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Nonstationary problems for equations of Fuchs-Borel type. (English. Russian original) Zbl 1001.35026
Dokl. Math. 59, No. 2, 189-192 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 365, No. 1, 22-25 (1999).
The authors are considering the Fuchs-Borel type second-order equation $\frac{\partial^2u}{\partial t^2}=H(t,x,y,ix^{k+1} \frac{\partial}{\partial x},-i{\partial\over\partial y})u$ with hyperbolic Hamiltonians $$H(t,x,y,E,p,q)$$ subject to the Cauchy data $$u|_{t=0}=u_0$$, $$u_t|_{t=0}=u_1$$. For solution of this problem the authors are applying Maslov’s perturbation theory and asymptotic methods of the WKB type. They are distinguishing two cases: the Fuchs-type degeneration $$(k=0)$$ and the Borel type degeneration $$(k>0)$$. For the Fuchs-type equations the initial data can be selected having asymptotic expansions defined by the “parameter” $$x^{-k}$$ different from the natural parameter $$\ln \frac 1x$$ and the solution can be written through the canonical Maslov operator.
##### MSC:
 35G10 Initial value problems for linear higher-order PDEs 35C20 Asymptotic expansions of solutions to PDEs