Introduction to Traveling Waves

Anna R. Ghazaryan is a Professor of Mathematics at Miami University, Oxford, OH. She received her Ph.D. in 2005 from the Ohio State University. She is an applied analyst with research interests in applied dynamical systems, more precisely, traveling waves and their stability. Stéphane Lafortune is Professor of Mathematics at the College of Charleston in South Carolina. He earned his Ph.D. in Physics from the Université de Montréal and Université Paris VII in 2000. He is an applied mathematician who works on nonlinear waves phenomena. More precisely, he is interested in the theory of integrable systems and in the problems of existence and stability of solutions to nonlinear partial differential equations. Vahagn Manukian is an Associate Professor of Mathematics at Miami University. He obtained a M.A. Degree Mathematics from SUNY at Buffalo and a Ph.D. in mathematics from the Ohio State University in 2005. Vahagn Manukian uses dynamical systems methods such as local and global bifurcation theory to analyze singularly perturbed nonlinear reaction diffusions systems that model natural phenomena.

• 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. 1.6. Other Partial Differential Equations that Support Waves. 2. Systems of Reaction-Diffusion Equations posed on Infinite Domains. 2.1. Systems of Reaction-Diffusion Equations. 2.2. Examples of Reaction-Diffusion Systems. 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. 3.4. Existence of Fronts in Solid Fuel Combustion Model. 3.5. Wavetrains. 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. 4.3. Location of the Point Spectrum. 4.4. Beyond Spectral Stability.