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 TMF, 2006, Volume 147, Number 2, Pages 240–256 (Mi tmf1961)

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

A. P. Chugainova

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity.

Keywords: nonlinear hyperbolic equations, shock waves, dissipation, dispersion

DOI: https://doi.org/10.4213/tmf1961

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English version:
Theoretical and Mathematical Physics, 2006, 147:2, 646–659

Bibliographic databases:

Citation: A. P. Chugainova, “Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation”, TMF, 147:2 (2006), 240–256; Theoret. and Math. Phys., 147:2 (2006), 646–659

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tmf1961
• https://doi.org/10.4213/tmf1961
• http://mi.mathnet.ru/eng/tmf/v147/i2/p240

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. G. Kulikovskii, A. P. Chugainova, “Classical and Nonclassical Discontinuities and Their Structure in Nonlinear Elastic Media with Dispersion and Dissipation”, Proc. Steklov Inst. Math., 276, suppl. 2 (2012), S1–S68
2. Chugainova A.P., “Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation”, J. Appl. Math. Mech., 71:5 (2007), 701–711
3. A. G. Kulikovskii, A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory”, Russian Math. Surveys, 63:2 (2008), 283–350
4. Kulikovskii A.G., Chugainova A.P., “On solution non-uniqueness in the nonlinear elasticity theory”, Reviews on Advanced Materials Science, 19:1–2 (2009), 93–97
5. A. G. Kulikovskii, A. P. Chugainova, “Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation”, Comput. Math. Math. Phys., 50:12 (2010), 2145–2156
6. Kulikovskii A.G., Chugainova A.P., “On the steady-state structure of shock waves in elastic media and dielectrics”, Journal of Experimental and Theoretical Physics, 110:5 (2010), 851–862
7. A. P. Chugainova, “Special discontinuities in nonlinearly elastic media”, Comput. Math. Math. Phys., 57:6 (2017), 1013–1021
8. A. G. Kulikovskii, A. P. Chugainova, “Long nonlinear waves in anisotropic cylinders”, Comput. Math. Math. Phys., 57:7 (2017), 1194–1200
9. A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113
10. Chugainova A.P., Il'ichev A.T., Shargatov V.A., “Stability of Shock Wave Structures in Nonlinear Elastic Media”, Math. Mech. Solids, 24:11 (2019), 3456–3471
11. Chugainova A.P., Kulikovskii A.G., “Longitudinal and Torsional Shock Waves in Anisotropic Elastic Cylinders”, Z. Angew. Math. Phys., 71:1 (2020), 17
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