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On a constructive method of construction of first and the second meromorphic Painlevé transcendents. (Russian) Zbl 0912.35009
Samoilenko, A. M. (ed.) et al., Symmetric and analytic methods in mathematical physics. Series: Mathematical physics. Dedicated to the memory of V. I. Fushchich. Kyïv: Instytut Matematyky NAN Ukraïny. Pr. Inst. Mat. Nats. Akad. Nauk Ukr., Mat. Zastos. 19, 155-165 (1998).
The interrelation between the Painlevé transcendents and the evolution equation is considered. The analytical approximation method for constructing the Painlevé transcendents of the first and the second order by application of the generalized power series is proposed. The mentioned algorithm contains three stages. The first stage is based on an idea of sequential solution of the initial and the inverse equations. Transition from the initial equation to the inverse one is executed under validation of the exact criterion of existence of movable points. Determination of the points is reduced to finding zeros of the inverse problem. The second stage is directed to the construction of the analytical expansion of the solution of the mentioned equations. The third stage determines the movable parameters for the first and the second Painlevé transcendents and coefficients of the regular and non-regular series. Examples demonstrate applications of the created algorithm.
For the entire collection see [Zbl 0902.00014].
35A25 Other special methods applied to PDEs
35C10 Series solutions to PDEs
35A35 Theoretical approximation in context of PDEs