Singular solutions of the BBM equation: analytical and numerical study

We show that the Benjamin-Bona-Mahony (BBM) equation admits stable travelling wave solutions representing a sharp transition from a constant state to a periodic wave train. The constant state is determined by the parameters of the periodic wave train: the wave length, amplitude and phase velocity, and satisfies both the generalized Rankine-Hugoniot conditions for the exact BBM equation and for its wave averaged counterpart. Such stable shock-like travelling structures exist if the phase velocity of the periodic wave train is not less than the solution wave averaged. To validate the accuracy of the numerical method, we derive the (singular) solitary limit of the Whitham system for the BBM equation and compare the corresponding numerical and analytical solutions. We find good agreement between analytical results and numerical solutions.

Automated summary

It is shown in this study that the Benjamin-Bona-Mahony (BBM) equation has stable travelling wave solutions, which represent a sharp transition from a constant state to a periodic wave train. The accuracy of the numerical method is validated by comparing the numerical and analytical solutions, and good agreement between the two is found.