Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

Abstract

Long time dynamics of the smoothed step initial value problem or dispersive Riemann problem for the Benjamin-Bona-Mahony (BBM) equation ut + uux = uxxt 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 ut + uux + uxxx = 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 → ∞ 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 Schrödinger 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.

Dynamic soliton-mean flow interaction with nonconvex flux

“…Such systems are abundant in geophysical fluids, describing magma and glacier flows [59,63] or wave-ice sheet interactions [30]. Indeed, recent works on DSWs in systems with nonconvex dispersion [51,17,61,27,3,10] reveal a plethora of unusual behaviours that can lead to new, interesting soliton-mean flow interactions.…”

“…It is important to stress that the existence of the simple wave solution leading to the conditions in eqs. ( 28) and ( 29) requires convexity (genuine nonlinearity) of the characteristic field (10) along the integral curve so that the conditions ( 13) are not violated.…”