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Instability of solitary waves on Euler’s elastica. (English) Zbl 1112.74026
Summary: We study stability of solitary waves in a thin inextensible unshearable rod of infinite length. 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 Evans function, an analytic function in 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 Evans function in some neighbourhood of the origin with its asymptotic at infinity. The explicit computation of leading coefficient of Taylor series of Evans function near the origin is performed by means of the symbolic computer language.

MSC:
74H55 Stability of dynamical problems in solid mechanics
74J35 Solitary waves in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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