Symmetries and integrability of generalized diffusion reaction equations

Abstract

Using Lie group methods and the Painleve test, the authors analyse nonlinear diffusion reaction equations ut=(D(u)u)+f(u) with power law diffusion coefficients (D approximately uv) and arbitrary nonlinear reaction terms f(u) which have a wide spectrum of applications in many areas of science. The Lie-group-based similarity method leads to a classification of the reaction terms according to its symmetry properties. With the help of the adjoint representation, the optimal system of similarity reductions is calculated. To check the integrability of the partial differential equation, the existence of generalized (Lie-Backlund) symmetries is investigated. Apart from three known cases, no further cases with third-order symmetries exist. Examining the integrability of the second-order ordinary differential equations resulting from the reductions, only a few parameter combinations can be found for which the Painleve property is given. However, they are able to construct unknown integrals of motion for a much larger range of parameter values. From the integrals, exact similarity solutions may be derived. This is demonstrated by examples corresponding to the important moving-wave reduction.