Date of Award

Spring 1-1-2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Mark J. Ablowitz

Second Advisor

Keith Julien

Third Advisor

Bengt Fornberg

Fourth Advisor

Willy Hereman

Fifth Advisor

Patrick Weidman

Abstract

Many physical phenomena are understood and modeled with nonlinear partial differential equations (PDEs). Unfortunately, nonlinear PDEs rarely have analytic solutions. But perturbation theory can lead to PDEs that asymptotically approximate the phenomena and have analytic solutions. A special subclass of these nonlinear PDEs have stable localized waves--called solitons--with important applications in engineering and physics. This dissertation looks at two such applications: dispersive shock waves and shallow ocean-wave soliton interactions. Dispersive shock waves (DSWs) are physically important phenomena that occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg-de Vries (KdV) equation is the universal model for phenomena with weak dispersion and weak quadratic nonlinearity. Here we show that the long-time asymptotic solution of the KdV equation for general step-like data is a single-phase DSW; this DSW is the `largest' possible DSW based on the boundary data. We find this asymptotic solution using the inverse scattering transform (IST) and matched-asymptotic expansions; we also compare it with a numerically computed solution. While multi-step data evolve to have multiphase dynamics at intermediate times, these interacting DSWs eventually merge to form a single-phase DSW at large time. We then use IST and matched-asymptotic expansions to find the modified KdV equation's long-time-asymptotic DSW solutions. Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these nonlinear interactions look like an X or a Y or two connected Ys; much less frequently, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. This dissertation reports that such interactions occur every day, close to low tide, on two flat beaches that are about 2,000 km apart. These interactions are related to the analytic, soliton solutions of the Kadomtsev-Petviashvili equation. On a much larger scale, tsunami waves can merge in similar ways.

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