Integrable and Non-Integrable Wave Models under the Effect of Nonzero Boundary Conditions

About This Grant

Many physical phenomena in fields such as water waves and optics naturally occur in regions with spatial boundaries, presenting significant mathematical challenges. These challenges stem from the complex ways in which boundaries influence the behavior of the underlying partial differential equations (PDEs) used to model such systems. This project will yield a universal methodology for analyzing the dynamics of PDEs in bounded domains, across both one and higher spatial dimensions. It will address fundamental questions about the existence, uniqueness, and stability of solutions, and provide new analytical and computational tools to study boundary-influenced behavior. The project will support the training of undergraduate and graduate students, contributing to the development of the next generation of mathematicians. The project will advance three interconnected directions: (1) the study of nonlinear PDEs in bounded domains in two or more spatial dimensions; (2) the analysis of multi-component systems with nonzero boundary conditions; and (3) the development of a framework for equations with nonzero boundary conditions at infinity, which is closely associated with complex phenomena such as modulational instability and rogue waves. Many of the systems investigated in this project arise as approximations to the fundamental Euler and Navier-Stokes equations in fluid dynamics. As such, the results produced will be of value to the broader area of hydrodynamics and other areas of applied mathematics, including concrete real-world problems from control theory, particularly in stabilization problems, and physical/laboratory experiments, such as the wavemaker problem. 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.