Dynamics of Solutions to the Nonlinear Dispersive PDEs

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The principal investigator will study the behavior of solutions to nonlinear dispersive partial differential equations, in particular, to the nonlinear Schrodinger equation (NLS). The latter equation is representative of a large class of dispersive models, of which it provides the simplest example. The NLS exhibits various types of solutions: some exist in finite time (nonlinear effects dominate over linear); other soliton-like solutions exist globally but may not decay with time (nonlinear and linear effects balance each other), and global-in-time solutions completely disperse with time (nonlinear effects are negligible compared with linear ones). The project will study the dynamics of all types of solutions for the class of dispersive equations with focusing nonlinearities. In particular, a new phenomenon of the contracting-ring blow-up solutions will be explored both analytically and numerically. The concentration phenomenon will be analyzed to obtain a better understanding of the blow-up behavior of such solutions. Results of this project will advance our understanding of various physical phenomena that arise in nonlinear optics, quantum and plasma physics, and fluid dynamics. Talks and mini-lecture series on methods of harmonic analysis and differential equations developed from the proposed research will be delivered at several scientific organizations. As an immediate educational consequence, the project will increase the scientific and mathematical awareness of the Arizona State University student body and thus, in the long run, will enhance the scientific awareness of society. A significant pool of female and minority students at Arizona State University will enable the principal investigator, with her previous experience, to educate a diverse scientific community for the future. She will engage graduate and undergraduate students in her research program. The education of a new generation of problem-solving mathematicians, the dissemination of new methods and techniques, and general scientific advancement will be some of the benefits of this project.