The Phase-plane Picture for a Class of Fourth-order Conservative Differential Equations
The Phase-plane Picture for a Class of Fourth-order Conservative Differential Equations
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DOI：
10.1006/jdeq.1999.3698
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发表时间：
2000-02
影响因子：
2.4
通讯作者：
J. B. Berg
中科院分区：
文献类型：
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作者：
J. B. Berg
研究了一类四阶方程−γu“+u”+f（u）=0，γ>0的有界解.我们证明了当γ不太大时，两个有界解在（u，u′）平面上的路径不相交.此外，与方程相关的守恒量对相平面中的有界解进行了排序，并且延拓定理表明它们填充了相平面的一部分。我们将这些结果应用到扩展的Fisher-Kolmogorov（EFK）方程，一个四阶模型方程的双稳态系统。唯一性和有序性结果表明，只要稳定平衡点是真实的鞍点，定常EFK方程的有界解与经典的二阶Fisher-Kolmogorov方程的有界解完全对应.此外，我们建立了EFK方程异宿解的渐近稳定性。
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.