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Dispersive Riemann problem for the Benjamin-Bona-Mahony equation

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HY2W01 - Modulation theory and dispersive shock waves

The Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ as a model for unidirectional, weakly nonlinear dispersive shallow water wave propagation is asymptotically equivalent to the celebrated Korteweg-de Vries (KdV) equation while providing more satisfactory short-wave behavior in the sense that the linear dispersion relation is bounded for the BBM equation, but unbounded for the KdV equation. However, the BBM dispersion relation is nonconvex, a property that gives rise to a number of intriguing features markedly different from those found in the KdV equation, providing the motivation for the study of the BBM equation as a distinct dispersive regularization of the Hopf equation. The dynamics of the smoothed step initial value problem or dispersive Riemann problem for BBM equation are studied using asymptotic methods and numerical simulations. I will present the emergent wave phenomena for this problem which can be split into two categories: classical and nonclassical. Classical phenomena include dispersive shock waves and rarefaction waves, also observed in convex KdV-type dispersive hydrodynamics. Nonclassical features are due to nonconvex dispersion and include the generation of two-phase linear wavetrains, expansion shocks, solitary wave shedding, dispersive Lax shocks, DSW implosion and the generation of incoherent solitary wavetrains. This presentation is based on a joint work with G. El, M. Shearer and M. Hoefer.

This talk is part of the Isaac Newton Institute Seminar Series series.

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