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Asymptotics of rapidly oscillating solutions of the generalized Korteweg-de Vries-Burgers equation. (English. Russian original) Zbl 1439.35428

Russ. Math. Surv. 74, No. 4, 755-757 (2019); translation from Usp. Mat. Nauk 74, No. 4, 181-182 (2019).
Summary: We consider the equation \[ \frac{\partial u}{\partial t} + \frac{\partial^3 u}{\partial x^3} + \Phi(u) \frac{\partial u}{\partial x} + F(u) = \mu \frac{\partial^2 u}{\partial x^2}, \tag{1} \] where \(\Phi(u) = \varphi_1 u + \varphi_2 u^2 + \varphi_3 u^3 + \varphi_4 u^4\) and \(F(u) = f_1 u + f_2 u^2 + f_3 uu^3 + f_4u^4 + f_5u^5\).
In this note we study the problem of the asymptotic properties of solutions of (1) which oscillate rapidly with respect to \(t\) and \(x\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] D. J. Korteweg and G. de Vries 1895 Philos. Mag. (5)39 240 422-443 · JFM 26.0881.02 · doi:10.1080/14786449508620739
[2] Б. А. Дубровин, В. Б. Матвеев, С. П. Новиков 1976 УМН31 1(187) 55-136 · Zbl 0326.35011
[3] English transl. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov 1976 Russian Math. Surveys31 1 59-146 · Zbl 0346.35025 · doi:10.1070/RM1976v031n01ABEH001446
[4] Н. А. Кудряшов 2010 Методы нелинейной математической физики, Уч. пособие Изд. дом “Интеллект”, Долгопрудный 368 pp.
[5] N. A. Kudryashov 2010 Methods of nonlinear mathematical physics, textbook Intellekt, Dolgoprudnyi 368 pp.
[6] S. A. Kaschenko 1996 Internat. J. Bifur. Chaos Appl. Sci. Engrg.6 6 1093-1109 · Zbl 0881.35057 · doi:10.1142/S021812749600059X
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