Mathematical Sciences: Nonlinear Localization
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The subject of Professor Faris' research is the relation                  between two frontier areas in wave propagation: random waves and                nonlinear waves.  The understanding of these types of waves and                 their interaction is of importance both for our knowledge of                    fundamental physical phenomena as well as technological                         applications in a wide variety of fields; for example,                          electromagnetic waves in plasma, optical fibers with impurities,                and doped superlattices.                                                                 When the medium through which a wave passes is complicated             and irregular, it may be treated as a random medium. The                        solutions, called random waves, of the equations that describe                  wave propagation in a highly random medium, are just beginning to               be sufficiently well understood to compare with experiment.  In                 some situations a remarkable phenomenon occurs, called                          localization, in which repeated scattering actually traps a wave.               The nature of the transition between localization and the more                  common case  of diffusion is not yet well understood, and                       mathematical progress here would be a conceptual advance in the                 study of complicated interactions of waves and matter.  In                      previous work, Professor Faris has found new situations, in                     dimension higher than one, in which localization takes place                    despite the fact that this is not the only possible behavior.  In               the current proposal, he treats localization in the presence of                 nonlinearities.  In this context, the wave interacts with itself                and produces solitary waves, or solitons.  These waves behave in                many ways like classical particles.   The goal is to obtain                     rigorous mathematical results for solitons in random media.  This               would be a major step in understanding the dynamics of complex                  random systems.
{'FirstName': 'William', 'LastName': 'Faris', 'PI_MID_INIT': 'G', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'William G Faris', 'EmailAddress': 'faris@math.arizona.edu', 'NSF_ID': '000433075', 'StartDate': '07/01/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Arizona', 'CityName': 'TUCSON', 'ZipCode': '85721', 'PhoneNumber': '5206266000', 'StreetAddress': '845 N PARK AVE RM 538', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Arizona', 'StateCode': 'AZ', 'CONGRESSDISTRICT': '07', 'CONGRESS_DISTRICT_ORG': 'AZ07', 'ORG_UEI_NUM': 'ED44Y3W6P7B9', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF ARIZONA', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126200', 'Text': 'MODERN ANALYSIS'}
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