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

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.

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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

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
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
3. Guido Schneider, C.Eugene Wayne, “Kawahara dynamics in dispersive media”, Physica D: Nonlinear Phenomena, 152-153 (2001), 384
4. Omel'yanov G., “Resonant Interaction of Short Waves: General Approach”, Russ. J. Math. Phys., 8:2 (2001), 239–243
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
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
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
8. A. BABIN, A. FIGOTIN, “LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS”, Rev. Math. Phys, 18:09 (2006), 971
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
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11. A. Babin, A. Figotin, “Wavepacket Preservation Under Nonlinear Evolution”, Comm Math Phys, 278:2 (2008), 329
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
13. Guido Schneider, “Bounds for the nonlinear Schrodinger approximation of the Fermi-Pasta-Ulam system”, GAPA, 89:9 (2010), 1523
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
15. Schneider G., “Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation”, Adv. Math. Phys., 2011, 854719
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
17. Dmitry Pelinovsky, Guido Schneider, “Rigorous justification of the short-pulse equation”, Nonlinear Differ. Equ. Appl, 2012
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
19. Dmitry Pelinovsky, Dmitry Ponomarev, “Justification of a nonlinear Schrödinger model for laser beams in photopolymers”, Z. Angew. Math. Phys, 2013
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
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
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
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