zbMATH — the first resource for mathematics

Alient differential calculus and resurgent equations. (English. Russian original) Zbl 0907.35024
Izv. Math. 61, No. 4, 843-875 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 4, 167-202 (1997).
Initiated by Frederic Pham, resurgent methods had been applied for different problems of mathematical physics. To the knowledge of the reviewer this was related mostly with one complex variable. In an earlier paper of the authors, resurgent theory built by Escal for one variable, had been enlarged for several complex variables. The so-called Stokes phenomenon appears as a kind of discontinuity (jumps) of the asymptotic expansions of solutions of differential equations. Due to this phenomenon, the interconnection between the real and formal monodromies of a resurgent function \(f(x)\) is complicated. So the asymptotic expansion of the analytic continuation of \(f(x)\) along a loop does not coincides with the analytic continuation of its asymptotic expansion. A general method for constructing the so-called resurgent equations is developed. These equations provide a global information about the structure of the Riemann surface of the hyperfunction \(F(s,x)\) corresponding to a given \(f(x)\). Alient derivatives are used to present the systems of resurgent equations as a kind of system of ODE. The corresponding “alient differential calculus” enables us to study the Stokes phenomenon. The best way to understand this paper is to study hard the given variety of examples.
Reviewer: S.Dimiev (Sofia)
35B40 Asymptotic behavior of solutions to PDEs
58J15 Relations of PDEs on manifolds with hyperfunctions
35C20 Asymptotic expansions of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
47A40 Scattering theory of linear operators
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text: DOI