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Existence and uniqueness of heteroclinic orbits for the equation \(\lambda u'''+u'=f(u)\). (English) Zbl 0673.34022

From author’s summary: “If f is a continuous even function which is decreasing on (0,\(\infty)\) and such that \(\pm \alpha\) are its only zeros and are simple, then in three-dimensional phase space the unstable manifold of the equilibrium \(u=-\alpha\) and stable manifold of \(u=\alpha\) are both two dimensional. If \(\lambda <0\) it is shown that there is a unique bounded orbit of the equation \(\lambda u'''+u'=f(u)\), 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,\(\infty).''\) In the special case \(f\equiv 1-u^ 2\), the author proves that if \(\lambda\geq 2/9\), there is no solution u of the equation such that \(\lim_{x\to \pm \infty}u(x)=\pm 1\) with \(u'\geq 0\) on \({\mathbb{R}}\).
Reviewer: J.O.C.Ezeilo

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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[1] DOI: 10.1016/0022-247X(86)90110-1 · Zbl 0633.34028 · doi:10.1016/0022-247X(86)90110-1
[2] DOI: 10.1063/1.865160 · Zbl 0565.76051 · doi:10.1063/1.865160
[3] Michelson, Physica 19D pp 89– (1986)
[4] DOI: 10.1143/PTP.55.356 · doi:10.1143/PTP.55.356
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