This article is cited in 2 papers

Instability of solitons under flexure and twist of an elastic rod

A. T. Il'ichev^a, V. Ja. Tomashpolskii<sup>b

a</sup> Steklov Mathematical Institute, RAS, Moscow, Russia
^b Bauman Moscow State Technical University, Moscow, Russia

Abstract: We study the stability of planar soliton solutions of equations describing the dynamics of an infinite inextensible unshearable rod under three-dimensional spatial perturbations. As a result of linearization about the soliton solution, we obtain an inhomogeneous scalar equation. This equation leads to a generalized eigenvalue problem. To establish the instability, we must verify the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of the instability is done using a local construction of the Evans function depending only on the spectral parameter. This function is analytic in the right half of the complex plane and has at least one zero on the positive real axis coinciding with an unstable eigenvalue of the generalized spectral problem.

Keywords: elastic rod, soliton, linearization, unstable spectrum, Evans function.

Received: 26.12.2011

DOI: 10.4213/tmf6959

 English version: Theoretical and Mathematical Physics, 2012, 172:3, 1206–1216

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