UTA Department of Mathematics

Special Mathematics Colloquium

Date/Time/Room: Wednesday (2/20/2002) at 2:30pm in 304 Pickard Hall

Speaker: Fei-Ran Tian, Department of Mathematics, Ohio State University

``Phase Transition in Dispersive Oscillations''

Abstract: Many dynamics in nature undergo dispersive processes, while the dissipative or diffusive mechanisms are negligible. When the dispersive term is small, there appear regions in space-time which are filled with small scale oscillations. One fascinating feature in the propagation of oscillations is that the number of phases in the oscillations can change in space and time. In this talk, we shall focus on the prototypical dispersive oscillations, which are governed by the KdV equation. The solution of the KdV has a weak limit when the dispersion tends to zero. This limit is described by the Whitham equations, which are (2g+1)quasilinear hyperbolic equations where g is the number of phases. We shall review the Lax-Levermore-Venakides theory of the KdV zero dispersion limit and Novikov-Dubrovin-Krichever-Tsarev theory of the Whitham equations. We shall discuss the intrinsic connection between the Whitham equations and Euler-Poisson-Darboux equations. We shall show how to use the solution of the latter equations to understand the generation and propagation of phases in space and time.