RUI: Solitary structures in nonlinear wave equations

该提案涉及浅水波非线性波动现象的研究，非线性光学，等离子体和其他非线性物理领域。对这种现象产生的孤立波的全面表征是一个重要的问题，但还远远没有完成。因此，本研究项目的目标是通过分析相关解空间的基本数学结构，对描述这种局部波形和波型的非线性方程进行广泛的研究。由于该理论被各种物理系统所共享，因此预计所提出的工作的结果将为其他领域提供见解，例如非线性光学中的光波和磁性薄膜中的自旋波。第二个目标是应用这个项目的理论结果来研究产生在公海经常观察到的极高海拔的“流氓”波的可能机制，以及浅水中的表面波模式。了解这种极端波浪的性质和动力学，并最终预测在人口密集的沿海地区或航运繁忙的港口入口处的海洋中出现的这种波浪现象，是极其重要的任务。拟议的研究活动将涉及几个本科生在这个项目的过程中，因为它将在一个本科院校进行。学生将获得应用数学的第一手研究经验，这可能会激励他们攻读数学科学的研究生或其他职业机会。

The proposal concerns the investigation of nonlinear wave phenomena in shallow water waves, nonlinear optics, plasma and other areas of nonlinear physics. A comprehensive characterization of the solitary waves resulting from such phenomena is an important problem, which is far from complete. Therefore an objective of this research project is to conduct an extensive study of the nonlinear equations describing such localized wave-forms and wave-patterns by analyzing the underlying mathematical structures of the relevant solution spaces. Since the theory is shared by various physical systems, it is anticipated that the results from the proposed work would provide insights into other areas such as light waves in nonlinear optics, and spin waves in magnetic thin films. A second objective is to apply the theoretical results of this project to investigate possible mechanisms generating "rogue" waves of extremely high elevations frequently observed in open seas, and surface wave patterns in shallow water. Understanding the nature and dynamics of such extreme waves, and ultimately predicting such wave phenomena in oceans near highly populated coastal areas or at harbor entrances with heavy shipping traffic, are extremely important tasks. The proposed research activities will involve several undergraduate students over the course of this project since it will be carried out in an undergraduate institution. The students will gain firsthand research experience in applied mathematics, which may motivate them to pursue graduate studies or other career opportunities in mathematical sciences.