Mathematics

Applied Mathematics Seminars 

Summer 2017

All seminars (unless otherwise stated) will take place on Tuesdays at 3.00pm in Room 505 (25 Gordon Street). See the map for further details. There will be tea afterwards in Mathematics Room 606 (25 Gordon Street). If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

04 July 2018

Speaker: Prof Noel Smyth (University of Edinburgh)

Title: Water waves, undular bores and resonances

Abstract:

Undular bores, also termed dispersive shock waves, are a widespread and fundamental nonlinear, dispersive wave form which arises in fluid mechanics, nonlinear optics and Bose-Einstein condensates. An undular bore is the dispersive equivalent of a gas dynamic shock wave which links two flow states, with dispersion smoothing the transition between the two, rather than viscosity. These two flow states are linked by a modulated dispersive wavetrain with solitary waves at one edge and linear dispersive waves at the other. This standard undular bore wave form fundamentally changes when there is a resonance, that is there is a resonance between linear dispersive waves and the individual (nonlinear) waves of the bore, which occurs when the dispersion relation is non-convex. This talk will look at resonant undular bores in terms of a Korteweg-de Vries (KdV) equation with fifth order dispersion. This equation will be derived from the system governing the propagation of optical beams in nematic liquid crystals. The resonant undular bore solution of the fifth order KdV equation will be derived using Whitham modulation theory. It is found that the resonant bore is a three component structure consisting of a negative polarity solitary wave headed by a resonant wavetrain, which is brought down to the undisturbed level ahead by a partial undular bore.