PI: Grozdena Todorova, University of Tennessee, Knoxville DMS-0245578
The proposed project intends to investigate current issues in nonlinear wave equations involving the topics of asymptotic behavior of solutions, development of singularities, and instability. We intend to study problems that deal with the dynamics in the vicinity of the least energy standing waves, investigate the question of stability and instability of bound states and bound states standing waves for fundamental equations of Mathematical Physics: Nonlinear Klein-Gordon (NLKG), Nonlinear Schroedinger (NLS), Nonlinear Damped Wave (NLDW).
The interaction between the variational structure of stationary equations associated with hyperbolic problems and the evolutionary nature of the process makes this type of problems a very interesting area in nonlinear analysis. We also intend to study the stability/instability of the trivial equilibrium, i.e., the behavior of the small amplitude solutions of various important hyperbolic equations, with the purpose to derive sharp criteria for blow--up and global existence.
Further questions of interest are: the asymptotically parabolic structure of the hyperbolic equations due to the presence of damping terms; the influence of damping on the principal support of solutions; the influence of damping on the decay rate of the energy; and the asymptotic behavior of solutions in external domains where the delicate interaction between the geometrical condition on the boundary and the dissipation has to be taken into account.
In all of the proposed problems, the underlying balance between different kinds of forces, which affect the solution, will be considered. It is important to add that the solution of many of the problems in the proposal requires the development of new techniques as well as the refinement of more classical methods.
Nonlinear Klein-Gordon, Nonlinear Damped Wave and Nonlinear Schroedinger equations arise in fluid dynamics, classical mechanics, quantum mechanics, plasma physics and nonlinear optics. In particular nonlinear optics is a main tool in telecommunication technology, mainly based on the large use of fiber optics. There is no need here to emphasize the relevance of the fiber optics in large capacity connection nets, which are crucial in high-speed Internet applications. Since the Nonlinear Schroedinger equation is a useful design tool for realistic fiber communication systems, it is important to understand the stability/instability phenomena of its solutions.
The solutions of the problems outlined in the proposal will develop our understanding of the physical phenomena related with the above fundamental equations. Given the crucial role of these equations in many areas of technology, understanding the principles governing them is of basic importance for a society built on technology.
