Math and Computer Science Honors Theses

Numerical Approximations for Solitary Waves of the Korteweg-de Vries Equation

Date of Creation

5-8-2014

Document Type

Departmental Honors Thesis - Restricted Access

Department

Mathematics

First Advisor

Steven Levandosky

Abstract

The Korteweg-de Vries (KdV) equation serves as a mathematical model for water waves on shallow surfaces. In this paper, we consider the stability of solitary wave solutions of both the generalized KdV and fifth-order KdV equations subject to various nonlinear terms. Through numerical computation and analysis, we determine the range of wave speed c over which the solitary waves solutions, or "solitons", are stable for these equations. Our analysis utilizes a known theorem regarding stability conditions dependent on a function of wave speed d(c) and its derivatives. We go on to numerically approximate time dependent solutions using both finite difference and spectral methods. Further, we begin to consider the Rotation Modified KadomtsevPetviashvili (RMKP) equation, modeling solitary wave solutions in two spatial dimensions and again analyzing the solution stability.

Comments

Reader: Edward J. Soares

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