Date of Award

Spring 1-1-2015

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Harvey Segur

Second Advisor

James Meiss

Third Advisor

Diane Henderson

Fourth Advisor

Keith Julien

Fifth Advisor

Bengt Fornberg

Abstract

The resonant interaction of three wavetrains is the simplest form of nonlinear interaction for dispersive waves of small amplitude. Such interactions arise frequently in applications ranging from nonlinear optics to internal waves in the ocean through the study of the weakly nonlinear limit of a dispersive system. The slowly varying amplitudes of the three waves satisfy a set of integrable nonlinear partial differential equations known as the three-wave equations. If we consider the special case of spatially uniform solutions, then we obtain the three-wave ODEs. The ODEs have been studied extensively, and their general solution is known in terms of elliptic functions. Conversely, the universally occurring PDEs have been solved in only a limited number of configurations. For example, Zakharov and Manakov (1973, 1976) and Kaup (1976) used inverse scattering to solve the three-wave equations in one spatial dimension on the real line. Similarly, solutions in two or three spatial dimensions on the whole space were worked out by Zakharov (1976), Kaup (1980), and others. These known methods of analytic solution fail in the case of periodic boundary conditions, although numerical simulations of the problem typically impose these conditions.

To find the general solution of an nth order system of ordinary differential equations, it is sufficient to find a function that satisfies the ODEs and has n constants of integration. The general solution of a PDE, however, is not well defined and is usually difficult, if not impossible, to attain. In fact, only a small number of PDEs have known general solutions. We seek a general solution of the three-wave equations, which has the advantage of being compatible with a wide variety of boundary conditions and any number of spatial dimensions. Our work indicates that the general solution of the three-wave equations can be constructed using the known general solution of the three-wave ODEs. In particular, we try to construct the general solution of the three-wave equations using a Painleve-type analysis. For now, we consider a convergent Laurent series solution (in time), which contains two real free constants and three real-valued functions (in space) that are arbitrary except for some differentiability constraints. In order to develop a full general solution of the problem, the two free constants must also be allowed to have spatial dependence, and one more function must be introduced. That is, a full general solution of the problem would involve six of these real-valued functions.

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