Entire solutions of the KPP equation

This paper deals with the solutions defined for all time of the KPP equatio n u(t) = u(xx) + f(u), 0 < u(x,t) < 1, (x,t) is an element of R-2, where f is a KPP-type nonlinearity defined in [0, 1]. f(0) = f(1) = 0, f'(0 ) > 0, f'(1) < 0, f > 0 in (0, 1), and f'(s) less than or equal to f'(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-d imensional, one is 4-dimensional, and two are 3-dimensional. Some of these new solutions are obtained by considering two traveling waves that come fro m both sides of the real axis and mix. Furthermore, the traveling-wave solu tions are on the boundary of these four manifolds. (C) 1999 John Wiley & So ns, Inc.