Benjamin–Bona–Mahony equation

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

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

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

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

${\displaystyle u_{t}=-{\mathcal {D}}{\frac {\delta H}{\delta u}},\,}$ with Hamiltonian ${\displaystyle H=\int _{-\infty }^{+\infty }\left({\tfrac {1}{2}}u^{2}+{\tfrac {1}{6}}u^{3}\right)\,{\text{d}}x\,}$ and operator ${\displaystyle {\mathcal {D}}=\left(1-\partial _{x}^{2}\right)^{-1}\,\partial _{x}.}$

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

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

${\displaystyle v_{t}-v_{xxt}=v\,v_{x}.}$

The three conservation laws then are:[3]

${\displaystyle {\begin{aligned}v_{t}-\left(v_{xt}+{\tfrac {1}{2}}v^{2}\right)_{x}&=0,\\\left({\tfrac {1}{2}}v^{2}+{\tfrac {1}{2}}v_{x}^{2}\right)_{t}-\left(v\,v_{xt}+{\tfrac {1}{3}}v^{3}\right)_{x}&=0,\\\left({\tfrac {1}{3}}v^{3}\right)_{t}+\left(v_{t}^{2}-v_{xt}^{2}-v^{2}\,v_{xt}-{\tfrac {1}{4}}v^{4}\right)_{x}&=0.\end{aligned}}}$

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

The linearized version of the BBM equation is:

${\displaystyle u_{t}+u_{x}-u_{xxt}=0.}$

Periodic progressive wave solutions are of the form:

${\displaystyle u=a\,\mathrm {e} ^{i(kx-\omega t)},}$

with ${\displaystyle k}$ the wavenumber and ${\displaystyle \omega }$ the angular frequency. The dispersion relation of the linearized BBM equation is[2]

${\displaystyle \omega _{\mathrm {BBM} }={\frac {k}{1+k^{2}}}.}$

Similarly, for the linearized KdV equation ${\displaystyle u_{t}+u_{x}+u_{xxx}=0}$ the dispersion relation is:[2]

${\displaystyle \omega _{\mathrm {KdV} }=k-k^{3}.}$

This becomes unbounded and negative for ${\displaystyle k\to \infty ,}$ and the same applies to the phase velocity ${\displaystyle \omega _{\mathrm {KdV} }/k}$ and group velocity ${\displaystyle \mathrm {d} \omega _{\mathrm {KdV} }/\mathrm {d} k.}$ Consequently, the KdV equation gives waves travelling in the negative ${\displaystyle x}$-direction for high wavenumbers (short wavelengths). This is in contrast with its purpose as an approximation for uni-directional waves propagating in the positive ${\displaystyle x}$-direction.[2]

The strong growth of frequency ${\displaystyle \omega _{\mathrm {KdV} }}$ and phase speed with wavenumber ${\displaystyle k}$ posed problems in the numerical solution of the KdV equation, while the BBM equation does not have these shortcomings.[2]

• 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, S2CID 120673596
• 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 (1): 143–160, Bibcode:1979MPCPS..85..143O, doi:10.1017/S0305004100055572, S2CID 10840014
• 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, S2CID 122299686
• 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.)