Long wave asymptotics of asolution of a~hyperbolic system of equations

L. A. Kalyakin

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 91-114

Résumé

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,\dots,m;\quad\xi=\varepsilon x,\quad\tau=\varepsilon t. \end{gather*} It is assumed that the initial vector $\Phi(x,\xi)=(\varphi_1,\dots,\varphi_m)$ 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: $x$, $t$; $\xi$, $\tau$; $\sigma_\alpha$, $\tau$, $\alpha=1,\dots,m$; 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.