Tel-Aviv University
School of Mathematical Sciences
Department of Applied Mathematics

Applied Mathematics Seminar


Tuesday, December 21st, 2010

Schreiber 309, 15:10



Boaz Ilan

University of California, Merced



Dynamics of oblique dispersive shock waves

Abstract:

One of the hallmarks of supersonic flows is the formation of shock waves. When a superfluid flows at supersonic speed it can form a dispersive shock wave (DSW) that possesses an expanding, rapidly oscillating front. DSWs appear as special solutions of nonlinear dispersive equations and have been observed in Bose-Einstein condensates and nonlinear optical media. Their theory is much less developed than their classical (dissipative) counterparts due to their complex structures.

We consider two-dimensional oblique DSW solutions of the weakly dispersive (2+1)D Gross-Pitaevskii equation. The jump conditions across the shock front are obtained using the results Whitham averaging theory and yield the fundamental relations between the upstream and downstream flows. It turns out that DSWs behave markedly different than classical shocks. For instance, the fluid can flow into the DSW from both sides. Direct numerical simulations elucidate the complex dynamics. The transition between the absolute and convective instability regimes is analyzed in detail.

This is joint work with Mark Hoefer.