Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ are studied using asymptotic methods and numerical simulations. The catalog of solutions of the dispersive Riemann problem for the BBM equation is much richer than for the related, integrable, Korteweg-de Vries equation $u_t + uu_x + u_{xxx} =0.$ The transition width of the initial smoothed step is found to significantly impact the dynamics. Narrow width gives rise to rarefaction and dispersive shock wave (DSW) solutions that are accompanied by the generation of two-phase linear wavetrains, solitary wave shedding, and expansion shocks. Both narrow and broad initial widths give rise to two-phase nonlinear wavetrains or DSW implosion and a new kind of dispersive Lax shock for symmetric data. The dispersive Lax shock is described by an approximate self-similar solution of the BBM equation whose limit as $t \to \infty$ is a stationary, discontinuous weak solution. By introducing a slight asymmetry in the data for the dispersive Lax shock, the generation of an incoherent solitary wavetrain is observed. Further asymmetry leads to the DSW implosion regime that is effectively described by a pair of coupled nonlinear Schr\"{o}dinger equations. The complex interplay between nonlocality, nonlinearity and dispersion in the BBM equation underlies the rich variety of nonclassical dispersive hydrodynamic solutions to the dispersive Riemann problem.