Analysis and numerical methods in solid state physics and chemistry

Mathematical physics;Dynamical system;Frennkel-Kontorova model;Normally hyperbolic invariant manifolds;Parameterization method

Zhang, Lei ; de la Llave, Rafael Mathematics Bonetto, Federico Loss, Michael Uzer, Turgay Zeng, Chongchun ; de la Llave, Rafael

University:Georgia Institute of Technology

Department:Mathematics

关键词: Mathematical physics;    Dynamical system;    Frennkel-Kontorova model;    Normally hyperbolic invariant manifolds;    Parameterization method;   

Others  :  https://smartech.gatech.edu/bitstream/1853/58675/1/ZHANG-DISSERTATION-2017.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

In the first part of the paper, we consider an atomic model of deposition over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posterori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type and a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of equilibrium equation. In the second part of the paper, we consider a model related to Transition State Theory in chemical reactions. We consider a particle with an initial position on the reactant side of a time-dependent energy barrier and study the invariant manifolds and associated bundles using the parameterization method.