Dynamics of Dispersive PDE

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This project studies dispersive partial differential equations, which are equations describing wave-like behavior in physical systems. As such, dispersive partial differential equations can describe phenomena in optics, fluids, or plasmas, to name just a few areas of application. To determine the validity of a particular equation as a model for a given phenomenon, it can be helpful to determine whether the equation has a solution, and if it does, how this solution depends upon the physical parameters. For certain nonlinear Schrodinger equations, and for certain equations which can be used as models for waves in water, the principal investigator will study when these equations either lack solutions, or have solutions which depend discontinuously upon the data. This will indicate that these particular models have only limited validity for applications. As another part of the project, for some other families of dispersive partial differential equations, the principal investigator will determine when the equations either do or do not possess solutions with certain special forms. Understanding such solutions, such as waves of permanent form or time-periodic waves, can lead to a deeper understanding of the behavior of the corresponding physical phenomena over long intervals of time.The principal investigator will study well-posedness and ill-posedness for quasilinear Schrodinger equations and for truncated series water wave models. For quasilinear Schrodinger equations, well-posedness theorems have been proved under certain non-degeneracy hypotheses; in this work, ill-posedness will be studied when the non-degeneracy condition is violated. For truncated series models of water waves, the relationship between well-posedness/ill-posedness and the strength of dispersion will be investigated using complex variable methods and tools from paradifferential calculus. For nonlinear Schrodinger equations and for general families of nonlinear dispersive equations, the principal investigator will use small divisor estimates, normal forms, and dispersive smoothing estimates to demonstrate the non-existence of small-amplitude spatially periodic, time-periodic waves in certain parameter regimes. Also, existence of time-periodic, spatially-periodic waves for dispersive equations with high-derivative nonlinearities (especially for systems related to capillary waves in fluids) will be studied, using small divisor methods and techniques of Fourier analysis.