Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation ut = uxx + f(u), 0 < u(x,t) < 1, (x,t) ∈ ℝ2, where ƒ is a KPP-type nonlinearity defined in [0,1]: ƒ(0) = ƒ(1) = 0, ƒ′(0) > 0, ƒ′(1) < 0, ƒ > 0 in (0,1), and ƒ′(s) ≤ ƒ′(0) in [0,1]. This equation admits infinitely many traveling-wave-type solutions, increasing or decreasing in x. It also admits solutions that depend only on t. In this paper, we build four other manifolds of solutions: One is 5-dimensional, one is 4-dimensional, and two are 3-dimensional. Some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix. Furthermore, the traveling-wave solutions are on the boundary of these four manifolds. © 1999 John Wiley & Sons, Inc.