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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.
35G10 Initial value problems for linear higher-order PDEs
35C20 Asymptotic expansions of solutions to PDEs