Department of Mathematics

Speaker: Mark Hoefer
Title: Eulerian Dispersive Shock Waves and Instabilities
Affiliation: North Carolina State University
Date: Friday, January 18, 2013
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. Recent experimental and theoretical research in Bose-Einstein condensation and nonlinear optics have demonstrated novel supersonic, fluid-like phenomena. Shock waves in these and other systems are modeled by a dispersive regularization of Euler's equations, implemented by use of the Whitham averaging technique. Normal and oblique dispersive shock waves (DSWs) connecting two constant states are constructed. Numerical computations of supersonic, dispersive flow over a corner in the special case of systems modeled by the Nonlinear Schrodinger equation (NLS) exhibit stable pattern formation (oblique DSWs) or instability (turbulent-like behavior) depending on the flow parameters. A combination of analytical and computational approaches are used to demonstrate that this change in behavior can be identified with the transition from convective to absolute instability of dark solitons. The linearized NLS behavior about the dark soliton DSW trailing edge is studied in detail to identify the separatrix between convective and absolute instabilities.