Contemporary Mathematics

Volume 379, 2005

An Introduction to Wave Equations

Ronald E. Mickens

ABSTRACT. This paper provides a concise, but heuristic introduction to vari-

ous wave equations that serve important roles as mathematical models of many

interesting dynamical systems in the natural and engineering sciences. We be-

gin our discussion with a general introduction as to why the physical equations

should be reformulated, by means of scaling of their variables, to dimensionless

equations. This procedure is illustrated by applying it to a number of both lin-

ear and nonlinear ordinary and partial differential equations. The remainder

of the paper is devoted to analyzing certain of the mathematical properties of

some linear and nonlinear wave equations.

1.

Introduction

This paper is a summary of the topics presented by the author, in two lectures,

given at the Regional Research Conference, "Mathematical Methods in Nonlin-

ear Wave Propagation," held May 15-19, 2002 at the Mathematics Department of

North Carolina A and T State University in Greensboro, NC. My main task was

to provide general background information and physical explanations for many of

the issues discussed by the principal lecturer in his ten lectures. This person was

Professor

J.

Kenneth Shaw (Departments of Mathematics, and Computer and Elec-

trical Engineering, Virginia Polytechnic Institute and State University, Blacksburg,

VA). Following an exchange of emails between us, we agreed that whatever each of

us presented, it should not contradict what the other gave! This compact worked

quite well as evidenced by the good reception of both of our presentations.

In the material to follow, I give a quick overview of several important topics

required if one is to understand the nature of wave equations, especially their use

as mathematical models of important physical and engineering systems. The paper

is organized into {our main sections. Section 2 introduces the concept of scaling of

variables such that the original "physical equations" get transformed into dimen-

sionless "mathematical equations." These are the differential equations that can

be studied using standard procedures from the area of differential equations. The

1991

Mathematics Subject Classification.

Primary 35K15, 35K55, 35L05; Secondary 35L15,

35L45, 35L60.

Key words and phmses.

Nonlinear wave equations, scaling and dimensionless variables, soli-

tons, traveling waves.

The work reported here was funded in part by research grants from DOE and the MBRS-

SCORE Program at Clark Atlanta University.

@

2005 American Mathematical Society

http://dx.doi.org/10.1090/conm/379/07022