Solitary wave interaction for the extended BBM equation

Solitary wave interaction is examined, for the case of surface waves on shallow water, by using an extended Benjamin–Bona–Mahony (eBBM) equation. This equation includes higher–order nonlinear and dispersive effects, and hence is asymptotically equivalent to the extended Korteweg–de Vries (eKdV) equation. However, it has certain numerical advantages as it allows the modelling of steeper waves, which, due to numerical instability, is not possible using the eKdV equation. Numerical simulations of a number of collisions of varying nonlinearity are performed. The numerical results show evidence of inelastic behaviour at high order. For waves of small amplitude the evidence of inelastic behaviour is indirect; after collision a dispersive wavetrain of extremely small amplitude is found behind the smaller solitary wave. For steeper waves, however, direct evidence of inelastic behaviour is found; the larger wave is increased and the smaller wave is decreased in amplitude after the collision. Conservation laws for the mass and energy, exact and asymptotic, respectively, are derived for the eBBM equation, and numerically verified. Data from the collisions, such as the change in solitary wave amplitudes, the higher–order phase shifts and the mass and energy of the dispersive wavetrains are all tabulated. These are used to show that the change in solitary wave amplitude is of O(α4), verifying previously obtained theoretical predictions. A good comparison is also obtained between the numerically obtained phase shifts and existing asymptotic results. Lastly, three different eBBM solitary wave interactions are examined in detail and compared with existing numerical data obtained from alternative weakly nonlinear models and the Euler water–wave equations.

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