Geodesic
Bifurcations in the generalized Korteweg--de Vries equation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2018), pp. 54-68.

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We consider the generalized Korteweg–de Vries (KdV) equation and the Korteweg–de Vries–Burgers (KdVB) equation with boundary condition by space variable. For different values of the parameters in a sufficiently small neighborhood of the zero equilibrium state we construct the asymptotic behavior of periodic solutions and invariant tori. Separately we consider the case of the characteristic equation has a countable number of roots in the range of stability of the zero solution. In this situation we build a special nonlinear boundary-value problem, which plays the role of a normal form and determines the dynamics of the original problem.
Keywords: partial derivative differential equation, normal form method
Mots-clés : torus, bifurcation. 
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S. A. Kashchenko; M. M. Preobrazhenskaya. Bifurcations in the generalized Korteweg--de Vries equation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2018), pp. 54-68. http://geodesic.mathdoc.fr/item/IVM_2018_2_a5/

[1] Korteweg D. J., de Vries G., “On the change of form of long waves advancing in a rectangular canal and on a new tipe of long stationary waves”, Phil. Mag., 39 (1895), 422–443 | DOI | MR

[2] Kudryashov N. A., Metody nelineinoi matematicheskoi fiziki, Uch. posobie, Izd. Dom «Intellekt», Dolgoprudnyi, 2010

[3] Nikolenko N. A., “Invariantnye, asimptoticheski ustoichivye tory vozmuschennogo uravneniya Kortevega-de Friza”, UMN, 35:5 (1980), 121–180 | Zbl

[4] Burgers J. M., “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech., 1 (1948), 171–199 | DOI | MR

[5] Kaschenko S. A., “Normalnaya forma dlya uravneniya Kortevega–de Friza–Byurgersa”, Dokl. RAN, 468:4 (2016), 1–4 | DOI

[6] Bibikov Yu. N., “Bifurkatsiya tipa Khopfa dlya kvaziperiodicheskikh dvizhenii”, Differents. uravneniya, 16:9 (1980), 1539–1544 | MR | Zbl

[7] Bibikov Yu. N., “Bifurkatsiya ustoichivogo invariantnogo tora iz sostoyaniya ravnovesiya”, Matem. zametki, 48:1 (1990), 15–19 | MR | Zbl

[8] Khartman F., Obyknovennye differentsialnye uravneniya, Mir, M., 1970

[9] Mitropolskii Yu. A., Lykova O. B., Integralnye mnogoobraziya v nelineinoi mekhanike, Nauka, M., 1973

[10] Bryuno A. D., Lokalnyi metod nelineinogo analiza differentsialnykh uravnenii, Nauka, M., 1979

[11] Khessard B., Kazarinov N., Ven I., Teoriya i prilozheniya bifurkatsii rozhdeniya tsikla, Mir, M., 1985 | MR

[12] Gukenkheimer Dzh., Kholms F., Nelineinye kolebaniya, dinamicheskie sistemy i bifurkatsii vektornykh polei, In-t kompyut. issledov., M., 2002

[13] Rabinovich M. I., Trubetskov D. I., Vvedenie v teoriyu kolebanii, RKhD, Izhevsk, 2000

[14] Kudryashov N. A., “On “new travelling wave solutions” of the KdV and the KdV–Burgers equations”, Commun. Nonlinear Sci. Numer. Simul., 14:5 (2009), 1891–1900 | DOI | MR | Zbl

[15] Kaschenko S. A., “O kvazinormalnykh formakh dlya parabolicheskikh uravnenii s maloi diffuziei”, Zhurn. vychisl. matem i matem. fiz., 299:5 (1988), 1049–1053

[16] Kashchenko S. A., “Normalization in the systems with small diffusion”, Int. J. of Bifurcations and Chaos, 6:7 (1996), 1093–1109 | DOI | MR | Zbl

[17] Kaschenko I. S., Kaschenko S. A., “Kvazinormalnye formy dvukhkomponentnykh singulyarno vozmuschennykh sistem”, Dokl. RAN, 447:4 (2012), 376–381 | MR

[18] Kaschenko I. S., “Multistabilnost v nelineinykh parabolicheskikh sistemakh s maloi diffuziei”, Dokl. RAN, 435:2 (2010), 164–167 | MR