|
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)
References:
PDF file
HTML file
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}
Linking options:
http://mi.mathnet.ru/eng/msb1891 http://mi.mathnet.ru/eng/msb/v174/i4/p470
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:
-
L. A. Kalyakin, “Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems”, Russian Math. Surveys, 44:1 (1989), 3–42
   -
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
-
Guido Schneider, C.Eugene Wayne, “Kawahara dynamics in dispersive media”, Physica D: Nonlinear Phenomena, 152-153 (2001), 384
-
Omel'yanov G., “Resonant Interaction of Short Waves: General Approach”, Russ. J. Math. Phys., 8:2 (2001), 239–243
-
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
-
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
-
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
 -
A. BABIN, A. FIGOTIN, “LINEAR SUPERPOSITION IN NONLINEAR WAVE DYNAMICS”, Rev. Math. Phys, 18:09 (2006), 971
-
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
-
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
-
A. Babin, A. Figotin, “Wavepacket Preservation Under Nonlinear Evolution”, Comm Math Phys, 278:2 (2008), 329
-
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
-
Guido Schneider, “Bounds for the nonlinear Schrodinger approximation of the Fermi-Pasta-Ulam system”, GAPA, 89:9 (2010), 1523
-
Schneider G., Wayne C.E., “Justification of the NLS Approximation for a Quasilinear Water Wave Model”, J. Differ. Equ., 251:2 (2011), 238–269
-
Schneider G., “Justification of the NLS Approximation for the KdV Equation Using the Miura Transformation”, Adv. Math. Phys., 2011, 854719
-
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
-
Dmitry Pelinovsky, Guido Schneider, “Rigorous justification of the short-pulse equation”, Nonlinear Differ. Equ. Appl, 2012
-
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
-
Dmitry Pelinovsky, Dmitry Ponomarev, “Justification of a nonlinear Schrödinger model for laser beams in photopolymers”, Z. Angew. Math. Phys, 2013
-
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
-
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
-
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
|
| Number of views: |
| This page: | 525 | | Full text: | 84 | | References: | 52 |
|
Terms of Use