# Benjamin-Bona-Mahony equation

BBM equation, regularized long wave equation

The model equation

$$\tag{a1} u _ { t } + u _ { x } + u u _ { x } - u _ { xxt } = 0,$$

where $u ( x , t ) : \mathbf{R} \times \mathbf{R} \rightarrow \mathbf{R}$ and the subscripts denote partial derivatives with respect to time $t$ and the position coordinate $x$. The Benjamin–Bona–Mahony equation serves as an approximate model in studying the dynamics of small-amplitude surface water waves propagating unidirectionally, while suffering non-linear and dispersive effects. (a1) was introduced in [a5] as an alternative of the famous Korteweg–de Vries equation

$$\tag{a2} u _ { t } + u _ { x } + u u _ { x } + u _ { xxx } = 0.$$

Unlike the Korteweg–de Vries equation, the Benjamin–Bona–Mahony equation is not integrable by the inverse scattering method [a10], [a12]. As indicated by several numerical experiments, (a1) has no multi-soliton solutions. It has been proved by A.C. Bryan and A.E.G. Stuart [a8] that (a1) has no analytic two-soliton solution. The equation has three independent invariants (conservation laws):

$D ( u ) = \int _ { \mathbf{R} } u d x$;

$E ( u ) = \int _ { \mathbf{R} } ( u ^ { 2 } + u _ { x } ^ { 2 } ) d x$; and

$F ( u ) = \int _ { \mathbf{R} } \left( u ^ { 2 } + \frac { 1 } { 3 } u ^ { 3 } \right) d x$. These quantities are time-independent during the time evolution of the solution $u$. The correctness of the initial value problem $u ( x , 0 ) = g ( x )$ (the Cauchy problem) for (a1) in Sobolev spaces $W _ { 2 } ^ { s } ( \mathbf R _ { x } ) = H ^ { s } ( \mathbf R _ { x } )$, $s \geq 1$ (cf. also Sobolev space), was investigated in [a5].

Equation (a1) has a solitary wave solution $u ( x , t ) = \phi ( x - v t - c )$, where $\phi ( \overline{x} ) = 3 ( v - 1 ) \operatorname { sech } ^ { 2 } \{ \overline{x} \sqrt { ( v - 1 ) / ( 4 v ) } \}$ (cf. also Soliton), provided that the wave velocity $v$ satisfies $v \notin [ 0,1]$. The non-linear stability of the wave $\phi$ with respect to the pseudo-metric $d ( u , \phi ) ( t ) = \operatorname { inf } \{ \| u - \phi ( x - v t - c ) \| _ { 1 } : c \in \mathbf{R} \}$ was established in [a3] and [a7] by a clever modification of Lyapunov's direct method in combination with a spectral decomposition technique. Here, $\| . \| _ { 1 }$ is the norm in the Sobolev space $H ^ { 1 } ( {\bf R} _ { x } )$. This means that the form of the solitary wave is stable under small perturbations in the form of the initial wave.

## Generalizations.

The generalized Benjamin–Bona–Mahony equation is an equation of the form

$$\tag{a3} u _ { t } + a ( u ) _ { x } - u _ { x x t } = 0,$$

where $a : \mathbf{R} \rightarrow \mathbf{R}$ is a differentiable function. (a3) allows two types of solitary waves: kink-shaped and bell-shaped ones. Depending on the concrete form of the non-linearity, these solitary waves can be stable or unstable with respect to the metric $d ( u , \phi )$. For more concrete results concerning (a3), see [a11], Chap. 4. The generalized Benjamin–Bona–Mahony equation in higher dimensions reads

$$\tag{a4} u _ { t } - \Delta u _ { t } + \operatorname { div } \varphi ( u ) = 0,$$

where $\Delta$ is the Laplace operator in ${\bf R} ^ { n }$ and $\varphi \in C ^ { 1 } ( \mathbf{R} ; \mathbf{R} ^ { n } )$. The uniqueness and global existence of a solution in Sobolev spaces to the initial boundary value problem for (a4) in $\Omega \times [ 0 , T]$, with Dirichlet (or more general) boundary conditions, was proved in [a2], [a9]. Here, $\Omega \subset \mathbf{R} ^ { n }$ is a bounded domain with smooth boundary. The Cauchy problem for (a4) is studied in [a1].

Non-local generalizations of the Benjamin–Bona–Mahony equation can be obtained after one writes (a1) in the form

\begin{equation*} M u _ { t } + u _ { x } + u u _ { x } = 0. \end{equation*}

Here, $M$ is a pseudo-differential operator (in fact, a Fourier multiplier operator), acting as $\widehat { M u } ( \xi ) = m ( \xi ) \hat { u } ( \xi )$, where $\widehat{\square}$ denotes the Fourier transform in the space variable. For the original Benjamin–Bona–Mahony equation one has $m ( \xi ) = 1 + \xi ^ { 2 }$. In general, one takes for $m ( \xi )$ a positive even function such that its negative power $m ( \xi ) ^ { - 1 }$ is monotone decreasing on $( 0 , \infty )$ and belongs to $L ^ { 1 } ( \mathbf{R} )$. See [a4], [a5] and the references therein for more details.

The variable-coefficient Benjamin–Bona–Mahony equation

\begin{equation*} u _ { t } + a ( t ) u _ { x } + b ( t ) u ^ { p } u _ { x } - u _ { xxt } = 0 \end{equation*}

describes the propagation of long weakly non-linear water waves in a channel of variable depth. This equation was studied in [a6].

#### References

 [a1] J. Avrin, "The generalized Benjamin–Bona–Mahony equation in ${\bf R} ^ { n }$ with singular initial data" Nonlin. Anal. Th. Meth. Appl. , 11 (1987) pp. 139–147 [a2] J. Avrin, J.A. Goldstein, "Global existence for the Benjamin–Bona–Mahony equation in arbitrary dimensions" Nonlin. Anal. Th. Meth. Appl. , 9 (1985) pp. 861–865 [a3] T.B. Benjamin, "The stability of solitary waves" Proc. Royal Soc. London A , 328 (1972) pp. 153–183 [a4] T.B. Benjamin, "Lectures on nonlinear wave motion" A.C. Newell (ed.) , Nonlinear Wave Motion , Lectures in Applied Math. , 15 , Amer. Math. Soc. (1974) pp. 3–47 [a5] T.B. Benjamin, J.L. Bona, J.J. Mahony, "Model equations for long waves in nonlinear dispersive systems" Philos. Trans. Royal Soc. London A , 272 (1972) pp. 47–78 [a6] V. Bisognin, G. Perla Menzala, "Asymptotic behaviour of nonlinear dispersive models with variable coefficients" Ann. Mat. Pura Appl. , 168 (1995) pp. 219–235 [a7] J.L. Bona, "On the stability theory of solitary waves" Proc. Royal Soc. London A , 344 (1975) pp. 363–374 [a8] A.C. Bryan, A.E.G. Stuart, "Solitons and the regularized long wave equation: a nonexistence theorem" Chaos, Solitons, Fractals , 7 (1996) pp. 1881–1886 [a9] B. Calvert, "The equation $A ( t , u ( t ) ) ^ { \prime } + B ( t , u ( t ) ) = 0$" Math. Proc. Cambridge Philos. Soc. , 79 (1976) pp. 545–561 [a10] C.S. Gardner, J.M. Greene, M.D. Kruskal, R.M. Miura, "Method for solving the Korteweg–de Vries equation" Phys. Rev. Lett. , 19 (1967) pp. 1095–1097 [a11] I.D. Iliev, E. Khristov, K.P. Kirchev, "Spectral methods in soliton equations" , Pitman Monographs and Surveys Pure Appl. Math. , 73 , Longman (1994) [a12] P.D. Lax, "Integrals of nonlinear equations of evolution and solitary waves" Commun. Pure Appl. Math. , 21 (1968) pp. 467–490
How to Cite This Entry:
Benjamin-Bona-Mahony equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Benjamin-Bona-Mahony_equation&oldid=50184
This article was adapted from an original article by Iliya D. Iliev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article