The proposal focuses on the stability of waves for several models both in one and higher spatial dimensions. The common feature of these equations is that they can be viewed as dynamical systems on an infinite-dimensional space. These Hamiltonian PDEs support coherent structures such as solitary waves as well as periodic wave solutions. The PI will address open questions by developing a systematic approach to the spectral stability of waves for second order in time PDEs. An advantage of this method is that it treats both spatially periodic and solitary waves, which provides a link between the well studied results for solitary waves and the mostly open questions on periodic waves. These models are at the frontiers of current research and present lots of challenges at the spectral and linear stability level, but virtually nothing is known for their asymptotic stability.
The PI studies will provide for better theoretical understanding of various nonlinear systems such as the beam equation, water waves and the Klein-Gordon model. Any progress on these questions will not only be important mathematically, but will find immediate applications in physical sciences. The PI will train the future researches by working with graduate and undergraduate students on the topics of the proposal. A special freshman seminar on nonlinear waves is being designed by the PI as part of a pilot First Year Seminar program at the University of Kansas. The PI has worked with students from local schools and is the main organizer for several mathematics competitions for school age kids. The students preparing for the competitions have been exposed to applied mathematics problems and have gained a better understanding of the role mathematics plays in modeling real life phenomena.