{"title":"Instability of supersonic solitary waves in a generalized elastic electrically conductive medium","authors":"V. I. Erofeev, A. T. Il’ichev","doi":"10.1007/s00161-023-01249-1","DOIUrl":null,"url":null,"abstract":"<div><p>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.</p></div>","PeriodicalId":525,"journal":{"name":"Continuum Mechanics and Thermodynamics","volume":"35 6","pages":"2313 - 2323"},"PeriodicalIF":1.9000,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Continuum Mechanics and Thermodynamics","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s00161-023-01249-1","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
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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This interdisciplinary journal provides a forum for presenting new ideas in continuum and quasi-continuum modeling of systems with a large number of degrees of freedom and sufficient complexity to require thermodynamic closure. Major emphasis is placed on papers attempting to bridge the gap between discrete and continuum approaches as well as micro- and macro-scales, by means of homogenization, statistical averaging and other mathematical tools aimed at the judicial elimination of small time and length scales. The journal is particularly interested in contributions focusing on a simultaneous description of complex systems at several disparate scales. Papers presenting and explaining new experimental findings are highly encouraged. The journal welcomes numerical studies aimed at understanding the physical nature of the phenomena.
Potential subjects range from boiling and turbulence to plasticity and earthquakes. Studies of fluids and solids with nonlinear and non-local interactions, multiple fields and multi-scale responses, nontrivial dissipative properties and complex dynamics are expected to have a strong presence in the pages of the journal. An incomplete list of featured topics includes: active solids and liquids, nano-scale effects and molecular structure of materials, singularities in fluid and solid mechanics, polymers, elastomers and liquid crystals, rheology, cavitation and fracture, hysteresis and friction, mechanics of solid and liquid phase transformations, composite, porous and granular media, scaling in statics and dynamics, large scale processes and geomechanics, stochastic aspects of mechanics. The journal would also like to attract papers addressing the very foundations of thermodynamics and kinetics of continuum processes. Of special interest are contributions to the emerging areas of biophysics and biomechanics of cells, bones and tissues leading to new continuum and thermodynamical models.
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