Spatially quasi-periodic water waves

Water waves are ubiquitous in nature, occurring everywhere from oceans and lakes to swimming pools and bathtubs. Their study has led to important engineering applications, such as tsunami detection, wave energy harvesting, and coastal protection using breakwaters. Water waves are typically studied under the assumption that they are periodic (i.e., possessing a spatially repeating pattern) or that they decay to zero at infinity (i.e., no waves are coming from the far away). However, these assumptions are often too restrictive for exploring more complex and realistic phenomena. For instance, large bodies of water such as oceans are often covered with waves of different wavelengths, propagating in all directions over vast distances. Furthermore, these wavelengths can be incommensurate. To accurately model the interactions of water waves, the investigator studies them in a spatially quasi-periodic setting. This project will advance the modeling and simulation of water waves, with broad applications in ocean wave forecasting and shoreline protection. The project will also offer research and training opportunities for students. One of the main goals of the project is to develop accurate and stable numerical methods for studying spatially quasi-periodic water waves in both two and three dimensions. These methods will then be used to simulate such waves over flat or variable bottom topography. Traveling wave solutions will be computed, and their stability will be analyzed. In addition, the project will contribute to the study of periodic water waves. The quasi-periodic framework will enable the investigation of the nonlinear instability of classical periodic traveling waves under subharmonic perturbations, whose wavelengths are incommensurate with those of the original wave. Beyond the numerical aspect, the project also aims to advance the theoretical understanding of spatially quasi-periodic water waves and address the local well-posedness of these solutions. 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.