PublisherThe University of Arizona.
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AbstractThe modulational behavior of exact oscillatory solutions to a family of non-linear systems of coupled differential equations is studied both numerically and analytically. The family of lattice systems investigated has applications ranging from theoretical biology to numerical methods. The goal is to obtain a description, given by a system of partial differential equations valid on long spatial and temporal scales, of the microscopic vibrations in the lattice. A theory of simple harmonic plane wave modulation is given for the entire family of microscopic systems, and the structure of the corresponding modulation equations is analyzed; particular utility is gained by casting the modulation equations in Riemann invariant form. Although difficulties are encountered in extending this theory to more complicated oscillatory modes in general, the special case of the integrable Ablowitz-Ladik system allows the program of describing more complicated modulated oscillations to be carried out virtually to completion. An infinite hierarchy of multiphase wavetrain solutions to these equations is obtained exactly using methods of algebraic geometry, and the complete set of equations describing the modulational behavior of each kind of multiphase wavetrain is written down using the same machinery. The distinguishing features of modulation theory in the presence of resonance are described, and an unusual set of modulation equations is derived in this case. The results of this dissertation can be interpreted in the context of nonequilibrium thermodynamics of regular oscillations in nonlinear lattices; instabilities in the modulation equations correspond to predictable phase transitions.
Degree ProgramApplied Mathematics