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Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation. (English) Zbl 1177.74207
Theor. Math. Phys. 147, No. 2, 646-659 (2006); translation from Teor. Mat. Fiz. 147, No. 2, 240-256 (2006).
Summary: 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.

74J30 Nonlinear waves in solid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35Q74 PDEs in connection with mechanics of deformable solids
Full Text: DOI
[1] A. G. Kulikovskii, Sov. Phys. Dokl., 29, 283–285 (1984).
[2] A. G. Kulikovskii and A. P. Chugainova, Comput. Math. Math. Phys., 44, 1062–1068 (2004).
[3] A. G. Kulikovskii and N. I. Gvozdovskaya, Proc. Steklov Math. Inst., 223, 55–65 (1998).
[4] A. G. Kulikovskii and E. I. Sveshnikova, Nonlinear Waves in Elastic Media, CRC Press, Boca Raton, Fl. (1995). · Zbl 0865.73004
[5] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems (Monographs and Surveys in Pure and Applied Mathematics, Vol. 118), Chapman and Hall/CRC, Boca Raton, Fl. (2001).
[6] N. S. Bakhvalov and M. E. Eglit, Dokl. Math., 61, No. 1, 1–4 (2000).
[7] A. A. Samarskii and Yu. P. Popov, Difference Methods for Solving Gas Dynamics Problems [in Russian], Nauka, Moscow (1978). · Zbl 0382.76052
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