Geodesic
Nonuniqueness of a Self-similar Solution to the Riemann Problem for Elastic Waves in Media with a Negative Nonlinearity Parameter
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern Methods of Mechanics, Tome 322 (2023), pp. 251-265.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study self-similar solutions of the Riemann problem in the nonuniqueness region for weakly anisotropic elastic media with a negative nonlinearity parameter. We show that all discontinuities contained in the solutions in the nonuniqueness region have a stationary structure. We also show that in the nonuniqueness region one can construct two types of self-similar solutions.
Keywords: shock waves, Riemann problem, nonuniqueness of self-similar solutions. 
@article{TM_2023_322_a19,
     author = {A. P. Chugainova and R. R. Polekhina},
     title = {Nonuniqueness of a {Self-similar} {Solution} to the {Riemann} {Problem} for {Elastic} {Waves} in {Media} with a {Negative} {Nonlinearity} {Parameter}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {251--265},
     publisher = {mathdoc},
     volume = {322},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2023_322_a19/}
}
TY  - JOUR
AU  - A. P. Chugainova
AU  - R. R. Polekhina
TI  - Nonuniqueness of a Self-similar Solution to the Riemann Problem for Elastic Waves in Media with a Negative Nonlinearity Parameter
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2023
SP  - 251
EP  - 265
VL  - 322
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2023_322_a19/
LA  - ru
ID  - TM_2023_322_a19
ER  - 
%0 Journal Article
%A A. P. Chugainova
%A R. R. Polekhina
%T Nonuniqueness of a Self-similar Solution to the Riemann Problem for Elastic Waves in Media with a Negative Nonlinearity Parameter
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2023
%P 251-265
%V 322
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2023_322_a19/
%G ru
%F TM_2023_322_a19
A. P. Chugainova; R. R. Polekhina. Nonuniqueness of a Self-similar Solution to the Riemann Problem for Elastic Waves in Media with a Negative Nonlinearity Parameter. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern Methods of Mechanics, Tome 322 (2023), pp. 251-265. http://geodesic.mathdoc.fr/item/TM_2023_322_a19/

[1] A. I. Akhiezer, G. Ia. Liubarskii, and R. V. Polovin, “The stability of shock waves in magnetohydrodynamics”, Sov. Phys. JETP, 8:3 (1959), 507–511 | MR | Zbl

[2] D. R. Bland, Nonlinear Dynamic Elasticity, Blaisdell, Waltham, MA, 1969 | MR | Zbl

[3] A. P. Chugainova, “The formation of a selfsimilar solution for the problem of non-linear waves in an elastic half-space”, J. Appl. Math. Mech., 52:4 (1988), 541–545 | DOI | MR

[4] A. P. Chugainova, “The interaction of non-linear waves in a slightly anisotropic viscoelastic medium”, J. Appl. Math. Mech., 57:2 (1993), 375–381 | DOI | MR | Zbl

[5] Chugainova A.P., Il'ichev A.T., Kulikovskii A.G., Shargatov V.A., “Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: Selection conditions for a unique solution”, IMA J. Appl. Math., 82:3 (2017), 496–525 | MR | Zbl

[6] Chugainova A.P., Polekhina R.R., “The effect of small-scale processes in the structure of discontinuities on the solution of the Riemann problem”, Wave Motion, 114 (2022), 102996 | DOI | MR

[7] V. F. D'jačenko, “Cauchy's problem for quasi-linear systems”, Sov. Math., Dokl., 2 (1961), 7–8 | MR | Zbl

[8] G. Ya. Galin, “Shock waves in media with arbitrary equations of state”, Sov. Phys., Dokl., 3 (1958), 244–247 | MR | Zbl

[9] G. Ya. Galin, “A theory of shock waves”, Sov. Phys., Dokl., 4 (1960), 757–760 | MR | Zbl

[10] A. G. Kulikovskii, “Equations describing the propagation of non-linear quasitransverse waves in a weakly non-isotropic elastic body”, J. Appl. Math. Mech., 50:4 (1986), 455–461 | DOI | Zbl

[11] A. G. Kulikovskii and A. P. Chugainova, “Disintegration conditions for a nonlinear wave in a viscoelastic medium”, Comput. Math. Math. Phys., 38:2 (1998), 305–312 | MR | Zbl

[12] A. G. Kulikovskii and A. P. Chugainova, “Simulation of the influence of small-scale dispersion processes in a continuum on the formation of large-scale phenomena”, Comput. Math. Math. Phys., 44:6 (2004), 1062–1068 | MR | Zbl

[13] A. G. Kulikovskii and A. P. Chugainova, “Classical and non-classical discontinuities in solutions of equations of non-linear elasticity theory”, Russ. Math. Surv., 63:2 (2008), 283–350 | DOI | DOI | MR | Zbl

[14] A. G. Kulikovskii, A. P. Chugainova, and V. A. Shargatov, “Uniqueness of self-similar solutions to the Riemann problem for the Hopf equation with complex nonlinearity”, Comput. Math. Math. Phys., 56:7 (2016), 1355–1362 | DOI | MR | Zbl

[15] A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyperbolic Systems, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., 118, Chapman Hall/CRC, Boca Raton, FL, 2001 | MR | Zbl

[16] A. G. Kulikovskii and E. I. Sveshnikova, Nonlinear Waves in Elastic Media, CRC Press, Boca Raton, FL, 1995 | MR | Zbl

[17] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, v. 6, Fluid Mechanics, Pergamon Press, Oxford, 1987 | MR | Zbl

[18] Lax P.D., “Hyperbolic systems of conservation laws. II”, Commun. Pure Appl. Math., 10:4 (1957), 537–566 | DOI | MR | Zbl

[19] O. A. Oleĭnik, “Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation”, Am. Math. Soc. Transl., Ser. 2, 33 (1963), 285–290 | MR | Zbl | Zbl

[20] L. I. Sedov, Mechanics of Continuous Media, v. 1, World Scientific, River Edge, NJ, 1997 | MR | MR | Zbl

[21] Toro E.F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, Springer, Berlin, 2009 | MR | Zbl

[22] N. D. Vvedenskaya, “An example of nonuniqueness of a generalized solution of a quasilinear system of equations”, Sov. Math., Dokl., 2 (1961), 89–90 | MR | Zbl