Benjamin–Bona–Mahony equation

An animation of the overtaking of two solitary waves according to the Benjamin–Bona–Mahony (BBM) equation. The wave heights of the solitary waves are 1.2 and 0.6, respectively, and their celerities are 1.4 and 1.2.
The upper graph is for a frame of reference moving with the average celerity of the solitary waves. The envelope of the overtaking waves is shown in grey: note that the maximum wave height reduces during the interaction.
The lower graph (with a different vertical scale and in a stationary frame of reference) shows the oscillatory tail produced by the interaction.^[1] Thus, the solitary wave solutions of the BBM equation are not solitons.

Benjamin–Bona–Mahony equation animation

The Benjamin–Bona–Mahony equation (or BBM equation) – also known as the regularized long-wave equation (RLWE) – is the partial differential equation

$u_t+u_x+uu_x-u_{xxt}=0.\,$

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.^[2][3]

Before, in 1966, this equation was introduced by Peregrine, in the study of undular bores.^[4]

A generalized n-dimensional version is given by^[5][6]

$u_t-\nabla^2u_t+\operatorname{div}\,\varphi(u)=0.\,$

where $\varphi$ is a sufficiently smooth function from $\mathbb R$ to $\mathbb R^n$. Avrin & Goldstein (1985) proved global existence of a solution in all dimensions.

Solitary wave solution

The BBM equation possesses solitary wave solutions of the form:^[3]

$u = 3 \frac{c^2}{1-c^2} \operatorname{sech}^2 \frac12 \left( cx - \frac{ct}{1-c^2} + \delta \right),$

where sech is the hyperbolic secant function and $\delta$ is a phase shift (by an initial horizontal displacement). For $|c|<1$, the solitary waves have a positive crest elevation and travel in the positive $x$-direction with velocity $1/(1-c^2).$ 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.^[1][3]

Hamiltonian structure

The BBM equation has a Hamiltonian structure, as it can be written as:^[7]

$u_t = - \mathcal{D} \frac{\delta H}{\delta u},\,$   with Hamiltonian   $H = \int_{-\infty}^{+\infty} \left( \tfrac12 u^2 + \tfrac16 u^3 \right)\, \text{d}x\,$   and operator   $\mathcal{D} = \left( 1 - \partial_x^2 \right)^{-1}\, \partial_x.$

Here $\delta H/\delta u$ is the variation of the Hamiltonian $H(u)$ with respect to $u(x),$ and $\partial_x$ denotes the partial differential operator with respect to $x.$

Conservation laws

The BBM equation possesses exactly three independent and non-trivial conservation laws.^[3] First $u$ is replaced by $u=-v-1$ in the BBM equation, leading to the equivalent equation:

$v_t - v_{xxt} = v\, v_x.$

The three conservation laws then are:^[3]

$\begin{align} v_t &- \left( v_{xt} + \tfrac12 v^2 \right)_x = 0, \\ \left( \tfrac12 v^2 + \tfrac12 v_x^2 \right)_t &- \left( v\, v_{xt} + \tfrac13 v^3 \right)_x = 0 \qquad \text{and} \\ \left( \tfrac13 v^3 \right)_t &+ \left( v_t^2 - v_{xt}^2 - v^2\, v_{xt} - \tfrac14 v^4 \right)_x = 0. \end{align}$

Which can easily expressed in terms of $u$ by using $v=-u-1.$

Notes

1. Bona, Pritchard & Scott (1980)
2. Benjamin, Bona, and Mahony (1972)
3. Olver (1979)
4. Peregrine (1966)
5. Goldstein & Wichnoski (1980)
6. Avrin & Goldstein (1985)
7. 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

References

• 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.)