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

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Teoreticheskaya i Matematicheskaya Fizika, 2006, Volume 147, Number 2, Pages 240–256
DOI: https://doi.org/10.4213/tmf1961 (Mi tmf1961)

This article is cited in 13 scientific papers (total in 13 papers)

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

A. P. Chugainova

Steklov Mathematical Institute, Russian Academy of Sciences

Full-text PDF (635 kB) Citations (13)

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

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.

Received: 01.09.2005

English version:
Theoretical and Mathematical Physics, 2006, Volume 147, Issue 2, Pages 646–659
DOI: https://doi.org/10.1007/s11232-006-0067-8

Bibliographic databases:

Document Type: Article

Language: Russian

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

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https://doi.org/10.4213/tmf1961

https://www.mathnet.ru/eng/tmf/v147/i2/p240

This publication is cited in the following 13 articles:

A. P. Chugainova, R. R. Polekhina, “The effect of small-scale processes in the structure of discontinuities on the solution of the Riemann problem”, Wave Motion, 114 (2022), 102996–11
Chugainova A.P., Kulikovskii A.G., “Longitudinal and Torsional Shock Waves in Anisotropic Elastic Cylinders”, Z. Angew. Math. Phys., 71:1 (2020), 17
Chugainova A.P., “Special Discontinuities Depending on Dispersion Processes”, AIP Conference Proceedings, 2302, ed. Todorov M., Amer Inst Physics, 2020, 100002
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
A. G. Kulikovskii, A. P. Chugainova, “Shock waves in anisotropic cylinders”, Proc. Steklov Inst. Math., 300 (2018), 100–113
A. P. Chugainova, “Special discontinuities in nonlinearly elastic media”, Comput. Math. Math. Phys., 57:6 (2017), 1013–1021
A. G. Kulikovskii, A. P. Chugainova, “Long nonlinear waves in anisotropic cylinders”, Comput. Math. Math. Phys., 57:7 (2017), 1194–1200
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
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
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
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
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
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

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

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