Dissertation
Nanopteron-stegoton traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou lattices
Doctor of Philosophy (Ph.D.), Drexel University
May 2018
DOI:
https://doi.org/10.17918/D8B95H
Abstract
We study the existence of traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. These are infinite, one-dimensional lattices of particles connected by nonlinear springs, in which either the masses alternate (the mass dimer or diatomic lattice) or the spring forces alternate (the spring dimer). Under the classical "long wave" scaling, the lattice equations of motion turn out to be singularly perturbed. In response to this complication, we apply a method of Beale to produce nanopteron traveling wave solutions with wave speed slightly greater than the lattice's speed of sound. The nanopteron wave profiles are the superposition of an exponentially decaying term (which itself is a small perturbation of a KdV soliton) and a periodic term of very small amplitude. This dissertation builds on the previous work of Faver and Wright on mass dimer lattices to treat spring dimer lattices. Further generalizing the spring forces from the mass dimer case, we allow the springs' nonlinearity to contain higher order terms beyond the quadratic. This necessitates the use of composition operators to phrase the long wave problem, and these operators require delicate estimates due to the characteristic superposition of different function types from Beale's ansatz. Additionally, the value of the leading order term in the spring dimer traveling wave profiles alternates between particle sites, so that, unlike in the mass dimer, the spring dimer traveling waves are also "stegotons."
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Details
- Title
- Nanopteron-stegoton traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou lattices
- Creators
- Timothy E. Faver - DU
- Contributors
- Jay Douglas Wright (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- xii, 159 pages
- Format
- Dissertation
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 8022; 991014632222904721