Mark Hoefer, University of Colorado--Boulder
Experiments on Solitons, Dispersive Shock Waves, and Their Interactions
A soliton is a localized traveling wave solution to a special class of partial differential equations (integrable equations). A defining property of solitons is their interaction behavior. In his seminal work of 1968 introducing a notion of integrability (the Lax pair), Peter Lax also proved that the Korteweg-de Vries (KdV) equation admits two soliton solutions whose interaction behavior is quite remarkable. Two solitons interact elastically, i.e., each soliton maintains the same speed and shape post-interaction as they had pre-interaction. Moreover, Lax classified the interaction geometry into three categories depending on the soliton amplitude ratio. This talk will present a physical medium (corn syrup and water) modeled by the KdV equation in the weakly nonlinear regime that supports approximate solitons. Numerical analysis and laboratory experiments will be used to show that the three Lax categories persist into the strongly nonlinear regime, beyond the applicability of the KdV model. Additionally, a wavetrain of solitons called a dispersive shock wave in this medium will be described and investigated using a nonlinear wave averaging technique (Whitham theory) and experiment. Interactions of dispersive shock waves and solitons reveal remarkable behavior including soliton refraction, soliton absorption, and two-phase dynamics.