Nonstationary problems for equations of Fuchs-Borel type (Q1594317)

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.