Keywords
nonlinear dynamics, soliton, reversible system, bifurcation theory, differential equations
Abstract
In this thesis, we apply a recently developed technique to comprehensively categorize all possible families of solitary wave solutions in two models of topical interest. The models considered are: a) the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods, and b) a generalized microstructure PDE. Limited analytic results exist for the occurrence of one family of solitary wave solutions for each of these equations. Since, as mentioned above, solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions of both models here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves for each model, we find a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. For the microstructure equation, the new family of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including the dynamics of each family of solitary waves using exponential asymptotics techniques, are also mentioned.
Notes
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Graduation Date
2008
Advisor
Choudhury, S. Roy
Degree
Master of Science (M.S.)
College
College of Sciences
Department
Mathematics
Degree Program
Mathematical Science
Format
application/pdf
Identifier
CFE0002151
URL
http://purl.fcla.edu/fcla/etd/CFE0002151
Language
English
Release Date
June 2008
Length of Campus-only Access
None
Access Status
Masters Thesis (Open Access)
STARS Citation
Leto, Jonathan, "Solitary Wave Families In Two Non-integrable Models Using Reversible Systems Theory" (2008). Electronic Theses and Dissertations. 3684.
https://stars.library.ucf.edu/etd/3684