Nonlinear Wave Models in Domains with a Boundary

水波、光学等科学领域中的物理现象自然发生在空间受限的区域。在这种现象的建模中，捕捉区域边界的行为是一项基本的挑战，这在数学上转化为色散方程解的存在和唯一性问题，以及这些解对初始和边界条件变化的敏感性问题。这个项目将开发一种方法来破译建模光学和有界区域中的水波现象的方程的复杂行为，并将提供工具来理解和设计物理和数值实验，特别是在捕获新的未探索行为的方向上。调查员将邀请本科生和研究生参与该项目，以符合对培养未来一代数学家和促进数学及其他领域的多样性的承诺。该项目包括三个主要部分。第一部分考虑一维以上空间中的非线性薛定谔方程。第二部分讨论高维有界域上的方程，重点是完全可积模型。这一方向将利用可积方程的丰富数学结构，并在目前大部分未探索的领域提供严格的结果。第三章研究了带边界区域上的多分量非线性浅水波动方程组。这类系统的出现是对欧拉流体动力学方程的近似；因此，在这一方向上取得的成果将对更广泛的水动力学领域具有价值。该奖项反映了NSF的法定使命，并已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。

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.