Tu, May 13, 2025, 9:00-10:00 am PDT:
Nikolas Eptaminitakis (Leibniz University Hannover, Germany).
Title: Inverse Problems for Nonlinear Hyperbolic PDEs with Geometric Optics — the Westervelt equation and the DC Kerr system.
Abstract: Inverse problems for nonlinear hyperbolic PDEs have gained significant attention over the past decade, particularly following the pioneering work of Kurylev, Lassas, and Uhlmann (2018) introducing the high-order linearization method. In this talk, we present two examples of a different approach to forward and inverse problems for quasilinear PDEs that does not make use of linearization. Instead, we construct highly oscillatory asymptotic solutions using geometric optics. The first example concerns the non-diffusive Westervelt equation, a second order scalar quasilinear PDE that models the time evolution of acoustic pressure in a medium relative to its equilibrium state. The second example focuses on the Maxwell system with a cubic Kerr-type nonlinearity, which models the DC Kerr effect in nonlinear optics—a phenomenon used in the design of ultra-fast optical switches. In both cases, we describe how to construct and rigorously justify asymptotic solutions whose behavior is consistent with experimental observations, and we demonstrate how these solutions can be used to recover the unknown nonlinear parameters appearing in the equations. Based on joint work with Plamen Stefanov.