Non-generic connections corresponding to front solutions

A classification of special 'nonlinear' front solutions for certain one-time and one-space reaction-diffusion equations is presented, using the method of Weiss, Tabor and Carnevale (WTC). These results are related to known stability criteria, in particular the steepness criterion of van Saarloos (1989). The WTC method is shown to be equivalent to a special first-order reduction, and both of these methods are shown to work for reaction-diffusion equations with special nonlinearities. Of particular interest is the fact that the special first-order reduction is shown to give separatrices in appropriate phase spaces. An example of reaction-diffusion equation is presented without these special nonlinearities. While this equation is shown to have a special 'nonlinear' connection and resulting stability properties, it is intractable for either a singular manifold expansion of a first-order reduction. A Lie symmetry analysis is carried out, and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques. However, the 'rescaling ansatz' of Cariello and Tabor (1991) suggests that some symmetries are present.