Mathematical Sciences: Stability,Instability and SingularityFormation in Nonlinear Partial Differential Equations
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
Nonlinear partial differential equations arising in models                 of physical phenomena and in geometry will be the focus of this                 research.  Studies on the stability and instability of solitary                 traveling waves arising in laser propagation, plasma waves and                  water waves will be carried out.                                                     Solitary waves are localized finite energy solutions of                    nonlinear equations that result from a balance of dispertion and                a focusing nonlinearity.  Important questions to be addressed in                this work are concerned with the stability of solitary waves and                their role in the general structure of solutions.  The model                    equation to be analyzed is the nonlinear Schroedinger equation.                 Many of the phenomena which hold for this equation also hold for                other nonlinear dispersive systems which arise as envelope                      equations.  These include the generalized Korteweg-de Vries,                    Benjamin-Ono, Intermediate Long Wave and Benjamin-Bona-Mahoney                  equations.                                                                           The ground state solitary wave solution plays a critical                   role in that it is an interface between stable and unstable                     solutions.  Two goals of this work are to determine the sense in                which blowing up solutions are attracted to the ground state and                to quantify the loss of compactness in these functions.                              Work related to differential geometry concerns the motion of               curves governed by their curvature.  Flows of this type arise in                the mathematical description of propagating fronts.  The initial                expectation was that one could obtain geodesics as the asymptotic               limit of some gradient flow on the space of curves.  This has                   turned out to be impossible in general.  Efforts will be made to                understand the global behavior of these flows and to determine                  the final form of flows on spaces of curves lying in unstable                   manifolds.
{'FirstName': 'Michael', 'LastName': 'Weinstein', 'PI_MID_INIT': 'I', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Michael I Weinstein', 'EmailAddress': 'miw2103@columbia.edu', 'NSF_ID': '000154899', 'StartDate': '07/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Regents of the University of Michigan - Ann Arbor', 'CityName': 'ANN ARBOR', 'ZipCode': '481091079', 'PhoneNumber': '7347636438', 'StreetAddress': '1109 GEDDES AVE, SUITE 3300', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Michigan', 'StateCode': 'MI', 'CONGRESSDISTRICT': '06', 'CONGRESS_DISTRICT_ORG': 'MI06', 'ORG_UEI_NUM': 'GNJ7BBP73WE9', 'ORG_LGL_BUS_NAME': 'REGENTS OF THE UNIVERSITY OF MICHIGAN', 'ORG_PRNT_UEI_NUM': None}
{'Name': None, 'CityName': None, 'StateCode': None, 'ZipCode': None, 'StreetAddress': None, 'CountryCode': None, 'CountryName': None, 'StateName': None, 'CountryFlag': None, 'CONGRESSDISTRICT': None, 'CONGRESS_DISTRICT_PERF': '""'}
[{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '126600', 'Text': 'APPLIED MATHEMATICS'}]
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8801857.xml'}