CNA Seminar

Marcus Wunsch
ETH, Zurich
Title: The Hunter-Saxton System and geodesics on (pseudo-) spheres

Abstract: In this talk, I will review some recent work on the Hunter-Saxton system, $$\partial_{txx}u+ 2 \partial_{x}u \partial_{xx}u + u \partial_{xxx}u = \kappa \rho \partial_{x}\rho \\ \partial_{t}\rho + \partial_{x}(u \rho) = 0$$ subject to initial conditions and periodic boundary conditions.

The Hunter-Saxton system is a generalization of the (single-component) Hunter-Saxton equation, which describes the propagation of weakly nonlinear orientation waves in a massive director field of a nematic liquid crystal. Moreover, the system above is the high-frequency limit of the two-component Camassa-Holm equation arising in the theory of shallow water waves, and it has also been proposed as a model for the nonlinear dynamics of dark matter.

After preparing the analytic foundations for this coupled nonlinear system, I will prove that classical solutions have explicit representations in terms of their Lagrangian coordinates. The latter, it turns out, describe the geodesics on an infinite-dimensional sphere ($\kappa=1$) or pseudosphere ($\kappa=-1$), which, a posteriori, reveals why there are explicit solution formulae. I will elaborate further on the geometry of the Hunter-Saxton system and show how the geometric intuition guides us naturally to the construction of weak solutions. The talk will conclude with some open questions.

Recording (avi): http://mm.math.cmu.edu/recordings/cna/CNA-WunschMarcus-Nov-01-2012.avi
Recording (mp4): http://mm.math.cmu.edu/recordings/cna/CNA-WunschMarcus-Nov-01-2012.mp4
Date: Thursday, November 1, 2012
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  David Kinderlehrer