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 — Painlev´e test. The resulting representation of the
solution allows an efficient numerical implementation.