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Dispersive Shock Wave Interactions And Two-Dimensional Oceanwave Soliton Interactions

Douglas Baldwin

Applied Mathematics Ph.D. Program,Ìý

Date and time:Ìý

Thursday, April 11, 2013 - 12:30pm

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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. So
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 oceanwave
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.