File:BBM equation - overtaking solitary waves animation.gif

English: 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 shows the oscillatory tail produced by the interaction, showing that the solitary wave solutions of the BBM equation are not solitons.
The presented solutions ${\displaystyle u(x,t)}$ are for the non-dimensional BBM equation of the form: ${\displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0.}$ The associated solitary waves are of the form:
      ${\displaystyle u=3(\nu -1)\operatorname {sech} ^{2}\left({\frac {1}{2}}{\sqrt {\frac {\nu -1}{\nu }}}\left(x-\nu t-x_{0}\right)\right),}$
with ${\displaystyle \nu }$ the velocity of the solitary wave, ${\displaystyle \nu \notin [0,1],}$ sech the hyperbolic secant function and ${\displaystyle x_{0}}$ a horizontal shift of the solitary wave.