Singular solutions of the BBM equation: analytical and numerical study

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