Geodesic
Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2261-2274 
A. G. Kulikovskii; A. P. Chugainova. Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 12, pp. 2261-2274. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_12_a14/
@article{ZVMMF_2010_50_12_a14,
     author = {A. G. Kulikovskii and A. P. Chugainova},
     title = {Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {2261--2274},
     year = {2010},
     volume = {50},
     number = {12},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_12_a14/}
}
TY  - JOUR
AU  - A. G. Kulikovskii
AU  - A. P. Chugainova
TI  - Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2010
SP  - 2261
EP  - 2274
VL  - 50
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_12_a14/
LA  - ru
ID  - ZVMMF_2010_50_12_a14
ER  - 
%0 Journal Article
%A A. G. Kulikovskii
%A A. P. Chugainova
%T Self-similar asymptotics describing nonlinear waves in elastic media with dispersion and dissipation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2010
%P 2261-2274
%V 50
%N 12
%U http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_12_a14/
%G ru
%F ZVMMF_2010_50_12_a14

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

Solutions of problems for the system of equations describing weakly nonlinear quasi-transverse waves in an elastic weakly anisotropic medium are studied analytically and numerically. It is assumed that dissipation and dispersion are important for small-scale processes. Dispersion is taken into account by terms involving the third derivatives of the shear strains with respect to the coordinate, in contrast to the previously considered case when dispersion was determined by terms with second derivatives. In large-scale processes, dispersion and dissipation can be neglected and the system of equations is hyperbolic. The indicated small-scale processes determine the structure of discontinuities and a set of admissible discontinuities (with a steady-state structure). This set is such that the solution of a self-similar Riemann problem constructed using solutions of hyperbolic equations and admissible discontinuities is not unique. Asymptotics of non-self-similar problems for equations with dissipation and dispersion were numerically found, and it appeared that they correspond to self-similar solutions of the Riemann problem. In the case of nonunique self-similar solutions, it is shown that the initial conditions specified as a smoothed step lead to a certain self-similar solution implemented as the asymptotics of the unsteady problem depending on the smoothing method.

[1] Kulikovskii A. G., “Ob uravneniyakh, opisyvayuschikh rasprostranenie nelineinykh kvazipoperechnykh voln v slaboanizotropnom uprugom tele”, Prikl. matem. i mekhan., 50:4 (1986), 597–604

[2] Kulikovskii A. G., Sveshnikova E. I., Nelineinye volny v uprugikh sredakh, Mosk. litsei, M., 1998

[3] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2001 | MR

[4] Sveshnikova E. I., “Prostye volny v nelineino uprugoi srede”, Prikl. matem. i mekhan., 46:4 (1982), 642–646

[5] Kulikovskii A. G., Sveshnikova E. I., “Issledovanie udarnoi adiabaty kvazipoperechnykh udarnykh voln v predvaritelno napryazhennoi uprugoi srede”, Prikl. matem. i mekhan., 46:5 (1982), 831–840

[6] Chugainova A. P., “Asimptoticheskoe povedenie nelineinykh voln v uprugikh sredakh s dispersiei i dissipatsiei”, Teor. i matem. fiz., 147:2 (2006), 240–256 | MR | Zbl

[7] Chugainova A. P., “Avtomodelnye asimptotiki volnovykh voln i struktury neklassicheskikh razryvov v nelineino-uprugikh sredakh s dispersiei i dissipatsiei”, Prikl. matem. i mekhan., 71:5 (2007), 775–787 | MR

[8] Kulikovskii A. G., Chugainova A. P., “Klassicheskie i neklassicheskie razryvy v resheniyakh uravnenii nelineinoi teorii uprugosti”, Uspekhi matem. nauk, 63:2 (2008), 85–152 | MR

[9] Gvozdovskaya N. I., Kulikovskii A. G., “Kvazipoperechnye udarnye volny v uprugikh sredakh s vnutrennei strukturoi”, Prikl. mekhan. i tekhn. fiz., 40:2 (1999), 174–180 | MR | Zbl

[10] Bakhvalov N. S., Eglit M. E., “Effektivnye uravneniya s dispersiei dlya rasprostraneniya voln v periodicheskikh sredakh”, Dokl. RAN, 370:1 (2000), 1–4 | MR | Zbl

[11] Kulikovskii A. G., Chugainova A. P., “O statsionarnoi strukture udarnykh voln v uprugikh sredakh i dielektrikakh”, Zh. eksperim. i teor. fiz., 137:4 (2010), 973–985

[12] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 6, Gidrodinamika, Nauka, M., 1986 | MR

[13] Gelfand I. M., “Nekotorye zadachi teorii kvazilineinykh uravnenii”, Uspekhi matem. nauk, 14:2 (1959), 87–158 | MR

[14] Kulikovskii A. G., Chugainova A. P., “Modelirovanie vliyaniya melkomasshtabnykh dispersionnykh protsessov v sploshnoi srede na formirovanie krupnomasshtabnykh yavlenii”, Zh. vychisl. matem. i matem. fiz., 44:6 (2004), 1119–1126 | MR | Zbl