Qualitative Behavior of Nonlinear Hyperbolic Waves

该奖项的研究致力于研究模拟波（如光波和声波）传播的非线性偏微分方程的解。这些现象不具有粘性效应，这意味着微分方程是双曲型的。数学分析为短波长的精确解和近似解提供了新的严格的误差估计。特别强调非线性光学的问题。解决了四个主要问题领域。第一部分是对三个非线性模型的定性性质的研究。最基本的问题是这些模型能否预测奇点的形成。在第二个领域，将构建非线性光学问题的可靠近似解，该解适用于比以前可能的更大的振幅。要做到这一点，需要对非线性相互作用项进行假设，这些假设在许多实际模型中得到了验证。第三个问题是研究焦点处强非线性的影响。第四个问题领域是用误差估计构造描述超短激光传播方程的近似解。该奖项支持的研究重点是波传播现象的数学模型，特别是光波和声波等高频现象。本研究的结果可用于验证此类现象的现有模型，预测新的效果，并有助于新设备的设计。这项工作的大部分是由超强激光器和光学参量振荡器的问题所激发的，这些问题在建模和设计方面遇到了困难。数学研究将允许预测光学器件变得不稳定或损坏的情况。

The research under this award is devoted to the study of solutions of nonlinearpartial differential equations which model the propagation of waves such as light waves and sound waves. These phenomena do not have viscous effects which means that the differential equations are of hyperbolic type. The mathematicalanalysis provides new rigorous error estimates for exact solutions and approximate solutions that are valid in the limit of short wavelengths. Particular emphasis is placed on problems of nonlinear optics. Four main problem areas are addressed. The first is the investigation of qualitative properties of three nonlinearmodels. The basic question is whether these models predict singularity formation or not.In the second area, reliable approximate solutions for nonlinear optics problemswill be constructed that are valid for larger amplitudes than previously possible.To do so requires hypotheses for the nonlinear interaction terms which are verifiedin many practical models. The third problem is to investigate the effects of strongnonlinearity at focal points. The fourth problem area is to construct approximate solutions with error estimates for equations that describe ultra short laser propagation.The research that is supported by this award focuses on mathematical models for wave propagation phenomena, specifically on high frequency phenomena such as light waves andacoustic waves. The results from this research can be used to validate current models for such phenomena, predict new effects, and help in the design of new equipment. Much of this work is motivated by problems for ultra intense lasers and optical parametric oscillators where difficulties with modeling and design have been encountered.The mathematical investigations will allow to predict situations in which optical devicesbecome unstable or are damaged.