Research on Dispersive Partial Differential Equations

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DMS-9800879 Gigiola Staffilani Abstract of the Research Project My research involves the study of certain Dispersive Partial Differential Equations. I concentrate my work on the Cauchy problem for two particular dispersive equations: the nonlinear Schr\"odinger equation and the generalized Korteweg-de Vries equation. The first question I address in my project is the following: given a dispersive equation, how much regularity does one have to assume for the initial profile (initial data condition) in order to be able to insure existence and uniqueness of the wave solution at later times? Such a question becomes more complicated to answer when one also imposes boundary conditions. At the moment, the techniques available (Harmonic and Fourier Analysis) only apply to the case of periodic boundary conditions. The second question I address is how certain regularity properties of the wave solution are preserved in time, once it is assumed that the wave ``lives'' for all times and it is smooth. More precisely, under these conditions, I study the asymptotic behavior of the size of the wave measured taking into account the order of differentiability. This is a very different problem than the one described above. Nevertheless the techniques I rely upon to answer both questions are very similar in their nature. Dispersive partial differential equations are introduced to model many wave phenomena that occur in nature. Some examples are, for instance, propagation of signals in optic fibers, nonlinear ionic-sonic waves in plasma in a magnetic field and long waves in plasma. In my work I improve and develop some abstract mathematical tools in order to obtain from general properties of the initial profile of the wave information about the evolution of the wave itself. This information regards the time of existence of the wave, the persistence of certain properties, the behavior near singular points, and the energy carried by the wave. In some cases part of this information can be recovered by conducting experiments in laboratories. Conversely, the results I obtain provide mathematical justifications for certain empirical observations.