Geodesic
Numerical study of solitary waves and reversible shock structures in tubes with controlled pressure
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 11, pp. 1921-1936 Cet article a éte moissonné depuis la source Math-Net.Ru

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Methods for computing and analyzing solutions for a model of a tube with elastic walls in the case of controlled internal pressure is developed. A membrane model or a plate model is used for the tube walls. Numerical methods are applied. The Boussinesq equations are used to describe waves near the transition to the instability zone of homogeneous states and to verify the numerical methods. Solitary waves and soliton shock structures for these equations are studied. The Boussinesq equations are analyzed and generalized. Next, the same methods are applied to the complete equations. Solitary waves and reversible shock structures (generalized kinks) are studied. The stability of the solitary waves is analyzed by finding an eigenfunction. The kinks are studied using general methods of the theory of reversible shocks. 
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I. B. Bakholdin. Numerical study of solitary waves and reversible shock structures in tubes with controlled pressure. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 11, pp. 1921-1936. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_11_a9/

[1] Fu Y. B., Pearce S. P., “Characterization and stability of localized bulging/necking in inflated membrane tubes”, IMA J. Appl. Math., 75 (2010), 581–602 | DOI | MR | Zbl

[2] Kulikovskii A. G., “Ob ustoichivosti odnorodnykh sostoyanii”, Prikl. matem. i mekhan., 30:1 (1966), 148–153 | Zbl

[3] Bakholdin I. B., Bezdissipativnye razryvy v mekhanike sploshnoi sredy, Fizmatlit, M., 2014

[4] Bakholdin I. B., Tomashpolskii V. Ya., “Uedinennye volny v modeli predvaritelno deformirovannogo nelineinogo kompozita”, Differents. ur-niya, 40:4 (2004), 527–538 | MR | Zbl

[5] Bakholdin I. B., “Metody issledovaniya struktur dissipativnykh i bezdissipativnykh razryvov v sistemakh s dispersiei”, Zh. vychisl. matem. i matem. fiz., 15:2 (2005), 330–343

[6] Bakholdin I. B., “Razryvy, opisyvaemye obobschennymi uravneniyami Kortevega-de Vriza”, Izv. RAN. Mekhan. zhidkosti i gaza, 1999, no. 1, 95–109

[7] Bakholdin I. B., “Statsionarnye i nestatsionarnye struktury razryvov dlya modelei, opisyvaemykh obobschennym uravneniem Kortevega–Byurgersa”, PMM, 75:2 (2011), 271–302 | MR | Zbl

[8] Bakholdin I. B., “Teoriya i klassifikatsiya obratimykh struktur razryvov v modelyakh gidrodinamicheskogo tipa”, PMM, 78:6 (2011)

[9] Rouch P. Dzh., Vychislitelnaya gidrodinamika, Mir, M., 1980

[10] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1977 | MR

[11] Fu Y. B., Il'ichev A., “Solitary waves in flulid-filled clastic tubes: existence, persistence, and the role of axial displacement”, IMA J. Appl. Math., 75 (2010), 257–268 | DOI | MR | Zbl

[12] Ilichev A. T., Fu I. B., “Stability of aneurism solutions in fluid-filled elastic membrane tube”, Acta Mechan. Sinica, 28 (2012), 1209–1218 | DOI | MR | Zbl

[13] Aydin A., Karasozen B., “Sympletic and multisimpletic Lobatto methods for the “good” Boussinesq equation”, J. Math. phys., 49 (2008), 083509 | DOI | MR | Zbl

[14] Alexander J. C., Sachs R., “Linear instability of solitary waves of Boussinesq-type equation: a computer assisted computation”, Nonlinear Word, 2 (1995), 471–507 | MR | Zbl

[15] Bona J. L., Sachs R. L., “Global existence of smooth solutions and stability of solitary waves for a generalized Bonssinesq equation”, Commun. Math. Phys., 118 (1988), 15–29 | DOI | MR | Zbl

[16] Bogdanov L. V., Zakharov V. E., “The Bonssinesq equation revisited”, Physica D, 165 (2002), 137–162 | DOI | MR | Zbl

[17] Miles J. W., “Resonantly interacting solitary waves”, J. Fluid Mech., 79:1 (1977), 157–169 | DOI | MR | Zbl

[18] Bakholdin I. B., “Razryvy peremennykh, kharakterizuyuschikh rasprostranenie uedinennykh voln v sloe zhidkosti”, Izv. AN SSSR. Mekhan. zhidkosti i gaza, 1984, no. 3, 87–93

[19] Kalantarov V. K., Ladyzhenskaya O. A., “O vozniknovenii kollapsov dlya kvazilineinykh uravnenii parabolicheskogo i giperbolicheskogo tipov”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii – 10, Zap. nauchn. sem. LOMI, 69, Nauka, L., 1977, 77–102 | MR | Zbl

[20] Linares F., “Global existence of small solutions for a generalized Bonssinesq equation”, J. Different. Equat., 106 (1993), 257–293 | DOI | MR | Zbl