The phase-plane picture for a class of fourth-order conservative differential equations

摘要

We study the bounded solutions of a class of fourth-order equations - gamma u''' + u' + f(u) = 0, gamma > 0 We show that when; is not tao large then the paths in the (u, u')-plant of two bounded solutions do nor cross. Moreover, the conserved quantity associated with the equation puts an ordering on the hounded solutions in the ph;phase-plane and a continuation theorem shows that they fill up part of the phase-plane. We apply these results to the Extended Fisher-Kolmogorov (EFK) equation. a fourth-order model equation for bi-stable systems. The uniqueness and ordering results imply that as long as the stable equilibrium points are real saddles: the bounded solutions of the stationary EFK; equation correspond exactly to those of the classical second-order Fisher-Kolmogorov equation. Besides, we establish the asymptotic stability of the heteroclinic solution of the EFK equation. (C) 2000 Academic Press. [References: 33]