Instability of solitary waves on Euler’s elastica |
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Authors: | Andrej T Il’ichev |
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Institution: | (1) Steklov Mathematical Insitute, Gubkina str. 8, 119991 Moscow, Russia |
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Abstract: | Stability of solitary waves in a thin inextensible and unshearable rod of infinite length is studied. Solitary-wave profile
of the elastica of such a rod without torsion has the form of a planar loop and its speed depends on a tension in the rod.
The linear instability of a solitary-wave profile subject to perturbations escaping from the plane of the loop is established
for a certain range of solitary-wave speeds. It is done using the properties of the Evans function, an analytic function on
the right complex half-plane, that has zeros if and only if there exist the unstable modes of the linearization around a solitary-wave
solution. The result follows from comparison of the behaviour of the Evans function in some neighbourhood of the origin with
its asymptotic at infinity. The explicit computation of the leading coefficient of the Taylor series of the Evans function
near the origin is performed by means of the symbolic computer language.
Received: April 6, 2004; revised: December 12, 2004 |
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Keywords: | 74K10 74J35 74H55 |
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