On solutions of traveling wave type for a nonlinear parabolic equation

We consider the Kolmogorov–Petrovsky–Piskunov equation which is a quasilinear parabolic equation of second order appearing in the flame propagation theory and in modeling of certain biological processes. An analytical construction of self-similar solutions of traveling wave kind is presented for the special case when the nonlinear term of the equation is the product of the argument and a linear function of a positive power of the argument. The approach to the construction of solutions is based on the study of singular points of analytic continuation of the solution to the complex domain and on applying the Fuchs–Kovalevskaya–Painlevé test. The resulting representation of the solution allows an efficient numerical implementation.