Inverse Problems for Nonlinear Wave Phenomena

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The project will study nonlinear wave phenomena and related inverse problems. Of particular interest are topics related to nonlinear optics, including the static (DC) and the optical (AC) Kerr effects, where the medium changes its index of refraction either under the influence of a strong external electric field or by self-modulation. The project will study nonlinear acoustics and nonlinear elasticity as well. One of the main goals is to understand the underlying models and the solutions well in the asymptotic high-frequency regime. The next goal is to solve the associated inverse problems: to determine the parameters of the medium from remote measurements. Ultrasound and elastography are known to work in the nonlinear regime, which is one of the motivations. In addition, the project will study the propagation of singular waves in case of a linear elastic-fluid interaction across an interface, and the associated inverse problem of recovery the parameters of the parameters of the medium, inspired by geophysical applications. The project will also study a question on the proper discretization of the geodesic X-ray transform and similar Radon transforms. The project will provide research training opportunities for graduate students.The investigator plans to analyze propagation of waves for the nonlinear wave fundamental phenomena: nonlinear optics, acoustics and elasticity. Such phenomena are described by quasilinear partial differential equations, for which the geometric optics theory is less than complete. The PI plans to exploit the specific nature of those models, combined with physics intuition. The PI plans to explain the DC and the AC Kerr effects for the nonlinear Maxwell equations, in particular, and to find the right asymptotic regimes under which they occur: the relationship between the wavelength and the intensity of the waves. Then the PI will study the associated inverse problems of recovery of the parameters of the medium from remote observations. The PI also plans to further study propagation of elastic and pressure waves in liquid-solid media: reflection, transmission and mode conversion across a smooth surface separating the two media. This is inspired by the model of Earth, where the Crust and the Mantle are solid, the Upper Core is believed to be liquid but the core is still solid. Finally, the PI plans to study sampling and proper discretization and inversion of the geodesic X-ray transform and similar Radon type of transforms. This is related to the recent work by the PI connecting sampling theory with semiclassical analysis, relating the sampling requirements to the semi-classical wave front set rather than looking into the “band limit” (the support of the Fourier transform).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.