Geodesic
Stability of nonstationary solutions of the generalized KdV-Burgers equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 2, pp. 253-266 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The stability of nonstationary solutions to the Cauchy problem for a model equation with a complex nonlinearity, dispersion, and dissipation is analyzed. The equation describes the propagation of nonlinear longitudinal waves in rods. Previously, complex behavior of traveling waves was found, which can be treated as discontinuity structures in solutions of the same equation without dissipation and dispersion. As a result, the solutions of standard self-similar problems constructed as a sequence of Riemann waves and shocks with a stationary structure become multivalued. The multivaluedness of the solutions is attributed to special discontinuities caused by the large effect of dispersion in conjunction with viscosity. The stability of special discontinuities in the case of varying dispersion and dissipation parameters is analyzed numerically. The computations performed concern the stability analysis of a special discontinuity propagating through a layer with varying dispersion and dissipation parameters. 
@article{ZVMMF_2015_55_2_a9,
     author = {A. P. Chugainova and V. A. Shargatov},
     title = {Stability of nonstationary solutions of the generalized {KdV-Burgers} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {253--266},
     year = {2015},
     volume = {55},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_2_a9/}
}
TY  - JOUR
AU  - A. P. Chugainova
AU  - V. A. Shargatov
TI  - Stability of nonstationary solutions of the generalized KdV-Burgers equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 253
EP  - 266
VL  - 55
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_2_a9/
LA  - ru
ID  - ZVMMF_2015_55_2_a9
ER  - 
%0 Journal Article
%A A. P. Chugainova
%A V. A. Shargatov
%T Stability of nonstationary solutions of the generalized KdV-Burgers equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 253-266
%V 55
%N 2
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_2_a9/
%G ru
%F ZVMMF_2015_55_2_a9
A. P. Chugainova; V. A. Shargatov. Stability of nonstationary solutions of the generalized KdV-Burgers equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 2, pp. 253-266. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_2_a9/

[1] Gelfand I. M., “Nekotorye zadachi teorii kvazilineinykh uravnenii”, Uspekhi matem. nauk, 14:2(86) (1959), 87–158

[2] Godunov S. K., “O needinstvennosti “razmazyvaniya” razryvov v resheniyakh kvazilineinykh sistem”, Dokl. AN SSSR, 136:2 (1961), 272–273

[3] Godunov S. K., Romenskii E. I., Elementy mekhaniki sploshnykh sred i zakony sokhraneniya, Nauchnaya kniga, Novosibirsk, 1998

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

[5] Kulikovskii A. G., “O vozmozhnom vliyanii kolebanii v strukture razryva na mnozhestvo dopustimykh razryvov”, Dokl. AN SSSR, 275:6 (1984), 1349–1352

[6] Kulikovskii A. G., “O poverkhnostyakh razryva, razdelyayuschikh idealnye sredy s razlichnymi svoistvami: Volny rekombinatsii”, Prikl. matem. i mekhan., 32:6 (1968), 1125–1131

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

[8] 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

[9] Kasimov A. R., Faria L. M., Rosales R. R., “Model for shock wave chaos”, Phys. Rev. Lett., 110:10 (2013), 104104 | DOI

[10] Ilichev A. T., Tsypkin G. G., “Neustoichivosti odnorodnykh filtratsionnykh techenii s fazovym perekhodom”, Zh. eksperim. i teor. fiz., 134:4 (2008), 815–830

[11] Kulikovskii A. G., Gvozdovskaya N. I., “O vliyanii dispersii na mnozhestvo dopustimykh razryvov v mekhanike sploshnoi sredy”, Tr. MIAN, 223, 1998, 63–73

[12] 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

[13] Chugainova A. P., “Nestatsionarnye resheniya obobschennogo uravneniya Kortevega–de Vriza–Byurgersa”, Tr. MIAN, 281, 2013, 215–223

[14] Oleinik O. A., “O edinstvennosti i ustoichivosti obobschennogo resheniya zadachi Koshi dlya kvazilineinogo uravneniya”, Uspekhi matem. nauk, 14:2 (1959), 159–164

[15] Samsonov A. M., “Travelling wave solutions for nonlinear dispersive equations with dissipation”, Applicable Analysis, 57 (1995), 85–100 | DOI