Stability of waves in discrete and continuous dynamical systems

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The main theme of the proposed research will be the linear and asymptotic stability of special solutions (waves) of a wide class of partial differential equations. In a series of recent works, the PI and collaborators have characterized the linear stability of travelling and standing waves for second order in time equations and systems. Such results provide necessary information for the asymptotic stability of the same waves. The PI and collaborators will build on their previous work to show asymptotic stability of standing waves for the following models: the Klein-Gordon equation, the sine Gordon equation and the Dirac equation. A second goal of the proposal is to study the existence and stability of coherent structures, arising in spatially discrete models. More concretely, the project will deal, among other things, with Hertzian granular chains/crystals and the discrete nonlinear Schr\"odinger equation.The project deals with nonlinear dispersive equations, which model wavelike behavior of important physical processes. Important class of problems under consideration include the propagation of light in optical waveguides, motion of quantum particles, the mechanics of fluids to mention a few. The overarching theme of the investigation will be the stability of coherent configuration - that is, if one is initially close to such coherent structure, does it stay close to it forever? The mathematical formulation of such problems, as well as their analysis and predictions about their long time behavior will greatly enhance our understanding of these and related processes.