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Existence and uniqueness of heteroclinic orbits for the equation λu‴ + u′ = f(u)

Published online by Cambridge University Press:  14 November 2011

J. F. Toland
J. F. Toland
Affiliation:
School of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.
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Synopsis

Iffis a continuous even function which is decreasing on (0,∞) and such that±α are its only zeros and are simple, then in three-dimensional phase spacethe unstable manifold of the equilibrium u = −α and the stable manifold of u = α are both two dimensional. If λ<0 it is shown that there is a unique bounded orbit of the equation λu‴ + u′ = f(u), and that this is a heteroclinic orbit joining these two equilibria. Other results on the existence and uniqueness of heteroclinic orbits are also established when f is not even and when f is not monotone on (0, ∞).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 23 - 36
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Hooper, A. P. and Grimshaw, R.. Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28 (1985) 37–45.CrossRefGoogle Scholar
2Kuramoto, Y. and Tsuzuki, T.. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theoret. Phys. 55 (1976), 356–369.CrossRefGoogle Scholar
3McCord, C. K.. Uniqueness of connecting orbits in the equation y3=Y 2–1. j. Math. Anal. Appl. 114 (1986), 584—592.CrossRefGoogle Scholar
4Michelson, D.. Steady solutions of the Kuramoto-Sivasikinsky equation. Physica 19D (1986), 89–111.Google Scholar