Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equation u_t = u_xx + f(u),   0 < u(x,t) < 1, (x,t) ∈ ?², where ? is a KPP‐type nonlinearity defined in 0,1]: ?(0) = ?(1) = 0, ?′(0) > 0, ?′(1) < 0, ? > 0 in (0,1), and ?′(s) ≤ ?′(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.