LEAPS-MPS: Analysis of Initial and Boundary Value Problems

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2)Differential equations have played a central role in the study of physical phenomena since Sir Isaac Newton described the motion of planetary objects using differential calculus, and are fundamental to the understanding of physics, engineering, biology, chemistry, and economics. Differential equations relate how quantities or systems change over space and time and are derived as models for real world situations. In applications such as optimal investing, infectious diseases, chemical reactions, and tsunamis, the differential models used are often of diffusive or dispersive type. The mathematical theory for diffusive and dispersive nonlinear partial differential equations is incomplete, even in basic setups such as when the region considered can be model as the half line or a bounded interval, which for example is the case when dealing with the ocean floor or a fiber optic cable. This project intends to further the mathematical study of these models, to aid scientists in their understanding of the underlining physical phenomena. The project will provide research and training opportunities for undergraduate and graduate students and for early career researchers, with a focus in promoting the participation of members of underrepresented groups in STEM. The central aim of this project is to study dispersive and diffusive equations on the half line and on bounded intervals using the Uniform Transform Method (UTM), which has recently been developed as an extension to the Fourier transform for initial and boundary data. The UTM allows to construct iterative maps via contour integrals in the complex plane but has not yet been successfully applied to the Nonlinear Schrödinger, Korteweg-de Vries, and Burgers equations, with initial and boundary value data in spaces of very low regularity. This project will use the UTM to extend to bounded intervals previous results obtained for unbounded domains in spaces of both very low and very high regularity, with the main goal of developing the techniques and theory necessary to study well-posedness and asymptotic behavior of the 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.