Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium 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: http://mm.math.cmu.edu/recordings/cna/CNA-WunschMarcus-Nov-01-2012.aviRecording: http://mm.math.cmu.edu/recordings/cna/CNA-WunschMarcus-Nov-01-2012.mp4Date: Thursday, November 1, 2012Time: 1:30 pmLocation: Wean Hall 7218Submitted by:  David Kinderlehrer