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Mat. Sb. (N.S.), 1987, Volume 132(174), Number 4, Pages 470–495 (Mi msb1891)  

This article is cited in 22 scientific papers (total in 22 papers)

Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium

L. A. Kalyakin


Abstract: The system of equations
$$ \partial_tU+A(U)\partial_xU+B(U)U=0,\qquad x\in\mathbf{R}^1,\quad t>0\quad (U\in\mathbf R^m), $$
is considered with initial data in the form of a wave packet of small amplitude
$$ U_{t=0}=\varepsilon\sum_{k=\pm1}\Phi_k(\xi)\exp(ikx),\quad \xi =\varepsilon x\quad(\Phi _k(\xi )=O((1+|\xi |)^{-N}) \forall N). $$
The asymptotics of the solution $U(x,t,\varepsilon)$ as $\varepsilon\to0$ which is uniform in the strip $x\in\mathbf R^1$, $0\leqslant t\leqslant O(\varepsilon^{-2})$, is constructed by the method of multiscale expansions. The coefficients of the asymptotics are a system of wave packets traveling with group velocities; the leading term is determined from a system of nonlinear equations of Schrödinger type.
Bibliography: 32 titles.

Full text: PDF file (1481 kB)
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English version:
Mathematics of the USSR-Sbornik, 1988, 60:2, 457–483

Bibliographic databases:

UDC: 517.956.226
MSC: 35L60, 35B20
Received: 09.12.1985

Citation: L. A. Kalyakin, “Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium”, Mat. Sb. (N.S.), 132(174):4 (1987), 470–495; Math. USSR-Sb., 60:2 (1988), 457–483

Citation in format AMSBIB
\Bibitem{Kal87}
\by L.~A.~Kalyakin
\paper Asymptotic decay of a~one-dimensional wave packet in a~nonlinear dispersive medium
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 132(174)
\issue 4
\pages 470--495
\mathnet{http://mi.mathnet.ru/msb1891}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=886641}
\zmath{https://zbmath.org/?q=an:0699.35135|0658.35047}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 60
\issue 2
\pages 457--483
\crossref{https://doi.org/10.1070/SM1988v060n02ABEH003181}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. L. A. Kalyakin, “Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems”, Russian Math. Surveys, 44:1 (1989), 3–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. L. A. Kalyakin, S. G. Glebov, “On the solvability of nonlinear equations of Shrödinger type in the class of rapidly oscillating functions”, Math. Notes, 56:1 (1994), 673–678  mathnet  crossref  mathscinet  zmath  isi
    3. Guido Schneider, C.Eugene Wayne, “Kawahara dynamics in dispersive media”, Physica D: Nonlinear Phenomena, 152-153 (2001), 384  crossref  mathscinet  zmath
    4. Omel'yanov G., “Resonant Interaction of Short Waves: General Approach”, Russ. J. Math. Phys., 8:2 (2001), 239–243  mathscinet  zmath  isi
    5. Bambusi D., Carati A., Ponno A., “The Nonlinear Schrodinger Equation as a Resonant Normal Form”, Discrete Contin. Dyn. Syst.-Ser. B, 2:1 (2002), 109–128  mathscinet  zmath  isi
    6. Guido Schneider, “Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances”, Journal of Differential Equations, 216:2 (2005), 354  crossref  mathscinet  zmath
    7. Wolf-Patrick Düll, Guido Schneider, “Validity of the resonant four-wave interaction system in a model for surface water waves on an infinite deep sea”, Nonlinear Analysis: Real World Applications, 7:5 (2006), 1243  crossref  mathscinet  zmath  elib
    8. A. BABIN, A. FIGOTIN, “LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS”, Rev. Math. Phys, 18:09 (2006), 971  crossref  mathscinet  zmath
    9. Martina Chirilus-Bruckner, Guido Schneider, Hannes Uecker, “On the interaction of NLS-described modulating pulses with different carrier waves”, Math Meth Appl Sci, 30:15 (2007), 1965  crossref  mathscinet  zmath  isi
    10. Wolf-Patrick Düll, “The validity of phase diffusion equations and of Cahn–Hilliard equations for the modulation of pattern in reaction–diffusion systems”, Journal of Differential Equations, 239:1 (2007), 72  crossref  mathscinet  zmath
    11. A. Babin, A. Figotin, “Wavepacket Preservation Under Nonlinear Evolution”, Comm Math Phys, 278:2 (2008), 329  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. Wolf-Patrick Düll, Guido Schneider, “Validity of Whitham’s Equations for the Modulation of Periodic Traveling Waves in the NLS Equation”, J Nonlinear Sci, 2009  crossref  mathscinet  zmath  isi
    13. Guido Schneider, “Bounds for the nonlinear Schrodinger approximation of the Fermi-Pasta-Ulam system”, GAPA, 89:9 (2010), 1523  crossref  mathscinet  zmath
    14. Schneider G., Wayne C.E., “Justification of the NLS Approximation for a Quasilinear Water Wave Model”, J. Differ. Equ., 251:2 (2011), 238–269  crossref  mathscinet  zmath  isi
    15. Schneider G., “Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation”, Adv. Math. Phys., 2011, 854719  crossref  mathscinet  zmath  isi
    16. Martina Chirilus-Bruckner, Christopher Chong, Oskar Prill, Guido Schneider, “Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations”, DCDS-S, 5:5 (2012), 879  crossref  mathscinet  zmath
    17. Dmitry Pelinovsky, Guido Schneider, “Rigorous justification of the short-pulse equation”, Nonlinear Differ. Equ. Appl, 2012  crossref  mathscinet
    18. C. Chong, G. Schneider, “Numerical evidence for the validity of the NLS approximation in systems with a quasilinear quadratic nonlinearity”, Z. angew. Math. Mech, 2013, n/a  crossref  mathscinet
    19. Dmitry Pelinovsky, Dmitry Ponomarev, “Justification of a nonlinear Schrödinger model for laser beams in photopolymers”, Z. Angew. Math. Phys, 2013  crossref  mathscinet
    20. Guido Schneider, D.A.li Sunny, Dominik Zimmermann, “The NLS Approximation Makes Wrong Predictions for the Water Wave Problem in Case of Small Surface Tension and Spatially Periodic Boundary Conditions”, J Dyn Diff Equat, 2014  crossref  mathscinet
    21. Martina Chirilus-Bruckner, Wolf-Patrick Düll, Guido Schneider, “NLS approximation of time oscillatory long waves for equations with quasilinear quadratic terms”, Math. Nachr, 2014, n/a  crossref  mathscinet
    22. D.A.li Sunny, Guido Schneider, Dominik Zimmermann, “The NLS and FWI approximation make wrong predictions for the water wave problem in case of small surface tension”, Proc. Appl. Math. Mech, 14:1 (2014), 747  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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