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Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations

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Abstract

We consider the simple pendulum equation with friction:

$${\begin{array}{ll}-u''(t) - \vert u'(t)\vert + g(u(t)) & = \lambda \sin u(t), \quad t \in I := (-T, T), \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad u(t) & > 0, \quad t \in I, \quad u(\pm T) = 0,\end{array}}$$

where T > 0 is a constant and λ > 0 is a parameter. The case without friction is known as the simple pendulum equation with self interaction, and the asymptotic shape of the solution as λ → ∞ is well understood. In this paper, we establish the asymptotic formula for the boundary layers of the solution u λ for the equation above, and show that its boundary slope is steeper than that of the solution without damping term | u′(t)|.

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  1. Department of Applied Mathematics, Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan

    Tetsutaro Shibata

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  1. Tetsutaro Shibata
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Correspondence to Tetsutaro Shibata.

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Shibata, T. Precise Asymptotics of Boundary Layers for Damped Simple Pendulum Equations. Results. Math. 58, 105–118 (2010). https://doi.org/10.1007/s00025-009-0009-5

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  • DOI: https://doi.org/10.1007/s00025-009-0009-5

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