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Coherent and focusing multidimensional nonlinear geometric optics. (English) Zbl 0836.35087
A symmetric-hyperbolic system with weak nonlinear and semilinear terms \[ \partial_t U + \sum A_j (t,x, \varepsilon U) \partial_j U = F(t,x, \varepsilon U,U),\;x \in \mathbb{R}^n \] is considered. The main goal is to construct the first terms of the asymptotics of oscillatory solution of the form \[ U(t,x, \varepsilon) = U_0 (t,x) + \varepsilon U_1 (t,x, \Phi (t,x)/ \varepsilon) + o (\varepsilon),\;\varepsilon \to 0 \] with finite number of the fast phases \(\Phi = \{\varphi_1, \dots, \varphi_m\}\). As usually the phase functions \(\varphi_j\) are determined from the eikonal equation \(\partial_t \varphi + \sum A_j (t,x,0) \partial_j \varphi = 0\). The main object which appears in construction is the linear span of the \(\varphi_j\). The problem is solved under a very strong assumption named coherence property in some domain \(\Omega\). That is each nonzero function from the linear span of the \(\varphi_j\) either satisfy the eikonal equation identically or does not satisfies it at every point \((t,x) \in \Omega\).
Reviewer: L.Kalyakin (Ufa)

MSC:
35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
35C20 Asymptotic expansions of solutions to PDEs
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