equation) – also known as the regularized long-wave equation (RLWE) – is the partial differential equation
This equation was studied in Benjamin, Bona, and Mahony (1972) as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions. They show the stability and uniqueness of solutions to the BBM equation. This contrasts with the KdV equation, which is unstable in its high wavenumber components. Further, while the KdV equation has an infinite number of integrals of motion, the BBM equation only has three.
where is a sufficiently smooth function from to . Avrin & Goldstein (1985) proved global existence of a solution in all dimensions.
Solitary wave solution
where sech is the hyperbolic secant function and is a phase shift (by an initial horizontal displacement). For , the solitary waves have a positive crest elevation and travel in the positive -direction with velocity These solitary waves are not solitons, i.e. after interaction with other solitary waves, an oscillatory tail is generated and the solitary waves have changed.
- with Hamiltonian and operator
Here is the variation of the Hamiltonian with respect to and denotes the partial differential operator with respect to
The three conservation laws then are:
Which can easily expressed in terms of by using
- Bona, Pritchard & Scott (1980)
- Benjamin, Bona, and Mahony (1972)
- Olver (1979)
- Peregrine (1966)
- Goldstein & Wichnoski (1980)
- Avrin & Goldstein (1985)
- Olver, P.J. (1980), "On the Hamiltonian structure of evolution equations", Mathematical Proceedings of the Cambridge Philosophical Society 88 (1): 71–88, Bibcode:1980MPCPS..88...71O, doi:10.1017/S0305004100057364
- Avrin, J.; Goldstein, J.A. (1985), "Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions", Nonlinear Analysis 9 (8): 861–865, doi:10.1016/0362-546X(85)90023-9, MR 0799889
- Benjamin, T. B.; Bona, J. L.; Mahony, J. J. (1972), "Model Equations for Long Waves in Nonlinear Dispersive Systems", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 272 (1220): 47–78, Bibcode:1972RSPTA.272...47B, doi:10.1098/rsta.1972.0032, ISSN 0962-8428, JSTOR 74079
- Bona, J. L.; Pritchard, W. G.; Scott, L. R. (1980), "Solitary‐wave interaction", Physics of Fluids 23 (3): 438–441, Bibcode:1980PhFl...23..438B, doi:10.1063/1.863011
- Goldstein, J.A.; Wichnoski, B.J. (1980), "On the Benjamin–Bona–Mahony equation in higher dimensions", Nonlinear Analysis 4 (4): 665–675, doi:10.1016/0362-546X(80)90067-X
- Olver, P. J. (1979), "Euler operators and conservation laws of the BBM equation", Mathematical Proceedings of the Cambridge Philosophical Society 85: 143–160, Bibcode:1979MPCPS..85..143O, doi:10.1017/S0305004100055572
- Peregrine, D.H. (1966), "Calculations of the development of an undular bore", Journal of Fluid Mechanics 25 (2): 321–330, Bibcode:1966JFM....25..321P, doi:10.1017/S0022112066001678
- Zwillinger, D. (1998), Handbook of differential equations (3rd ed.), Boston, MA: Academic Press, pp. 174 & 176, ISBN 978-0-12-784396-4, MR 0977062 (Warning: On p. 174 Zwillinger misstates the Benjamin–Bona–Mahony equation, confusing it with the similar KdV equation.)