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Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation. (English) Zbl 0794.35099

Summary: It is well known that a quasilinear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data \(\lambda^{-1}a_ 0(\lambda x\cdot\eta,x)r_ 1(\eta)\), \(\lambda>0\), \(x,\eta\in\mathbb{R}^ n\), \(\eta\neq 0\). Here \(r_ 1(\eta)\) is a characteristic vector, and \(a_ 0(\sigma,x)\) is a smooth scalar function of compact support. Under the additional requirements that \(n=2\) or 3 and that \(a_ 0(\sigma,x)\) have vanishing mean with respect to \(\sigma\), it is shown that a genuine solution exists in a time interval independent of \(\lambda\), and that the formal solution is asymptotic to the genuine solution as \(\lambda\to\infty\).

MSC:

35L60 First-order nonlinear hyperbolic equations
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41A63 Multidimensional problems
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