Travelling Waves in Nonlinear Diffusion-Convection Reaction

Travelling waves are observed in many natural processes, ranging from the spread of diseases to the combustion of fuels. A common feature of a multitude of these phenomena is that they can be similarly described in terms of a dispersive or diffusive mechanism, an advective or convective process, some kind of reaction, sorption, source or sink mechanism, or, an intricate balance between several of these factors. This book presents an overview of the occurence of travelling waves when these underlying phenomena can be mathematically modelled by a partial differential equation. The contents provide an up-to-date review of the state of the art in this field of research, touching on many different areas of application and including a large number of new results.

1 Introduction.- 2 General theory.- 2.1 Basic hypotheses.- 2.2 Integral equation theory.- 2.3 Proof of equivalence.- 2.4 Illustration of difficulties.- 2.5 Further properties.- 2.6 Classical results.- Bibliographical notes.- 3 Transformations.- Bibliographical notes.- 4 Travelling waves.- 4.1 Admissible wave speeds.- 4.2 Number of solutions.- Bibliographical notes.- 5 Convection-diffusion.- Bibliographical notes.- 6 Reaction-diffusion.- 6.1 Sink term.- 6.2 Source term.- 6.3 Smooth coefficients.- Bibliographical notes.- 7 Power-law equations.- Bibliographical notes.- 8 Wavefronts.- 8.1 Admissible wave speeds.- 8.2 Number of wavefronts.- 8.3 Illustrations.- 8.4 Multiple equilibria.- Bibliographical notes.- 9 Wavefronts for convection-diffusion.- Bibliographical notes.- 10 Wavefronts for reaction-diffusion.- 10.1 Fixed sign.- 10.2 One sign change.- 10.3 Smooth coefficients.- Bibliographical notes.- 11 Unbounded waves.- 12 Wavefronts and unbounded waves for power-law equations.- 12.1 Convection-diffusion.- 12.2 Reaction-diffusion with linear convection.- 12.3 Reaction-convection-diffusion.- Bibliographical notes.- 13 Explicit travelling-wave solutions.- 13.1 Power-law equations.- 13.2 Generalizations of the Fisher equation.- 13.3 Generating further explicit solutions.- Bibliographical notes.