New Perspectives on Nonlinear Waves: Taming Modulational Instability

非线性Schrödinger （NLS）方程是现代数学物理中的一个基本模型方程。例如，这个方程可以用来描述某些水波和光在光纤中的传播。然而，这个方程的非线性本质是一个障碍，它阻碍了大多数现有通用数学工具的直接应用。该项目将利用该方程的特殊结构来详细研究其解的两个特征，即调制不稳定性（MI）和超连续统生成（SCG）。虽然这些研究的出发点是基于NLS方程的数学性质，但MI和SCG是现实世界中的重要现象。例如，MI被认为是海洋中“异常浪”形成的重要因素；这些罕见但相对较大的自发波可能相当危险。此外，SCG在光学通信系统和光学相干断层扫描中也很重要，光学相干断层扫描是一种用于眼科的成像技术，用于显示详细的视网膜结构。因此，对这一数学现象的更好理解有可能影响电信系统和光学设备的设计和发展。本项目的MI和SCG研究是基于小色散区域聚焦NLS方程的初值问题。在这种情况下，非线性效应占主导地位，解表现出小尺度振荡和向更复杂的振荡模式的剧烈转变（非线性焦散）。MI和SCG在这种行为中都很突出。事实上，在消失色散的极限（也称为半经典极限）中，MI表明，一般平面波解可能会立即分裂成另一种可能更无序的形式。然而，最近的结果表明，确实存在明显的扰动，这些扰动不会激发已知存在于小色散状态中的急性调制不稳定性。这个项目将把这种特殊的扰动纳入聚焦NLS方程的零色散极限的更广泛的理论中。作为这项工作的一部分，将开发一种对这些扰动进行分类和描述的方案。一旦这样做了，就可以评估这种特殊扰动对应用的重要性。本研究的另一个组成部分将是将计算解的傅里叶功率谱的非标准方法扩展到更广泛的初始数据类别；一旦完成，该方法将提供一种新的、直接的方法来测量破波（非线性焦散）对SCG的影响。

The nonlinear Schrödinger (NLS) equation is a fundamental model equation in modern mathematical physics. This equation may serve, for example, to describe both certain water waves and the propagation of light in optical fibers. However, the nonlinear nature of this equation is an obstacle that prevents the straightforward application of most existing general-purpose mathematical tools. This project will use the special structure of this equation to examine in detail two features of its solutions, called modulational instability (MI) and supercontinuum generation (SCG). While the starting point of these investigations is based on the mathematical properties of the NLS equation, MI and SCG are important phenomena in the real world. For example, MI has been proposed as an important element in the formation of "rogue waves" in the ocean; these rare but relatively large and spontaneous waves may be quite dangerous. Furthermore, SCG is important in optical telecommunications systems and in optical coherence tomography, an imaging technique used in opthalmology to reveal detailed retinal structures. A better understanding of this mathematical phenomenon thus has the potential to influence the design and development of telecommunications systems and optical devices. This project's investigation of MI and SCG is based on the initial-value problem for the focusing NLS equation in the small-dispersion regime. In this regime, nonlinear effects are dominant, and solutions exhibit small-scale oscillations and violent transitions (nonlinear caustics) to even more complicated oscillatory patterns. Both MI and SCG figure prominently in this behavior. Indeed, in the limit of vanishing dispersion (also known as the semiclassical limit), MI suggests that a generic plane wave solution may be expected to break immediately into some other, presumably more disordered, form. However, recent results show that there do exist distinguished perturbations which do not excite the acute modulational instabilities known to be present in the small-dispersion regime. This project will incorporate such special perturbations into the broader theory of the zero-dispersion limit of the focusing NLS equation. As part of this effort, a scheme for classifying and describing these perturbations will be developed. Once this is done, the import of such special perturbations for applications can be assessed. Another component of this research will be to extend a non-standard method for computing the Fourier Power Spectrum of solutions to a broader class of initial data; once completed, this method will give a new, direct way to measure the impact of wave breaking (nonlinear caustics) on SCG.