INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 38 (2005) 3367–3379 doi:10.1088/0305-4470/38/15/009 A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions B H Gilding 1 and R Kersner 2 1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Al-Khodh, Oman 2 Department of Mathematics, Pollack Mih´ aly Faculty of Engineering, University of P´ ecs, P´ ecs, Hungary E-mail: gilding@squ.edu.om Received 12 September 2004, in final form 1 February 2005 Published 30 March 2005 Online at stacks.iop.org/JPhysA/38/3367 Abstract This paper concerns processes described by a nonlinear partial differential equation that is an extension of the Fisher and KPP equations including density- dependent diffusion and nonlinear convection. The set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized. There is a minimal wave speed. For this wave speed there is a unique wavefront which can be found explicitly. It displays a sharp propagation front. For all greater wave speeds there is a unique wavefront which does not possess this property. For such waves, the asymptotic behaviour as the equilibrium states are approached is determined. PACS numbers: 87.23.Cc, 47.54.+r, 82.40.Ck 1. Introduction A characteristic of a huge number of biological, chemical and physical phenomena is that in the course of time a spatio-temporal pattern develops from a state that does not initially exhibit any structure. In many instances, the population density or concentration will evolve into a spatial profile which does not appear to change shape with time, yet moves with a well-defined velocity. By its very nature, such a phenomenon indicates the formation of a travelling wave. One of the many challenges involved in mathematically modelling biological, chemical, physical and other processes is identifying whether or not the model can simulate the occurrence of such a wave. Should this be the case, there is the further challenge of predicting the shape and velocity of the travelling wave, and relating this to the spatio-temporal pattern being observed in practice. Many models in the form of nonlinear partial differential equations—the linear diffusion equation with logistic growth is an example par excellence—admit travelling-wave solutions 0305-4470/05/153367+13$30.00 © 2005 IOP Publishing Ltd Printed in the UK 3367