Asymptotics of the Evans function for subsonic solitary waves in a micropolar electrically conductive elastic medium

V. I. Erofeev^a, A. T. Il'ichev^b, V. Ya. Tomashpolskii<sup>c

a</sup> Mechanical Engineering Research Institute of RAS — Branch of the Federal Research Center "Gaponov--Grekhov Institute of Applied Physics," Russian Academy of Sciences, Nizhnii Novgorod, Russia
^b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
^c Bauman Moscow State Technical University, Moscow, Russia

Abstract: As a result of the linearization of nonlinear equations for displacements in a nonlinear model of elastically conductive micropolar medium in a magnetic field on the background of a soliton solution describing subsonic solitary waves, we obtain an inhomogeneous scalar linear equation. This equation leads to a generalized spectral problem. To establish the instability of the mentioned solitary waves, the existence of an unstable eigenvalue (with a positive real part) must be verified. The corresponding proof is carried out by constructing the Evans function that depends only on the spectral parameter. This function is analytic in the right complex half-plane, and its zeros coincide with the unstable eigenvalues. It is proved that the Evans function tends to unity at infinity. This property of the Evans function, for some of its local properties in a neighborhood of the origin, allows us to conclude that it has zeros on the positive real semi-axis and therefore the subsonic solitary wave is unstable.

Keywords: electrically conductive elastic medium, nonlinear wave displacements, solitary waves, spectral stability, Evans function.

Received: 10.03.2025
Revised: 10.04.2025

DOI: 10.4213/tmf10975

 English version: Theoretical and Mathematical Physics, 2025, 224:3, 1613–1624

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