Date of Award
Doctor of Philosophy (PhD)
A numerical procedure to solve the differential equation for solitary waves via iteration for a generalized Frenkel-Kontorova model is presented. The generalization consists of adding a constant force proportional to particle velocity. The equation of motion is cast into integral form using a Green's function. The introductory chapter briefly reviews recent progress in nonlinear dynamics using the sine-Gordon equation as an example. Emphasis is placed on the concept of "non-linear modes" and its relevance in theoretical descriptions of transport processes. After a second chapter defining the generalized Frenkel-Kontorova equation, its dimensionless reparametrization and an outline of the iteration scheme, the third chapter defines and solves the continuum approximation of a single kink transport problem. An exact functional relationship connecting applied force, friction and the resulting solitary wave velocity is given. Use is made of the sine-Gordon equation and a power balance formula in choosing a good trial solution. The numerical solution of the transport problem is given and compared with the work of others as well as the exact solutions of two similar problems. I also demonstrate how the calculation is related to a variational principle. In the fourth chapter the full discrete case is examined. How this differs from the continuum approximation is discussed as well as why the continuum approximation fails to adequately model aspects of transport processes in such applications as crystal defect propagation and superionic conductors. After presenting the solution to a related solvable model, I present results of my calculations. Throughout this work the Green's function used in the numerical analysis is shown to have physical meaning and is used in particular to clarify the role of discreteness in transport processes. No simplifying assumptions are made.
Adams, Louis William Jr, "The Solution to Single Kink Solitary Wave Transport for Systems Described by a Generalized Frenkel-Kontorova Equation." (1980). LSU Historical Dissertations and Theses. 3508.