View source for Benjamin-Bona-Mahony equation

''BBM equation, regularized long wave equation''

The model equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300902.png" /> and the subscripts denote partial derivatives with respect to time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300903.png" /> and the position coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300904.png" />. 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 [[#References|[a5]]] as an alternative of the famous [[Korteweg–de Vries equation|Korteweg–de Vries equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300905.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Unlike the Korteweg–de Vries equation, the Benjamin–Bona–Mahony equation is not integrable by the inverse scattering method [[#References|[a10]]], [[#References|[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 [[#References|[a8]]] that (a1) has no analytic two-soliton solution. The equation has three independent invariants (conservation laws):

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300906.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300907.png" />; and

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300908.png" />. These quantities are time-independent during the time evolution of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b1300909.png" />. The correctness of the initial value problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009010.png" /> (the [[Cauchy problem|Cauchy problem]]) for (a1) in Sobolev spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009012.png" /> (cf. also [[Sobolev space|Sobolev space]]), was investigated in [[#References|[a5]]].

Equation (a1) has a solitary wave solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009014.png" /> (cf. also [[Soliton|Soliton]]), provided that the wave velocity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009015.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009016.png" />. The non-linear stability of the wave <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009017.png" /> with respect to the [[Pseudo-metric|pseudo-metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009018.png" /> was established in [[#References|[a3]]] and [[#References|[a7]]] by a clever modification of Lyapunov's direct method in combination with a spectral decomposition technique. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009019.png" /> is the norm in the Sobolev space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009020.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009021.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009023.png" />. For more concrete results concerning (a3), see [[#References|[a11]]], Chap. 4. The generalized Benjamin–Bona–Mahony equation in higher dimensions reads

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009025.png" /> is the [[Laplace operator|Laplace operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009027.png" />. The uniqueness and global existence of a solution in Sobolev spaces to the initial boundary value problem for (a4) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009028.png" />, with Dirichlet (or more general) boundary conditions, was proved in [[#References|[a2]]], [[#References|[a9]]]. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009029.png" /> is a bounded domain with smooth boundary. The Cauchy problem for (a4) is studied in [[#References|[a1]]].

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009030.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009031.png" /> is a [[Pseudo-differential operator|pseudo-differential operator]] (in fact, a Fourier multiplier operator), acting as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009033.png" /> denotes the [[Fourier transform|Fourier transform]] in the space variable. For the original Benjamin–Bona–Mahony equation one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009034.png" />. In general, one takes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009035.png" /> a positive even function such that its negative power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009036.png" /> is monotone decreasing on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009037.png" /> and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009038.png" />. See [[#References|[a4]]], [[#References|[a5]]] and the references therein for more details.

The variable-coefficient Benjamin–Bona–Mahony equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009039.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Avrin,   "The generalized Benjamin–Bona–Mahony equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009040.png" /> with singular initial data"  ''Nonlin. Anal. Th. Meth. Appl.'' , '''11'''  (1987)  pp. 139–147</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.B. Benjamin,   "The stability of solitary waves"  ''Proc. Royal Soc. London A'' , '''328'''  (1972)  pp. 153–183</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V. Bisognin,   G. Perla Menzala,   "Asymptotic behaviour of nonlinear dispersive models with variable coefficients"  ''Ann. Mat. Pura Appl.'' , '''168'''  (1995)  pp. 219–235</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.L. Bona,   "On the stability theory of solitary waves"  ''Proc. Royal Soc. London A'' , '''344'''  (1975)  pp. 363–374</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  B. Calvert,   "The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130090/b13009041.png" />"  ''Math. Proc. Cambridge Philos. Soc.'' , '''79'''  (1976)  pp. 545–561</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  I.D. Iliev,   E. Khristov,   K.P. Kirchev,   "Spectral methods in soliton equations" , ''Pitman Monographs and Surveys Pure Appl. Math.'' , '''73''' , Longman  (1994)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  P.D. Lax,   "Integrals of nonlinear equations of evolution and solitary waves"  ''Commun. Pure Appl. Math.'' , '''21'''  (1968)  pp. 467–490</TD></TR></table>