Instability of supersonic solitary waves in a generalized elastic electrically conductive medium

Abstract

We study the spectral instability of supersonic solitary waves taking place in a nonlinear model of an elastic electrically conductive micropolar medium. As a result of linearization about the soliton solution, an inhomogeneous scalar equation is obtained. This equation leads to a generalized spectral problem. To establish instability, it is necessary to make sure of the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of instability is carried out using the local construction at the origin and the asymptotics at infinity of the Evans function, which depends only on the spectral parameter. This function is analytic in the right complex half-plane and has at least one zero on the positive real half-axis for a certain range of physical parameters of the problem in question. This zero coincides with the unstable eigenvalue of the generalized spectral problem.