Nonlinear Wave Models in Domains with a Boundary

水波、光学和其他科学领域中的物理现象自然地发生在空间受限的区域中。在这种现象的建模中捕捉区域边界处的行为提出了基本的挑战，用数学术语来说，这转化为色散方程解的存在性和唯一性问题，以及这些解对初始和边界条件变化的敏感性问题。该项目将开发一种方法，用于破译有界区域光学和水波现象建模方程的复杂行为，并将提供工具来理解和设计物理和数值实验，特别是在捕捉新的未探索行为的方向。研究者将让本科生和研究生参与该项目，以致力于培养未来几代数学家和促进数学及其他领域的多样性。该项目包括三个主要组成部分。第一个分量考虑高于一个空间维度的非线性Schrödinger-type方程。第二部分着重于高维有界域中的方程，重点是完全可积模型。这个方向将利用可积方程丰富的数学结构，并在目前很大程度上未开发的领域提供严格的结果。第三部分研究了带边界域的非线性浅水波动方程的多分量系统。这种系统产生于流体力学欧拉方程的近似；因此，在这个方向上取得的成果将对更广泛的流体力学领域有价值。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Physical phenomena in water waves, optics, and other fields of science naturally occur in spatially confined regions. Capturing in the modeling of such phenomena the behavior at the boundary of the region presents fundamental challenges, which in mathematical terms translates into questions of existence and uniqueness of solutions of dispersive equations and of sensitivity of these solutions to changes in initial and boundary conditions. This project will develop a methodology for deciphering the complex behavior of the equations modeling optics and water waves phenomena in bounded regions and will provide tools to understand and design physical and numerical experiments, especially in the direction of capturing novel unexplored behaviors. The investigator will engage both undergraduate and graduate students in the project, in line with a commitment to the training of future generations of mathematicians and the promotion of diversity in the field of mathematics and beyond.The project involves three main components. The first component considers nonlinear Schrödinger-type equations in higher than one space dimensions. The second focuses on equations in higher dimensional bounded domains with an emphasis on completely integrable models. This direction will exploit the rich mathematical structure of integrable equations and provide rigorous results on a currently largely unexplored area. The third studies multi-component systems of nonlinear shallow water wave equations in domains with a boundary. Such systems arise as approximations to the Euler equations of hydrodynamics; therefore, results achieved in this direction will be of value to the broader area of hydrodynamics.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.