Dynamics of Solutions to the Nonlinear Dispersive PDEs

主要的研究人员将研究非线性色散偏微分方程解的行为，特别是非线性薛定谔方程(NLS)。后一个方程代表了一大类色散模型，它提供了最简单的例子。NLS表现出各种类型的解：一些在有限时间内存在(非线性效应超过线性)；其他类孤子解全局存在但可能不随时间衰减(非线性和线性效应相互平衡)，以及全局时间解完全随时间分散(与线性效应相比，非线性效应可以忽略不计)。该项目将研究一类具有聚焦非线性的色散方程的所有类型解的动力学。特别地，将从解析和数值两方面探讨收缩环爆破解的一种新现象。将对集中现象进行分析，以更好地了解此类解的爆破行为。这个项目的结果将促进我们对非线性光学、量子和等离子体物理以及流体动力学中出现的各种物理现象的理解。将在几个科学组织举办关于调和分析和微分方程式方法的讲座和小型系列讲座。作为一个直接的教育结果，该项目将提高亚利桑那州立大学学生的科学和数学意识，因此，从长远来看，将提高社会的科学意识。亚利桑那州立大学有一大批女性和少数族裔学生，这将使首席研究员凭借她以前的经验，为未来培养一个多样化的科学界。她将让研究生和本科生参与她的研究项目。新一代解决问题的数学家的教育、新方法和新技术的传播以及普遍的科学进步将是该项目的一些好处。

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.