Mathematical Sciences: Research in Nonlinear Wave Motion
{'Code': '03040100', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'Michael Steuerwalt', 'PO_EMAI': None, 'PO_PHON': None}
8822444                                                                         Segur                                                                                This project includes three separate lines of research in                  nonlinear wave motion in evolutionary systems.  The three ideas                 are largely independent.  Part A concerns the development of an                 effective method to determine whether or not a particular                       solution of an evolution equation is stable to small changes in                 initial data.  Stability theory advanced significantly after                    the discovery by Arnol'd (1965) and others of a systematic                      method to construct Lyapunov functionals.  Even so, the methods                 currently available are unable to determine whether a flow as                   simple as stably statified, paralled shear flow is a stable                     solution of Euler's equations in two dimensions.  One goal is                   to devise a generalization of current methods that might settle                 the question.                                                                                                                                                      The focus in part B is on the Kadomstev-Peviashvili (KP)                     equation with periodic boundary conditions.  The equation is                    known to be completely integrable, so there is reason to                        believe that the initial value problem for the KP equation with                 periodic boundary conditions is solvable.  The KP equation also                 provides accurate models of a class of water waves, and it is                   intimately connected with the theory of Riemann surfaces in                     algebraic geometry.  In fact, the problem is so rich that                       progress in almost any direction would valuable.                                    The objective in part C is to construct a new kinetic                       theory for a large collection of wavepackets that interact                      primarily through resonant triads.  If successful, the theory                   would result in a  Boltzmann-type equation, whose equilibrium                   solutions would provide an equilibrium spectrum of wave                         energies, analogous to the Maxwell-Boltzmann distribution for                   molecules.  An earlier version of such a theory was given by                    Segur (1984); the work in this part of the project would                        correct a deficiency in that model.
{'FirstName': 'Harvey', 'LastName': 'Segur', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Harvey Segur', 'EmailAddress': 'segur@colorado.edu', 'NSF_ID': '000310754', 'StartDate': '06/01/1989', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
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{'Code': '1266', 'Text': 'APPLIED MATHEMATICS'}
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