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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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This publication is cited in the following articles:
-
L. A. Kalyakin, “Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersive medium”, Math. USSR-Sb., 60:2 (1988), 457–483
-
Kaliakin L., “Matching Method for the Problem of Asymptotic Decomposition of Plane-Wave Packet in a Dispersive Medium”, 301, no. 5, 1988, 1048–1052
-
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, “Asymptotic decay of solutions of the Liouville equation under perturbations”, Math. Notes, 68:2 (2000), 173–184
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L A Kalyakin, Inverse Probl, 17:4 (2001), 879
-
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
-
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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