LEAPS-MPS: Long-time behavior for nonlinear dispersive equations

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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). Nonlinear dispersive partial differential equations arise in many physical settings and are characterized by the tendency of waves of different frequencies to travel at different velocities. In such models there is often a competition between dispersive and nonlinear effects, resulting in a rich set of possible solution behaviors. These include decay and scattering, the presence of coherent structures known as solitary waves, or even wave collapse (or blowup). This project includes consideration of problems related to the long-time behavior of solutions to nonlinear dispersive equations, including the stability properties of solitary waves, global decay estimates for low regularity solutions, and the behavior of solutions living at or near certain sharp scattering thresholds. The project focuses on several specific models that are physically meaningful but still simple enough to admit deep analysis. Such choices allow for the distillation of the essential mathematical difficulties underlying some important problems in the field of dispersive equations. In this way, the proposed research has the potential to pave the way for future progress even beyond the specific problems under consideration in this project. The project contains problems that are suitable for the involvement of students at the undergraduate, Masters, and PhD levels. The project includes several activities to encourage participation of underrepresented or rural students in STEM via outreach to public schools, organization of meetings and mentoring of undergraduate research projects.The project will first address asymptotic stability properties for solutions to the one-dimensional nonlinear Schrodinger equation (NLS) in the presence of an attractive delta potential, a simple model arising in nonlinear optics. Some of the main goals include establishing asymptotic stability for the entire family of stable solitary waves, as well as the construction of stable manifolds in the unstable regime. Next, the project will address the problem of global space-time estimates for low regularity solutions to completely integrable models, including the 1d cubic NLS. The project seeks to develop virial and Morawetz-type estimates adapted to the novel microscopic conservation laws that have recently played a key role in the low-regularity well-posedness theory for such equations. Third, the project will address several problems related to threshold behaviors for solutions to NLS models with broken symmetries, including the inhomogeneous NLS and the NLS with external potentials. In addition to classifying the possible solution dynamics at the sharp scattering threshold, the project will involve the construction of solutions with traveling wave behavior for models that lack a nonlinear ground state. Finally, the project seeks to increase participation from underrepresented groups in mathematics by fostering student interest in STEM subjects, beginning at the high school level, as well as developing a supportive community for mathematics students at both the undergraduate and graduate level. Specific steps towards this goal include outreach to public high schools, the organization of regular meetings and presentations for undergraduate math majors, the supervision of undergraduate research, and the continued organization of research seminars and invitation of research visitors.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.