Stability Theory of Solitary Loops Propagating Along Euler’s Elastica

Abstract

The problem of dynamic stability of twist free solitary wave solutions of the equations describing oscillations of an inextensible elastic rod (Euler’s elastica) is treated. The governing equations describe sufficiently large displacements, though we are restricted to small strains. We show that under the condition of well-posedness of the initial value problem (in some specific sense) the family of solitary wave solutions is nonlinearly stable for two-dimensional perturbations not coming out from the plane of principal bending. The framework of the analysis is largely based on the spectral properties of the “linearized Hamiltonian” \({\mathscr {H}}\). We show that for planar perturbations \({\mathscr {H}}\) is positively semidefinite subject to a certain constraint, which implies the orbital stability. We consider also the case of perturbing the solitary wave by three-dimensional spatial perturbations. As a result of linearization about the solitary wave 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.