Geodesic
Nonstationary contrast structures in adjacency of the special point
Matematičeskoe modelirovanie, Tome 26 (2014) no. 8, pp. 107-125.

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

Evolution equations of internal transition layer (ITL) are obtained for reaction-diffusion equation and pseudoparabolic third order equation with the small parameter near high derivatives, which describes different processes in physics, chemistry, biology, particularly, the process of magnetic field generation in the turbulent medium. The case of existence of the point with zero drift velocity (special point) is considered, so that drift velocity doesn't change the sign from the left and from the right of this point. It's shown that drift velocity of the first order of asymptotic expansion for cube balanced nonlinearity, which is widely used in physic applications, equals zero too, second order approximation allows to find drift velocity in the special point. It's shown that ITL runs the special point in limited time.
Keywords: internal transition layer (ITL), contrasting structure (CS), asymptotic method. 
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A. A. Bykov; A. S. Sharlo. Nonstationary contrast structures in adjacency of the special point. Matematičeskoe modelirovanie, Tome 26 (2014) no. 8, pp. 107-125. http://geodesic.mathdoc.fr/item/MM_2014_26_8_a7/

[1] Pao C. V., Nonlinear parabolic and elliptic equations, Plenum, New York, 1992 | MR

[2] Zeldovich Ya. B., Ruzmaikin A. A., Sokolov D. D., Magnitnye polya v astrofizike, In-t khaotich. dinam., M.–Izhevsk, 2006

[3] Barenblatt G. I., Zeldovich Ya. B., “Promezhutochnye asimptotiki v matematicheskoi fizike”, Uspekhi matematicheskikh nauk, 26:2(158) (1971), 115–129 | MR | Zbl

[4] Rozhdestvenskii B. L., Yanenko N. N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, 2-e izd., Nauka, M., 1978 | MR

[5] Martinson L. K., Malov Yu. I., Differentsialnye uravneniya matematicheskoi fiziki, Izd-vo MGTU im. Baumana, M., 2002

[6] Davydov A. S., Biologiya i kvantovaya mekhanika, Naukova dumka, Kiev, 1979 | MR

[7] Hideo Ikeda, Masayasu Mimura, Tohru Tsuijikawa, “Singular Perturbation Approach to Travelling Wave Solutions of the Hodgkin–Huxley Equations and Its Application to Stability Problems”, North-Holland Mathematics Studies, 148, 1987, 1–73 | DOI

[8] Volpert V., Petrovskii S., “Reaction-diffusion waves in biology”, Physics of Life Reviews, 6 (2009), 267–310 | DOI | MR

[9] Kolmogorov A., Petrovsky I., Piskounoff N., “Etude de L'Equations de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique”, Bull Univ. Moskou, Ser. Internat. 1A, 1937, 1–25

[10] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, 2007

[11] Korpusov M. O., Pletner Yu. D., Sveshnikov A. G., “O kvazistatsionarnykh protsessakh v provodyaschikh sredakh bez dispersii”, ZhVMiMF, 40:8 (2000), 1237–1249 | MR | Zbl

[12] Korpusov M. O., Sveshnikov A. G., “O “razrushenii” za konechnoe vremya reshenii nachalno-kraevykh zadach dlya uravnenii psevdoparabolicheskogo tipa s psevdolaplassianom”, ZhVMiMF, 45:2 (2005), 272–286 | MR | Zbl

[13] Ma W. X., Fuchssteiner B., “Explicit and exact solutions to a Kolmogorov–Petrovskii–Piskunov equation”, Int. J. Non-Linear Mechanics, 31:3 (1996), 329–338 | DOI | MR | Zbl

[14] Wei J., Yang J., “Solutions with transition layers and spike in an inhomogeneous phase transition models”, J. Differential Equations, 246 (2009), 3642–3667 | DOI | MR | Zbl

[15] Bozhevolnov Yu. V., Nefedov N. N., “Dvizhenie fronta v parabolicheskoi zadache reaktsiya-diffuziya”, ZhVMiMF, 50:2 (2010), 276–285 | MR | Zbl

[16] Nefedov N. N., “Nestatsionarnye kontrastnye struktury v sisteme reaktsiya-diffuziya”, Matematicheskoe modelirovanie, 4:8 (1992), 58–65 | MR | Zbl

[17] Kozhanov A. I., “Nachalno-kraevaya zadacha dlya uravnenii tipa obobschennogo uravneniya Bussineska s nelineinym istochnikom”, Matematicheskie zametki, 65:1, Yanvar (1999), 70–75 | DOI | MR | Zbl

[18] Vasileva A. B., Butuzov V. F., Nefedov N. N., “Kontrastnye struktury v singulyarno vozmuschennykh zadachakh”, Fundamentalnaya i prikladnaya matematika, 4:3 (1998), 799–851 | MR | Zbl

[19] Bykov A. A., Popov V. Yu., “O vremeni zhizni odnomernykh nestatsionarnykh kontrastnykh struktur”, ZhVMiMF, 309:2 (1999), 280–288 | MR