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Uspekhi Mat. Nauk, 2020, Volume 75, Issue 5(455), Pages 59–100 (Mi umn9953)  

Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes

A. T. Il'ichev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: This survey presents results on the stability of elevation solitary waves in axisymmetric elastic membrane tubes filled with a fluid. The elastic tube material is characterized by an elastic potential (elastic energy) that depends non-linearly on the principal deformations and describes the compliant elastic media. Our survey uses a simple model of an inviscid incompressible fluid, which nevertheless makes it possible to trace the main regularities of the dynamics of solitary waves. One of these regularities is the spectral stability (linear stability in form) of these waves. The basic equations of the ‘axisymmetric tube – ideal fluid’ system are formulated, and the equations for the fluid are averaged over the cross-section of the tube, that is, a quasi-one-dimensional flow with waves whose length significantly exceeds the radius of the tube is considered. The spectral stability with respect to axisymmetric perturbations is studied by constructing the Evans function for the system of basic equations linearized around a solitary wave type solution. The Evans function depends only on the spectral parameter $\eta$, is analytic in the right-hand complex half-plane $\Omega^+$, and its zeros in $\Omega^+$ coincide with unstable eigenvalues. The problems treated include stability of steady solitary waves in the absence of a fluid inside the tube (the case of constant internal pressure), together with the case of local inhomogeneity (thinning) of the tube wall, the presence of a steady fluid filling the tube (the case of zero mean flow) or a moving fluid (the case of non-zero mean flow), and also the problem of stability of travelling solitary waves propagating along the tube with non-zero speed.
Bibliography: 83 titles.

Keywords: axisymmetric elastic tube, membrane, elastic energy, ideal fluid, quasi-one-dimensional motion, internal pressure, bifurcation, spectral parameter, spectral stability, Evans function.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 075-15-2019-1614
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).


DOI: https://doi.org/10.4213/rm9953

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English version:
Russian Mathematical Surveys, 2020, 75:5, 843–882

Bibliographic databases:

UDC: 532.59
PACS: 74J35
MSC: Primary 74B20; Secondary 76B15
Received: 24.05.2020

Citation: A. T. Il'ichev, “Dynamics and spectral stability of soliton-like structures in fluid-filled membrane tubes”, Uspekhi Mat. Nauk, 75:5(455) (2020), 59–100; Russian Math. Surveys, 75:5 (2020), 843–882

Citation in format AMSBIB
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