Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. (English) Zbl 0863.35106

Ma, W. X.; Fuchssteiner, B.

Int. J. Non-Linear Mech. 31, No. 3, 329-338 (1996).

The paper deals with explicit traveling wave solutions of the Kolmogorov-Petrovskii-Piskunov equation \[ u_t-u_{xx}+\mu u+\nu u^2+\delta u^3=0,\tag{1} \] where \(\mu\), \(\nu\), \(\delta\) are real constants. The authors decompose the equation (1) into two problems and solve these problems, in order to analyze the possibilities of solutions corresponding to two Riccati equations as well as to give explicitly the number of exact solutions of (1). In the further investigations, the authors use Cole-Hopf transformation and Bäcklund transformation.

Reviewer: E.Minchev (Sofia)

MSC:

35Q72	Other PDE from mechanics (MSC2000)
58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35C05	Solutions to PDEs in closed form

Keywords:

explicit traveling wave solutions of the Kolmogorov-Petrovskii-Piskunov equation; Riccati equations; number of exact solutions; Cole-Hopf transformation; Bäcklund transformation