Long-wave asymptotics of the solution of a hyperbolic system of equations. (Russian) Zbl 0566.35066

Kalyakin, L. A.

Mat. Sb., N. Ser. 124(166), No. 1(5), 96-120 (1984).

The long-wave asymptotics of the solution U(x,t,\(\epsilon)\) at \(\epsilon\) \(\to 0\) of the hyperbolic system of equations \[ [\partial_ t+\lambda_ i(\xi,\tau)\partial_ x]u_ i=\epsilon [A_ i(U,\epsilon,\tau)\partial_ xU+b_ i(U,\xi,\tau)] \] with initial conditions \(u_ i(x,t,\epsilon)|_{t=0}=\phi_ i(x,\xi),\) \(i=1,...,m\) uniformly in some region \(0\leq | x| +t\leq M\epsilon^{-1}\) are constructed. Here x,t are independent variables, \(x\in R^ 1\), \(t\geq 0\) is a small parameter; \(\xi =\epsilon x\), \(\tau =\epsilon t\) are slow variables; \(\lambda_ i(\xi,\tau)\) are scalar functions; \(b_ i\), \(\phi_ i\) are known and \(u_ i\) unknown functions.

Reviewer: L.G.Vulkov

MSC:

35L60	First-order nonlinear hyperbolic equations
35B20	Perturbations in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs

Keywords:

long-wave asymptotics; small parameter