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

This article is cited in 7 scientific papers (total in 7 papers)

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 $\varepsilon$:
\begin{gather*} [\partial_t+\lambda_i(\xi,\tau)\partial_x]u_i=\varepsilon[A_i(U,\xi,\tau)\partial_xU+b_i(U,\xi,\tau)],\qquad t>0;
u_i(x,t,\varepsilon)|_{t=0}=\varphi_i(x,\xi),\quad x\in\mathbf R^1;\quad i=1,…,m;\quad\xi=\varepsilon x,\quad\tau=\varepsilon t. \end{gather*}
It is assumed that the initial vector $\Phi(x,\xi)=(\varphi_1,…,\varphi_m)$ has asymptotics
$$ \Phi(x,\xi)=\Phi^\pm(\xi)+O(x^{-N}),\qquad x\to\pm\infty,\quad\forall N,\quad\forall |\xi|\leqslant M_0. $$
A`complete 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
Received: 05.04.1983

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  mathnet  crossref  mathscinet  zmath
    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  mathscinet  isi
    3. 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
    4. L. A. Kalyakin, “Asymptotic decay of solutions of the Liouville equation under perturbations”, Math. Notes, 68:2 (2000), 173–184  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. L A Kalyakin, Inverse Probl, 17:4 (2001), 879  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath  adsnasa  isi
    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  crossref  mathscinet  zmath  isi  elib
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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