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 Mat. Sb. (N.S.), 1984, Volume 124(166), Number 1(5), Pages 96–120 (Mi msb2042)

Long wave asymptotics of asolution of a hyperbolic system of equations

L. A. Kalyakin

Abstract: The Cauchy problem is considered for a hyperbolic system of equations with a small parameter $\varepsilon$:
It is assumed that the initial vector $\Phi(x,\xi)=(\varphi_1,…,\varphi_m)$ has asymptotics
$$\Phi(x,\xi)=\Phi^\pm(\xi)+O(x^{-N}),\qquad x\to\pm\infty,\quad\forall N,\quad\forall |\xi|\leqslant M_0.$$
Acomplete asymptotic expansion of the solution $U(x,t,\varepsilon)$ as $\varepsilon\to0$ which is uniform in a large domain $0\leqslant|x|$, $t\leqslant O(\varepsilon^{-1})$ is constructed by the method of matching. Several subdomains are distinguished in which the expansion can be represented in the form of various series. The following pairs of variables are characteristic in these subdomains: $x$, $t$; $\xi$, $\tau$; $\sigma_\alpha$, $\tau$, $\alpha=1,…,m$; here $\sigma_\alpha=\varepsilon^{-1}\omega_\alpha(\xi,\tau)$, $\partial_\tau\omega_\alpha+\lambda_\alpha\partial_\xi\omega_\alpha=0$, and $\omega_\alpha(\xi,0)=\xi$.
Bibliography: 20 titles.

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English version:
Mathematics of the USSR-Sbornik, 1985, 52:1, 91–114

Bibliographic databases:

UDC: 517.956
MSC: 35L45, 35B25

Citation: L. A. Kalyakin, “Long wave asymptotics of asolution of a hyperbolic system of equations”, Mat. Sb. (N.S.), 124(166):1(5) (1984), 96–120; Math. USSR-Sb., 52:1 (1985), 91–114

Citation in format AMSBIB
\Bibitem{Kal84} \by L.~A.~Kalyakin \paper Long wave asymptotics of asolution of a~hyperbolic system of equations \jour Mat. Sb. (N.S.) \yr 1984 \vol 124(166) \issue 1(5) \pages 96--120 \mathnet{http://mi.mathnet.ru/msb2042} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=743059} \zmath{https://zbmath.org/?q=an:0599.35098|0566.35066} \transl \jour Math. USSR-Sb. \yr 1985 \vol 52 \issue 1 \pages 91--114 \crossref{https://doi.org/10.1070/SM1985v052n01ABEH002879} `

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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, “Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium”, Math. USSR-Sb., 60:2 (1988), 457–483
2. Kaliakin L., “Matching Method for the Problem of Asymptotic Decomposition of Plane-Wave Packet in a Dispersive Medium”, 301, no. 5, 1988, 1048–1052
3. L. A. Kalyakin, “Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems”, Russian Math. Surveys, 44:1 (1989), 3–42
4. L. A. Kalyakin, “Asymptotic decay of solutions of the Liouville equation under perturbations”, Math. Notes, 68:2 (2000), 173–184
5. L A Kalyakin, Inverse Probl, 17:4 (2001), 879
6. Le U.V., “A Semilinear Wave Equation with Space-Time Dependent Coefficients and a Memory Boundarylike Antiperiodic Condition: a Low-Frequency Asymptotic Expansion”, J. Math. Phys., 52:2 (2011), 023510
7. Le U.V., “On a Low-Frequency Asymptotic Expansion of a Unique Weak Solutions of a Semilinear Wave Equation with a Boundary-Like Antiperiodic Condition”, Manuscr. Math., 138:3-4 (2012), 439–461
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