Numerical investigations of the Korteweg-de Vries (KdV) equation

Abstract

This thesis will develop material regarding the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which has soliton solutions. We introduce the equation with its history and establish some preliminaries in §1. In §2, we will examine the soliton solutions and the uniqueness of such. We will also speak of the construction of multiple soliton solutions, as well as other solutions. Next, the conservation properties of the KdV equation will be visited, then the properties of interacting solitons. In §3 we will discuss the historical numerical schemes for the KdV equation, including finite difference methods, pseudospectral methods, collocation, and finite element methods. We will comment on their accuracy and efficiency. Contained within §4 is a selection of numerical schemes which were implemented (and in one case, improved!) by the author.