Contents

Preface ix

Chapter 1. Diﬀerential Equations 1

§1.1. Classiﬁcation of Diﬀerential Equations 4

§1.2. Can we write solutions explicitly? 5

§1.3. Diﬀerential Equations as Models of Reality

§1.4. Named Equations 8

§1.5. When are two equations equivalent? 9

§1.6. Evolution in Time 12

Problems 18

Suggested Reading 22

Chapter 2. Developing PDE Intuition 23

§2.1. The Structure of Linear Equations 23

§2.2. Examples of Linear Equations 30

§2.3. Examples of Nonlinear Equations 35

Problems 41

Suggested Reading 43

Chapter 3. The Story of Solitons 45

§3.1. The Observation 45

§3.2. Terminology and Backyard Study 46

§3.3. A Less-than-enthusiastic Response 47

§3.4. The Great Eastern 49

§3.5. The KdV Equation 49

§3.6. Early 20th Century 52

v

and Unreality 7