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Uspekhi Mat. Nauk, 2000, Volume 55, Issue 2(332), Pages 95–120 (Mi umn268)  

This article is cited in 16 scientific papers (total in 17 papers)

The buffer property in resonance systems of non-linear hyperbolic equations

A. Yu. Kolesova, E. F. Mishchenkob, N. Kh. Rozovc

a P. G. Demidov Yaroslavl State University
b Steklov Mathematical Institute, Russian Academy of Sciences
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We study hyperbolic boundary-value problems for systems of telegraph equations with non-linear boundary conditions at the endpoints of a finite interval. The buffer property is established, that is, the existence of an arbitrary given finite number of stable time-periodic solutions for appropriately chosen parameter values, for this class of systems. For the case of a resonance spectrum of eigenfrequencies, the study of self-induced oscillations in various systems is shown to lead to one of the following two model problems, which are a kind of invariant:
\begin{gather*} \frac{\partial^2w}{\partial t\partial x}=w+\lambda(1-w^2)\frac{\partial w}{\partial x} , \qquad w(t,x+1)\equiv-w(t,x), \qquad \lambda>0;
\frac{\partial w}{\partial t}+a^2\frac{\partial^3w}{\partial x^3}=w-w^3, \qquad w(t,x+1)\equiv-w(t,x), \qquad a\ne 0. \end{gather*}
Informative examples from radiophysics are considered.

DOI: https://doi.org/10.4213/rm268

Full text: PDF file (308 kB)
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English version:
Russian Mathematical Surveys, 2000, 55:2, 297–321

Bibliographic databases:

UDC: 517.926
MSC: Primary 35L70, 35L75, 35L20; Secondary 35L35, 35B10, 35C20, 35Q99, 35K60
Received: 05.01.2000

Citation: A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “The buffer property in resonance systems of non-linear hyperbolic equations”, Uspekhi Mat. Nauk, 55:2(332) (2000), 95–120; Russian Math. Surveys, 55:2 (2000), 297–321

Citation in format AMSBIB
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\pages 95--120
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Burkin, IM, “The buffer phenomenon in multidimensional dynamical systems”, Differential Equations, 38:5 (2002), 615  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    2. Kolesov, AY, “The buffer phenomenon in the Van Der Pol oscillator with delay”, Differential Equations, 38:2 (2002), 175  mathnet  crossref  mathscinet  zmath  isi  scopus  scopus
    3. A. Yu. Kolesov, N. Kh. Rozov, “Two-Frequency Autowave Processes in the Complex Ginzburg–Landau Equation”, Theoret. and Math. Phys., 134:3 (2003), 308–325  mathnet  crossref  crossref  mathscinet  isi
    4. A. Yu. Kolesov, N. Kh. Rozov, “The existence of countably many stable cycles for a generalized cubic Schrödinger equation in a planar domain”, Izv. Math., 67:6 (2003), 1213–1242  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. A. Yu. Kolesov, N. Kh. Rozov, “Optical Buffering and Mechanisms for Its Occurrence”, Theoret. and Math. Phys., 140:1 (2004), 905–917  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer Phenomenon in Nonlinear Physics”, Proc. Steklov Inst. Math., 250 (2005), 102–168  mathnet  mathscinet  zmath
    7. E. P. Belan, “O dinamike beguschikh voln v parabolicheskom uravnenii s preobrazovaniem sdviga prostranstvennoi peremennoi”, Zhurn. matem. fiz., anal., geom., 1:1 (2005), 3–34  mathnet  mathscinet  zmath  elib
    8. A. Yu. Kolesov, N. Kh. Rozov, “Smoothing the discontinuous oscillations in the mathematical model of an oscillator with distributed parameters”, Izv. Math., 70:6 (2006), 1201–1224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. A. Yu. Kolesov, N. Kh. Rozov, “The nature of the bufferness phenomenon in weakly dissipative systems”, Theoret. and Math. Phys., 146:3 (2006), 376–392  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Buffer phenomenon in systems close to two-dimensional Hamiltonian ones”, Proc. Steklov Inst. Math. (Suppl.), 253, suppl. 1 (2006), S117–S150  mathnet  crossref  mathscinet  zmath  elib
    11. A. Yu. Kolesov, N. Kh. Rozov, “The buffer property in a non-classical hyperbolic boundary-value problem from radiophysics”, Sb. Math., 197:6 (2006), 853–885  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Buffer phenomenon in systems with one and a half degrees of freedom”, Comput. Math. Math. Phys., 46:9 (2006), 1503–1514  mathnet  crossref  mathscinet  elib  elib
    13. N. Kh. Rozov, “Fenomen bufernosti v matematicheskikh modelyakh estestvoznaniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2010, no. 3, 58–63  mathnet  elib
    14. A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Mnogochastotnye avtokolebaniya v dvukhmernykh reshetkakh svyazannykh ostsillyatorov”, Tr. IMM UrO RAN, 16, no. 5, 2010, 82–94  mathnet  elib
    15. A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Multifrequency self-oscillations in two-dimensional lattices of coupled oscillators”, Izv. Math., 75:3 (2011), 539–567  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. D. V. Anosov, S. M. Aseev, R. V. Gamkrelidze, S. P. Konovalov, M. S. Nikol'skii, N. Kh. Rozov, “Evgenii Frolovich Mishchenko (on the 90th anniversary of his birth)”, Russian Math. Surveys, 67:2 (2012), 385–402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    17. A. Yu. Kolesov, N. Kh. Rozov, “Invariant tori for a class of nonlinear evolution equations”, Sb. Math., 204:6 (2013), 824–868  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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