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Long-wave asymptotics of the solution of a nonlinear system of equations with small dissipation. (English. Russian original) Zbl 0672.35009

Sib. Math. J. 28, No. 3, 433-443 (1987); translation from Sib. Mat. Zh. 28, No. 3(163), 101-114 (1987).
The system under consideration is \[ \partial_ tU+A(U,\xi,\tau)\partial_ xU+\epsilon (B(\bar D_ x^{p- 1}U,U,\xi,\tau)\partial^ p_ xU+C(D\quad \bar {\;}_ x^{p- 1}U,U,\xi,\tau))=0, \] \(t>0\), \(\xi =\epsilon x\), \(\tau =\epsilon t\), \(p=2n\), with initial condition \[ U(x,t,\epsilon)|_{t=0}=\Phi^ 0(\xi)+\epsilon \Phi^ 1(x,\xi),\quad x\in R. \] Here \(x\in R\), \(t>0\) are independent variables; \(\epsilon >0\) is a small parameter; \(U,C,\Phi^ 0,\Phi^ 1\) are vector functions with values in \(R^ m\); A,B are matrices of type \(m\times m\); \(\bar D_ x^{p-1}=(\partial_ x,...,\partial_ x^{p-1})\) is a set of differential operators of order p-1\(\geq 1.\)
An asymptotic decomposition in \(\epsilon\) of the solution U(x,t,\(\epsilon)\) is constructed, uniform as \(\epsilon\) \(\to 0\) in the stripe \(\pi (M)=\{x\in R\); \(0\leq t\leq M\epsilon^{-1}\}\).
Reviewer: I.Ginchev

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B25 Singular perturbations in context of PDEs
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