Project Grant 2348018
- This National Science Foundation Project Grant award in the amount of $311,024 provides funding from June 15, 2022 through May 31, 2025 to support research advancing the low-regularity theory of certain completely integrable dispersive partial differential equations. The primary goal of the research is to study these systems both for their intrinsic mathematical properties and as tools for understanding the statistical mechanics and dynamics of related physical systems. The awardee, the...
- This Project Grant from the National Science Foundation's Mathematical and Physical Sciences program (CFDA 47.049) provides $232,812 to The Johns Hopkins University from July 1, 2022 to June 30, 2025 to improve understanding of dispersive partial differential equations. Key products include research on the long-time behavior of solutions to critical scaling equations like the Schrödinger maps problem and focusing/mass-critical nonlinear Schrödinger equation. Additional work will analyze energy...
- This $224,621 Project Grant awarded by the National Science Foundation (NSF) under the Mathematical and Physical Sciences (CFDA 47.049) program supports research on nonlinear waves and solitons in integrable and non-integrable systems. The principal investigator and their team at The Research Foundation for the State University of New York (RF SUNY), Sponsored Projects Services, will conduct an in-depth investigation into the mathematical properties and physical applications of nonlinear...
- This $231,575 Project Grant was awarded by the National Science Foundation (NSF) under the Mathematical and Physical Sciences (CFDA 47.049) federal grant program. The overarching goal of the project is to establish stability results for solitons, further study their dynamics, and understand the soliton resolution conjecture for general solutions to dispersive equations. The research will utilize techniques from partial differential equations, harmonic analysis, asymptotic analysis, dynamical...
- This three-year, $295,778 Project Grant from the National Science Foundation's Division of Mathematical Sciences will support research addressing long-standing problems in the nonlinear propagation of waves. The principal investigator and collaborators at the University of Chicago will conduct mathematical studies of soliton resolution for energy critical nonlinear wave equations and related models, both with and without symmetries. Quantitative unique continuation properties with connections to...
- This $114,216 Project Grant was awarded by the National Science Foundation (NSF) Division of Mathematical Sciences under the Mathematical and Physical Sciences program (CFDA 47.049). The grant funds a research project at Oregon State University focused on understanding the long-time dynamics and evolutionary behavior of dispersive systems, which have applications in areas like numerical simulations, optics, condensed matter, fluid mechanics, and biology. The project consists of three main...
- This $282,599 Project Grant awarded by the National Science Foundation's (NSF) Division of Mathematical Sciences will support research into validated numerical methods for studying the stability of nonlinear wave phenomena. The investigators will focus on establishing the stability of periodic traveling wave solutions to various Hamiltonian partial differential equations (PDEs), including the generalized Korteweg-de Vries (KdV) and nonlinear Schrödinger equations. Key objectives include...
- The National Science Foundation awarded a $179,538 Project Grant to The Ohio State University under the Mathematical and Physical Sciences program (CFDA 47.049) to support research analyzing initial and boundary value problems for dispersive and diffusive partial differential equations. The award period is from October 1, 2022 through June 30, 2024. The University will use Uniform Transform Method techniques to extend previous results on unbounded domains to bounded intervals in very low and...
- This three-year National Science Foundation Project Grant of $168,087 will support research into nonlinear wave models in bounded domains. The principal investigator and their team at the University of Kansas Center for Research will develop a methodology for analyzing the behavior of dispersive equations modeling optical and water wave phenomena in confined regions. They will provide tools to understand novel behaviors and design physical and numerical experiments. Three research components are...
- This $388,536 project grant awarded by the National Science Foundation (NSF) Division of Mathematical Sciences, under the Mathematical and Physical Sciences (CFDA 47.049) program, supports fundamental research at the University of Massachusetts (UMass) to study the propagation of randomness in nonlinear wave phenomena. The key objectives of the 3-year project are to: Analyze the long-term dynamics and stability of dispersive flows from a probabilistic perspective in energy subcritical regimes....
WELL-POSEDNESS AND LONG-TIME BEHAVIOR OF DISPERSIVE INTEGRABLE SYSTEMS -INTEGRABLE SYSTEMS HAVE LONG SERVED AS GUIDES IN THE STUDY OF HAMILTONIAN PARTIAL DIFFERENTIAL EQUATIONS. THEY ARISE AS EFFECTIVE MODELS OF REAL PHYSICAL SYSTEMS, INCLUDING IN OPTICS AND MANY-BODY QUANTUM MECHANICS. IT IS IN THE SETTING OF COMPLETELY INTEGRABLE SYSTEMS THAT SOLITONS AND MULTISOLITONS WERE FIRST DISCOVERED. THESE STRUCTURES HAVE SINCE FOUND NUMEROUS APPLICATIONS IN THE APPLIED SCIENCES: FOR EXAMPLE, IN FIBER OPTICS, SOLITONS HAVE BEEN EMPLOYED IN THE TRANSMISSION OF DIGITAL SIGNALS OVER LONG DISTANCES, WHILE IN BIOLOGY, THEY ARE USED TO DESCRIBE SIGNAL PROPAGATION IN THE NERVOUS SYSTEM AND LOW-FREQUENCY COLLECTIVE MOTION IN PROTEINS. THIS PROJECT SEEKS TO INVESTIGATE BOTH LONGSTANDING AND NEWLY INTRODUCED INTEGRABLE MODELS. SPECIFICALLY, WE SEEK TO FIND THE MINIMAL CONDITIONS ON THE INITIAL STATE UNDER WHICH ONE CAN CONSTRUCT GLOBAL-IN-TIME DYNAMICS, INVESTIGATE THE (IN)STABILITY OF SPECIAL STRUCTURES (SUCH AS SOLITONS AND MULTISOLITONS), AND ELUCIDATE THE LONG-TIME BEHAVIOR OF GENERAL SOLUTIONS. THE PROJECT PROVIDES SIGNIFICANT RESEARCH TRAINING OPPORTUNITIES FOR GRADUATE STUDENTS, WHO ARE INTEGRATED INTO THE MAIN OBJECTIVES OF THE PROJECT. THE PROJECT INVESTIGATES THE FOLLOWING SPECIFIC QUESTIONS FOR THE NEWLY INTRODUCED CONTINUUM CALOGERO-MOSER EQUATIONS: (1) LARGE DATA WELL-POSEDNESS IN THE SCALING-INVARIANT SPACE, (2) SCATTERING FOR BOTH THE DEFOCUSING MODEL AND THE FOCUSING EQUATION FOR INITIAL DATA WITH MASS LESS THAN THAT OF THE GROUND STATE SOLITON, AND (3) THE DETERMINATION OF THE BLOWUP THRESHOLD IN THE FOCUSING CASE. FURTHER OBJECTIVES INCLUDE ORBITAL AND ASYMPTOTIC STABILITY OF MULTISOLITON SOLUTIONS TO THE BENJAMIN-ONO EQUATION IN OPTIMAL WELL-POSEDNESS SPACES, DISPERSIVE DECAY AWAY FROM THE SOLITON COMPONENT FOR LARGE SOLUTIONS TO THIS EQUATION, AND THE CONSTRUCTION OF GIBBS DYNAMICS FOR THE LANDAU-LIFSHITZ MODEL. 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.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.
Mod # | Description | ReasonForModification | Federal Obligation | Date |
|---|---|---|---|---|
| Not listed | $127.3k | 7/15/25 | ||
| Not listed | $130.7k | 4/1/24 |