Invariant solutions of certain nonlinear evolution type equations with small parameters

Abstract

The Fisher equation, which arises in the study of reaction diffusion waves in biology, does not display a high level of symmetry properties. Consequently, only travelling wave solutions are obtainable using the method of invariants. This has a direct bearing on studying perturbed forms of the equation which may arise from considering, e.g., damping or dissipative factors. We show, here, how one can get around this limitation by appending some unknown function to the perturbation and obtain interesting practical results using invariants. The ideas have significant consequences for equations which do not admit large class of symmetry properties. The method used in this analysis is then extended to other classes of evolution type equations that involve perturbations, for, e.g., the KdV type equations.