Dynamics and Stability of Nonlinear Waves

许多物理过程以数学模型的形式得到最好的描述，这些数学模型涉及控制特征物理量的相互作用和动力学的(偏微分方程式)，例如幅度、电压或浓度。人们经常观察到一种类似群体的行为，这归因于这些数学模型的所谓波解的存在和持久性。这种解决方案及其稳定性(或不稳定性)性质的确定在当代技术应用中具有极其重要的意义。另一方面，有关不稳定和异常行为(如爆炸)的准确信息提供了有关数学模型本身的有效性和适用性的有价值的信息。这项研究旨在为研究非线性色散方程提供新的数学工具，这些方程包括水波动力学、材料磁化、光在光学器件中的传播以及与量子力学相关的现象的数学模型等。特别是，首席调查员(PI)的目标是获得关于此类系统长期行为的准确、定量的信息。此外，PI将积极参与新一代科学家的培训，他们需要在几个领域拥有技术专业知识，以应对尖端物理系统及其建模的分析和计算方面的挑战。该奖项反映了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。

Many physical processes are best described mathematically in terms of mathematical models that involve (partial differential) equations governing the interactions and dynamics of characteristic physical quantities, such as amplitude, voltage, or concentration to name a few. Very often, one observes a group-like behavior, which is attributed to the existence and persistence of the so-called wave solutions of these mathematical models. The identification of such solutions and their stability (or instability) properties is of enormous importance in contemporary technological applications. On the other hand, precise information about instability and abnormal behavior (such as blow ups) provides valuable information about validity and applicability of mathematical models themselves. This research is aimed at providing new mathematical tools to study non-linear dispersive equations, which are mathematical models for water wave dynamics, magnetization of materials, propagation of light in optical devices and related phenomena of quantum mechanics, to name a few. In particular, the principal investigator (PI) aims obtaining precise, quantitative information about the long-time behavior of such systems. In addition, the PI will be actively engaged in the training of the new generation of scientists, who will need to have the technical expertise in several areas to meet the challenges in the analysis and computation of cutting-edge physical systems and their modeling.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.