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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

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