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Steady solutions of a nonlinear equation for epitaxial crystal growth. (English) Zbl 0932.34028

Summary: A third-order autonomous ordinary differential equation is studied that describes stationary solutions to a nonlinear partial differential equation. The PDE is related to the Kuramoto-Sivashinsky equation, but contains an additional nonlinear term. It describes the growth of an epitaxial film on misoriented crystal substrates. The fixed points and the periodic solutions to the ODE system are analyzed, and stability results are given. Parameter regions are identified where the fixed points and the periodic solutions are unstable, but other bounded solutions exist. Their phase portrait is a double focus that connects the stable and the unstable manifolds of the fixed points.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
82D25 Statistical mechanics of crystals
34C60 Qualitative investigation and simulation of ordinary differential equation models
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