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On the Cauchy problem in classes of resurgent functions. (English. Russian original) Zbl 0793.35019
Sov. Math., Dokl. 44, No. 2, 496-500 (1992); translation from Dokl. Akad. Nauk SSSR 320, No. 3, 551-555 (1991).
From the paper: We consider the Cauchy problem for differential equations of the form \[ -k^{m-1} {\partial \psi \over \partial t}+H \left( x,- {\partial \over \partial x},k \right) \psi=0,\quad \psi |_{t=0}=\psi_ 0(x). \] Here \(k\) is a parameter, and \(H(x,p,k)\) is a polynomial in \((p,k)\) of degree \(m\) with coefficients holomorphic in \(x\). Following the idea of Balian and Bloch, we seek solutions to this type of equation in the class of resurgent functions.
35G25 Initial value problems for nonlinear higher-order PDEs
35A20 Analyticity in context of PDEs
35C15 Integral representations of solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs