Inverse Problems for Nonlinear Wave Phenomena

该项目将研究非线性波动现象和相关的逆问题。特别感兴趣的是与非线性光学相关的主题，包括静态（DC）和光学（AC）克尔效应，其中介质在强外部电场或自调制的影响下改变其折射率。该项目还将研究非线性声学和非线性弹性。其中一个主要目标是了解基本模型和解决方案以及在渐近高频制度。下一个目标是解决相关的逆问题：从远程测量中确定介质的参数。已知超声和弹性成像在非线性区域中工作，这是动机之一。此外，该项目将研究奇异波在界面上线性弹性流体相互作用的情况下的传播，以及受地球物理应用启发的恢复介质参数的相关逆问题。该项目还将研究测地线X射线变换和类似Radon变换的适当离散化问题。本项目将为研究生提供研究培训机会。研究人员计划分析非线性波基本现象的波的传播：非线性光学，声学和弹性。这种现象是由准线性偏微分方程描述的，几何光学理论对此还不完全。PI计划利用这些模型的特定性质，结合物理直觉。PI计划解释非线性麦克斯韦方程的直流和交流克尔效应，特别是，并找到正确的渐近制度下，他们发生：波长和波的强度之间的关系。然后PI将研究从远程观测恢复介质参数的相关逆问题。PI还计划进一步研究弹性波和压力波在液体-固体介质中的传播：在分离两种介质的光滑表面上的反射，透射和模式转换。这是受到地球模型的启发，地壳和地幔是固体，上核被认为是液体，但核心仍然是固体。最后，PI计划研究采样和适当的离散化以及测地X射线变换和类似Radon型变换的反演。这与PI最近将采样理论与半经典分析联系起来的工作有关，将采样要求与半经典波前集联系起来，而不是研究“带限”（傅立叶变换的支持）。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.