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High frequency semilinear oscillations. (English) Zbl 0703.35103
Wave motion: theory, modelling, and computation, Proc. Conf. Hon. 60th Birthday P. D. Lax, Publ., Math. Sci. Res. Inst. 7, 202-216 (1987).
[For the entire collection see Zbl 0638.00016.]
The authors investigate nonlinear analogues of the oscillatory solutions studied by P. D. Lax [Duke Math. J. 24, 627-646 (1957; Zbl 0083.318)]. Considered is a semilinear hyperbolic system of order 1, of the form \[ (1)\quad Lu+f(x,u)=g\in C^{\infty}_{(0)}([-T,T]\times {\mathbb{R}}^ d), \] where L is strictly hyperbolic with respect to (1,0,...,0), f is a smooth function satisfying \[ f(x,0)=\nabla_ uf(x,0)=0\text{ for } x=(x_ 0,x')\in [-T,T]\times {\mathbb{R}}^ d, \] and \(D^{\alpha}_{x,u}f\) are all bounded in \([-T,T]\times {\mathbb{R}}^ d\times K\) for any \(\alpha\) and a compact \(K\subset {\mathbb{C}}^ k\). Let \(\underline u\) be the solution to (1) with \(\underline u|_{t=0}=0\), \(\underline u\in C([0,\bar T]\times {\mathbb{R}}^ d)\) and let \(q>d/2\) or \(d=1\) and \(q>0\). Then for small \(\epsilon\) the solution \(u^{\epsilon}\) to (1) with \(u^{\epsilon}|_{t=0}=\epsilon^ qa(x')e^{i\psi(x')/\epsilon}\) \((d\psi\neq 0)\) exists in \([0,\bar T]\times {\mathbb{R}}^ d\) and admits an expansion \[ u^{\epsilon}\sim \sum^{\infty}_{\ell =0}\epsilon^{q\ell}P_{\ell}(ae^{i\psi /\epsilon}), \] with \(P_ 0=\underline u\), \(P_{\ell}\) is \(\ell\)- linear: \(\Pi^{\ell}B\to\) some function space for each \(\ell\) where \(B=L^{\infty}\cap C\) when \(d=1\) or \(B=H^ s\) \((d/2<s<q)\) and that the expansion is asymptotic in the sense that \[ \sup_{[0,\bar T]\times {\mathbb{R}}^ d}| u^{\epsilon}-\sum_{\ell \leq N- 1}\epsilon^{q\ell}P_{\ell}| \leq C_ N\epsilon^ N. \] Also the propagation problem to (1) is studied with an arbitrary d for one mode oscillation \(u^{\epsilon}= A(x,\phi(x)/\epsilon)+O(\epsilon)\) by giving \(v^{\epsilon}|_{t\leq 0}\) and \(g^{\epsilon}\).
Reviewer: T.Kakita

35L60 First-order nonlinear hyperbolic equations
35L45 Initial value problems for first-order hyperbolic systems
35C20 Asymptotic expansions of solutions to PDEs