Speaker: Fioralba Cakoni, Rutgers University
Title: The Control of Scattered Field for Linear and Nonlinear Scattering Media
Abstract: Many imaging methods in inverse scattering rely on the ability to superimpose scattering data so that the resulting scattered field corresponds to that of a point source. In the frequency domain, for compact inhomogeneities, this leads to solving two elliptic PDEs in a bounded region with a prescribed difference in Cauchy data. The case of zero scattered field leads to the study of the transmission eigenvalue problem and the regularity of free boundaries. In this talk, we introduce this concept for linear media and review key results on the trans-mission eigenvalue problem, its resolvent, and non-scattering phenomena [1]. We then present recent results from [2] on the scattering problem for a nonlinear medium with compact support in the second-harmonic generation. When such a medium is probed with monochromatic light beams at a frequency ω, it generates additional waves at the frequency 2ω. The response of the medium is governed by a system of two coupled semilinear PDEs. We explore the possibility that the generated 2ω wave remains localized within the support of the medium, effectively rendering the nonlinear interaction with the probing wave invisible to an outside observer.
References:
[1] F. Cakoni, D. Colton, and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues,
CBMS-NSF SIAM publication, 2nd edition 98, 2023.
[2] F. Cakoni, N. Hovsepyan, M. Lassas and M. Vogelius, On the lack of external response of a
nonlinear medium in the second-harmonic generation process, SIAM J. Math Analysis 57 (2025).
In person attendance at this seminar is only open to Columbia Univesity affiliates. External guests are welcome to attend remotely. Please contact [email protected] if you need the Zoom link for this seminar.
