Geodesic
Stability of discontinuity structures described by a generalized KdV–Burgers equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 2, pp. 259-274 
A. P. Chugainova; V. A. Shargatov. Stability of discontinuity structures described by a generalized KdV–Burgers equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 2, pp. 259-274. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_2_a8/
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     author = {A. P. Chugainova and V. A. Shargatov},
     title = {Stability of discontinuity structures described by a generalized {KdV{\textendash}Burgers} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     language = {ru},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form $\varphi(u)=u^4-u^2$ is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.

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

[2] 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

[3] Kulikovskii A. G., Chugainova A. P., “Klassicheskie i neklassicheskie razryvy v resheniyakh uravnenii nelineinoi teorii uprugosti”, Uspekhi. matem. nauk, 63:2 (2008), 380 | MR

[4] Ilichev A. T., Shargatov V. A., Chugainova A. P., “Spektralnaya ustoichivost osobykh razryvov”, Dokl. AN. Matematika, 462:5 (2015), 512–516 | MR | Zbl

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

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

[7] Godunov S. K., Romenskii E. I., Elementy mekhaniki sploshnykh sred i zakony sokhraneniya, Nauchnaya kniga, Novosibirsk, 1998, 267 pp.

[8] Kulikovskii A. G., Pogorelov N. V., Semenov A. Yu., Matematicheskie voprosy chislennogo resheniya giperbolicheskikh sistem uravnenii, Fizmatlit, M., 2012

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

[10] Chugainova A. P., Shargatov V. A., “Ustoichivost nestatsionarnykh reshenii obobschennogo uravneniya Kortevega–de Vriza–Byurgersa”, Zh. vychisl. matem. i matem. fiz., 55:2 (2015), 253–266 | DOI | MR | Zbl

[11] Oleinik O. A., “O edinstvennosti i ustoichivosti obobschennogo resheniya zadachi Koshi dlya kvazilineinogo uravneniya”, Uspekhi matem. nauk, 14:2(86) (1959), 159–164 | MR | Zbl

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

[13] Pego R. L., Smereka P., Weinstein M. I., “Oscillatory instability of traveling waves for a KdV–Burgers equation”, Physica D, 67 (1993), 45–65 | DOI | MR | Zbl