Qualitative Behavior of Nonlinear Hyperbolic Waves

该奖项下的研究致力于研究非线性偏微分方程的解，该方程模拟光波和声波等波的传播。 这些现象没有粘性效应，这意味着微分方程是双曲型的。 理论分析为精确解和近似解提供了新的严格的误差估计，这些误差估计在短波长范围内是有效的。 特别强调的是放在非线性光学问题。 四个主要问题领域得到解决。 第一部分研究了三种非线性模型的定性性质。 基本的问题是这些模型是否预测奇异性的形成。在第二个领域，非线性光学问题的可靠的近似解将被构造为比以前可能的更大的振幅是有效的。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。 第四个问题领域是为描述超短激光传播的方程构造具有误差估计的近似解。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。 这项研究的结果可用于验证此类现象的当前模型，预测新的影响，并帮助设计新设备。 大部分的工作是出于对超强激光器和光学参量振荡器的建模和设计困难的问题，数学研究将允许预测的情况下，光学器件变得不稳定或损坏。

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.