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.
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References
Erofeeev, V.I., Malkhanov, A.O.: Spatially localized nonlinear magnetoelastic waves in an electrically conductive micropolar medium. ZAMM (2023). https://doi.org/10.1002/zamm.202100066
Evans, J.V.: Nerve axon equations, III: Stability of the nerve impulse. Indiana Univ. Math. J. 22, 577–594 (1972). https://doi.org/10.1512/iumj.1973.22.22048
Pego, R.L., Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. R. Soc. Lond. A340, 47–94 (1992). https://doi.org/10.1098/rsta.1992.0055
Alexander, J.C., Sachs, R.: Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation. In: Lakshmikantham, V., Hallam, T.G. (eds.) Nonlinear World 2, pp. 471–507. Walter de Gruyter & Co, Berlin (1995)
Pearce, S.P., Fu, Y.B.: Characterisation and stability of localized bulging/necking in inflated membrane tubes. IMA J. Appl. Math. 75, 581–602 (2010). https://doi.org/10.1093/imamat/hxq026
Il’ichev, A.T., Shargatov, V.A., Fu, Y.B.: Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube. Acta Mech. 231, 4095–4110 (2020). https://doi.org/10.1007/s00707-020-02754-z
Il’ichev, A.T., Shargatov, V.A.: Stability of the aneurysm-type solution in a membrane tube with localized wall thinning filled with a fluid with a non-constant velocity profile. J. Fluids Struct. 114, 103712 (2022). https://doi.org/10.1016/j.jfluidstructs.2022.10371
Eringen, A.C.: Microcontinuum Field Theory I. Foundations and Solids. Springer, New York (1999)
Eringen, A.C.: Microcontinuum Field Theory II. Fluent Media. Springer, New York (2001)
Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Altenbach, H., Berezovski, A., dell’Isola, F., Porubov, A. (eds) Sixty Shades of Generalized Continua. Advanced Structured Materials, vol. 170. Springer, Cham (1981). https://doi.org/10.1007/978-3-031-26186-2
Marin, M., Oechsner, A., Vlase, S.: A model of dual-phase-lag thermoelasticity for a Cosserat body. Continuum Mech. Thermodyn. 35, 1–16 (2023). https://doi.org/10.1007/s00161-022-01164-x
d’Agostino, M.V., Rizzi, G., Khan, H., et al.: The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters. Continuum Mech. Thermodyn. 34, 1393–1431 (2022). https://doi.org/10.1007/s00161-022-01126-3
Vilchevskaya, E.N., Mueller, W.H., Eremeyev, V.A.: Extended micropolar approach within the framework of 3M theories and variations thereof. Continuum Mech. Thermodyn. 34, 533–554 (2022). https://doi.org/10.1007/s00161-021-01072-6
Funding
The work of the first author (V.E) was supported by a government contract of Federal Research Center of A.V. Gaponov-Grekhov Institute of Applied Physics of the RAS for fundamental research for 2021–2015, project no. 0030-2021-0025, state registration number 1021060908990-9-2.3.2; the work of the second author (A. I.) was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2022-265). The authors declare that they have not known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Communicated by Andreas Öchsner.
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Erofeev, V.I., Il’ichev, A.T. Instability of supersonic solitary waves in a generalized elastic electrically conductive medium. Continuum Mech. Thermodyn. 35, 2313–2323 (2023). https://doi.org/10.1007/s00161-023-01249-1
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DOI: https://doi.org/10.1007/s00161-023-01249-1