Inverse Problems for Nonlinear Wave Phenomena

该项目将研究非线性波动现象和相关的反问题。特别令人感兴趣的是与非线性光学有关的主题，包括静态(DC)和光学(AC)克尔效应，其中介质在强大的外部电场的影响下或通过自调制改变其折射率。该项目还将研究非线性声学和非线性弹性。其主要目标之一是很好地理解渐近高频区域中的基本模型和解。下一个目标是解决相关的逆问题：从远程测量确定介质的参数。众所周知，超声波和弹性成像是在非线性区域工作的，这是其动机之一。此外，该项目将研究在线性弹性-流体相互作用情况下的奇异波在界面上的传播，以及受地球物理应用的启发，恢复介质参数的相关反问题。该项目还将研究测地线X射线变换和类似的Radon变换的适当离散化问题。该项目将为研究生提供研究培训机会。研究人员计划分析波的传播，研究非线性波的基本现象：非线性光学、声学和弹性。这种现象可用几何光学理论不完备的拟线性偏微分方程组来描述。PI计划利用这些模型的特定性质，结合物理直觉。PI计划特别解释非线性麦克斯韦方程的直流和交流克尔效应，并找到它们发生的正确的渐近机制：波长和波强度之间的关系。然后，PI将研究从远程观测中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-固介质中的传播：在分隔两种介质的光滑表面上的反射、传输和模式转换。这是受到地球模型的启发，在地球模型中，地壳和地幔是固体的，上核心被认为是液体的，但核心仍然是固体的。最后，PI计划研究测地线X射线变换和类似Radon类型变换的采样和适当的离散化和反演。这与PI最近将抽样理论与半经典分析相结合的工作有关，将抽样要求与半经典波前集联系起来，而不是研究“频带极限”(傅立叶变换的支持)。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.