On holomorphic regularization of strongly nonlinear singularly perturbed problems

Abstract

The method of holomorphic regularization, being a logical continuation of the method of S.A. Lomova, allows one to construct solutions to nonlinear singularly perturbed initial problems as series in powers of a small parameter converging in the usual sense. The method is based on a generalization of the Poincare decomposition theorem: in the regular case, solutions depend holomorphically on a small parameter, in the singular case the first integrals inherit this dependence. Having arised in the framework of the regularization method, S.A. Lomov's concept of a pseudo-analytic (pseudo-holomorphic) solution of singularly perturbed problems initiated the formation of the analytic theory of singular perturbations. This theory is designed to equalize the rights of regular and singular theories. In the first case, under sufficiently general assumptions, the series obtained in the solution of problems in powers of the small parameter converge in the usual sense, and in the second case they are basically asymptotic. A vivid example of the holomorphic dependence on a parameter of the solution to a differential equation is given by Poincare's decomposition theorem. In the present paper, the holomorphic regularization method is applied for constructing pseudo-holomorphic solutions to a singularly perturbed first order equation and to a second order Tikhonov system.