230 ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2018, Vol. 58, No. 2, pp. 230–237. © Pleiades Publishing, Ltd., 2018. Original Russian Text © S.V. Pikulin, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 2, pp. 244–252. Dedicated to the 100th birthday of Academician N.N. Moiseev Traveling-Wave Solutions of the Kolmogorov–Petrovskii–Piskunov Equation S. V. Pikulin Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, Moscow, 119333 Russia e-mail: spikulin@gmail.com Received July 12, 2017 Abstract—We consider quasi-stationary solutions of a problem without initial conditions for the Kolmogorov–Petrovskii–Piskunov (KPP) equation, which is a quasilinear parabolic one arising in the modeling of certain reaction–diffusion processes in the theory of combustion, mathematical biol- ogy, and other areas of natural sciences. A new efficiently numerically implementable analytical rep- resentation is constructed for self-similar plane traveling-wave solutions of the KPP equation with a special right-hand side. Sufficient conditions for an auxiliary function involved in this representation to be analytical for all values of its argument, including the endpoints, are obtained. Numerical results are obtained for model examples. Keywords: Kolmogorov–Petrovskii–Piskunov equation, generalized Fisher equation, Abel’s equation of the second kind, Fuchs–Kowalewski–Painlevé test, self-similar solutions, traveling waves, inter- mediate asymptotic regime. DOI: 10.1134/S0965542518020124 1. INTRODUCTION Equations of the Kolmogorov–Petrovskii–Piskunov (KPP) type [1] arise in the simulation of various autowave processes. Such processes are considered, for example, in the theory of combustion and explo- sion [2–5], nuclear reactors [6], gas discharges [7, 8], Rayleigh–Bénard convection [9, 10], mathematical ecology [11–13], neurophysiology [14–17], cardiology [18–21], and other fields of natural sciences [22–24] and engineering [25]. The original KPP equation has the form (1.1) where is time, is a point of space , , is the Laplacian in , and the right-hand side is positive on the interval and satisfies the conditions ; (1.2) moreover, for . It is well known [1] that the solution of this problem obeys the inequalities , provided that they are satisfied by the initial value. The problem formulated arises in modeling the transition from the unstable equilibrium to the stable equilibrium . A point of special interest in the study of models based on KPP equations is the so-called intermediate asymptotic regime [26], which represents a stage of an uncompleted process at which a particular form of initial data has not had a noticeable effect on its steady-state behavior. This regime is usually described by self-similar plane traveling-wave solutions of the form (1.3) ∂ -Δ , = , , ∂ ( ) (( )) u ut Fut t x x ∈ ℝ t = , , 1 ( ) n x …x x ℝ n ≥ 1 n Δ=∂ + +∂ ⋯ 2 2 1 n ℝ n ∈ , 1 ([0 1]) F C , (0 1) = = , := > 0 (0) (1) 0 '(0) 0 F F f F ξ≤ ξ 0 () F f ξ∈ , (0 1) = , ( ) u ut x ≤ ≤ 0 1 u , ≡ ( ) 0 ut x , ≡ ( ) 1 ut x , =ψη, , η, := , -ω, ω= > , ( ) (( )) ( ) ( ) const 0 ut t t t x x x xn