## Dispersive Shock Wave Interactions And Two-Dimensional Oceanwave Soliton Interactions

## Douglas Baldwin

## Applied Mathematics Ph.D. Program, University of Colorado Boulder

## Date and time:

Thursday, April 11, 2013 - 12:30pm

## 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. 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.