The Phase-plane Picture for a Class of Fourth-order Conservative Differential Equations

We study the bounded solutions of a class of fourth-order equations −γu′′′′+u″+f(u)=0, γ>0. We show that when γ is not too large then the paths in the (u, u′)-plane of two bounded solutions do not cross. Moreover, the conserved quantity associated with the equation puts an ordering on the bounded solutions in the 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.