Introduction to Traveling Waves is an invitation to research focused on traveling waves for undergraduate and masters level students. Traveling waves are not typically covered in the undergraduate curriculum, and topics related to traveling waves are usually only covered in research papers, except for a few texts designed for students. This book includes techniques that are not covered in those texts.
Through their experience involving undergraduate and graduate students in a research topic related to traveling waves, the authors found that the main difficulty is to provide reading materials that contain the background information sufficient to start a research project without an expectation of an extensive list of prerequisites beyond regular undergraduate coursework. This book meets that need and serves as an entry point into research topics about the existence and stability of traveling waves.
Features
- Self-contained, step-by-step introduction to nonlinear waves written assuming minimal prerequisites, such as an undergraduate course on linear algebra and differential equations.
- Suitable as a textbook for a special topics course, or as supplementary reading for courses on modeling.
- Contains numerous examples to support the theoretical material.
- Supplementary MATLAB codes available via GitHub.
Author(s): Anna R. Ghazaryan, Stéphane Lafortune, Vahagn Manukian
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 173
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
CHAPTER 1: Nonlinear traveling waves
1.1. TRAVELING WAVES
1.2. REACTION-DIFFUSION EQUATIONS
1.3. TRAVELING WAVES AS SOLUTIONS OF REACTION-DIFFUSION EQUATIONS
1.4. PLANAR WAVES
1.5. EXAMPLES OF REACTION-DIFFUSION EQUATIONS
1.5.1. Fisher-KPP equation
1.5.2. Nagumo equation
1.6. OTHER PARTIAL DIFFERENTIAL EQUATIONS THAT SUPPORT WAVES
1.6.1. Nonlinear diffusion, convection, and higher order derivatives
1.6.2. Burgers equation
1.6.3. Korteweg-de Vries (KdV) equation
CHAPTER 2: Systems of Reaction-Diffusion Equations posed on infinite domains
2.1. SYSTEMS OF REACTION-DIFFUSION EQUATIONS
2.2. EXAMPLES OF REACTION-DIFFUSION SYSTEMS
2.2.1. FitzHugh-Nagumo system
2.2.2. Population models
2.2.3. Belousov-Zhabotinski reaction
2.2.4. Spread of infection disease
2.2.5. The high Lewis number combustion model
CHAPTER 3: Existence of fronts, pulses, and wavetrains
3.1. TRAVELING WAVES AS ORBITS IN THE ASSOCIATED DYNAMICAL SYSTEMS
3.2. DYNAMICAL SYSTEMS APPROACH: EQUILIBRIUM POINTS
3.3. EXISTENCE OF FRONTS IN FISHER-KPP EQUATION: TRAPPING REGION TECHNIQUE
3.3.1. Existence of fronts in Nagumo equation
3.3.2. Rotated vector fields and existence of a heteroclinic orbit between A and C for some c ≠ 0
3.4. EXISTENCE OF FRONTS IN SOLID FUEL COMBUSTION MODEL
3.5. WAVETRAINS
CHAPTER 4: Stability of fronts and pulses
4.1. STABILITY: INTRODUCTION
4.2. A HEURISTIC PRESENTATION OF SPECTRAL STABILITY FOR FRONT AND PULSE TRAVELING WAVE SOLUTIONS
4.2.1. Eigenvalue problem
4.2.2. Spectrum and spectral stability
4.3. LOCATION OF THE POINT SPECTRUM
4.3.1. Spectral Energy Estimates
4.3.2. Evans function
4.3.2.1. Definition of the Evans function
4.3.2.2. Gap Lemma
4.3.2.3. Evans function computation: scalar equations
4.3.2.4. Evans function computation: systems of equations
4.4. BEYOND SPECTRAL STABILITY
Bibliography
Index
