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Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 6, Pages 137–168 (Mi izv462)  

This article is cited in 3 scientific papers (total in 3 papers)

The existence of countably many stable cycles for a generalized cubic Schrödinger equation in a planar domain

A. Yu. Kolesov, N. Kh. Rozov


Abstract: We consider the boundary-value problem
$$ u_t+i\Delta u=\varepsilon(u-d|u|^2u), \qquad u|_{\partial \Omega}=0, $$
in the domain $\Omega=\{(x,y)\colon 0\leqslant x\leqslant 1,0\leqslant y\leqslant 1\}$, where $u$ is a complex-valued function, $\Delta$ is the Laplace operators, $0<\varepsilon\ll1$ and $d=1+ic_0$, $c_0\in\mathbb R$. We establish that it has countably many stable solutions that are periodic in $t$. We study the question of whether this phenomenon is preserved under a change of domain or boundary conditions.

DOI: https://doi.org/10.4213/im462

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English version:
Izvestiya: Mathematics, 2003, 67:6, 1213–1242

Bibliographic databases:

UDC: 517.926
MSC: 35B10, 35B25, 35B35, 35Q80, 35K50, 35K57, 35L75
Received: 17.06.2002

Citation: A. Yu. Kolesov, N. Kh. Rozov, “The existence of countably many stable cycles for a generalized cubic Schrödinger equation in a planar domain”, Izv. RAN. Ser. Mat., 67:6 (2003), 137–168; Izv. Math., 67:6 (2003), 1213–1242

Citation in format AMSBIB
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\by A.~Yu.~Kolesov, N.~Kh.~Rozov
\paper The existence of countably many stable cycles for a~generalized cubic Schr\"odinger equation in a~planar domain
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\pages 137--168
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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. A. Yu. Kolesov, N. Kh. Rozov, “On the theoretical explanation of the diffusion buffer phenomenon”, Comput. Math. Math. Phys., 44:11 (2004), 1922–1941  mathnet  mathscinet  zmath  elib
    2. 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
    3. 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
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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