Dispersive Riemann problems for the Benjamin–Bona–Mahony equation

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