Nonlinear Waves Seminar - Thibault Congy
Speaker:Ìý Dr. Thibault Congy
Affiliation:Ìý University of Northumbria, Newcastle, UK
Title:Ìý Dispersive Riemann problem for the Benjamin-Bona-MahonyÌýequation
Abstract:
The Benjamin-Bona-Mahony (BBM) equation $u_t + uu_x = u_{xxt}$ as aÌýmodel for unidirectional, weakly nonlinear dispersive shallow waterÌýwave propagation is asymptotically equivalent to the celebratedÌýKorteweg-de Vries (KdV) equation while providing more satisfactoryÌýshort-wave behavior in the sense that the linear dispersion relationÌýis bounded for the BBM equation, but unbounded for the KdVÌýequation. However, the BBM dispersion relation is nonconvex, aÌýproperty that gives rise to a number of intriguing features markedlyÌýdifferent from those found in the KdV equation, providing theÌýmotivation for the study of the BBM equation as a distinct dispersiveÌýregularization of the Hopf equation.
The dynamics of the smoothed step initial value problem orÌýdispersive Riemann problem for BBM equation are studied usingÌýasymptotic methods and numerical simulations. I will present the emergent wave phenomenaÌýfor this problem which can be split into twoÌýcategories: classical and nonclassical. Classical phenomena includeÌýdispersive shock waves and rarefaction waves, also observed in convexÌýKdV-type dispersive hydrodynamics. Nonclassical features are dueÌýto nonconvex dispersion and include the generation of two-phase linearÌýwavetrains, expansion shocks, solitary wave shedding, dispersive LaxÌýshocks, DSW implosion and the generation of incoherent solitaryÌýwavetrains.
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This presentation is based on a joint work with G. A. El, M. Shearer and M. Hoefer, available at:Ìý