Stability Theory of the Euler Loop on Inextensible Elastic Rods

The stability of solitary waves in a thin inextensible elastic rod of infinite length is studied. In the absence of torsion, the profile of the elastica of such a rod that corresponds to a solitary wave has the form of a plane loop. The range of speeds of the loop depends on the tension in the rod. The orbital stability of solitary waves with respect to the plane perturbations of the form of the loop is established. The stability result follows from the fact that the orbit of a solitary wave provides a local minimum of a certain invariant functional. The minimum is attained on a nonlinear invariant submanifold of the basic space of solutions. For a certain range of speeds of a solitary wave, its linear instability with respect to nonplanar perturbations of the loop is proved. The instability result is obtained by using the properties of the Evans function, which is analytic in the right complex half-plane of the spectral parameter. This function has zeroes in the right half-plane if and only if there exists an unstable global mode. The instability follows directly from the comparison of the asymptotic behavior of the Evans function in the neighborhood of zero and at infinity. Expressions for the leading coefficients in the Taylor expansion of the Evans function in the neighborhood of the origin are obtained with the use of Mathematica 4.0 package.