Stability of nonlinear dispersive waves and wave collapse phenomena

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The purpose of this project is to study mathematical models of nonlinear dispersive waves that occur in fluids, plasmas, and are encounters in many physical phenomena. Research will be conducted on the stability and instability of solitary waves in nonlinear dispersive equations; wave breaking phenomena for such equations will also be investigated. In particular, the Degasperis-Procesi (DP) equation, the Ostrovsky (OS) equation and the Boussinesq (BQ) equations will be considered. At the heart of this field of inquiry are nonlinear wave interactions. These occur when waves, moving possibly with different speeds and in different directions, intersect. If the interaction is strong enough, then it may create new waves or even lead to wave breaking. Various methods of mathematical analysis will be employed in these investigations. The project will provide specific analysis and numerical simulations of the stability of solitary waves in the OS equations, the strong instability of solitary wave in the BQ equations, and global weak solutions, shock waves, and wave breaking phenomena as well as blow-up structures of the DP equation. The techniques developed in carrying out this project are expected to be useful for other nonlinear dispersive wave equations. Waves are ubiquitous in many different physical contexts, for example, in the ocean, the atmosphere, acoustics, telecommunications, and so on. The most complicated ones are nonlinear waves whose evolution is difficult to predict without performing very complex computations and modeling. In particular, it is known that some of nonlinear waves are not stable; that is, a small change will produce a large disturbance. On the other hand, wave breaking (the wave plunging or surging in short time) is another physical phenomenon of critical importance, especially for water waves. The study of stability of nonlinear waves is necessary for applications such as fusion energy research and fiber-optic communications. The objectives of this project are to develop a better mathematical understanding of which waves are stable and which are unstable, as well as how and when these waves form and break. Research will be focused on certain kinds of nonlinear waves that are well described by mathematical models, but quite difficult to solve. The analysis of these special solutions to the equations has potential impact on the fundamental understanding of ocean waves and currents. Although the equations which are the primary focus of this research are related to water waves and ocean dynamics, the new methods of analysis of special waves and wave breaking developed here could be extended to other areas of fluid mechanics, plasma physics, and other applications. Graduate students with emerging expertise in applied analysis or numerical differential equations will be involved into research and trained through this project.