Asymptotics of solutions to a third-order equation in a neighborhood of an irregular singular point

M. V. Korovina ; H. A. Matevossian ; I. N. Smirnov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 106-122

Résumé

The article is devoted to the construction of uniform asymptotics of solutions to a 3rd order equation with holomorphic coefficients with an arbitrary irregular singularity in the space of functions of exponential growth. In general, the problem of constructing asymptotics of solutions of differential equations in the neighborhood of irregular singular points was formulated by Poincaré in his articles devoted to the analytical theory of differential equations. The problem of constructing asymptotics for equations with degeneracies of arbitrary order, in the case of multiple roots, has been solved only for some special cases, for example, when the equation is of second order. The main method for solving the problem for equations with degenerations of higher orders is the re-quantization method based on the Laplace–Borel transform, which was created to construct asymptotics of solutions of differential equations in the neighborhood of irregular singular points in the case when the main symbol of the differential operator has multiple roots. The problem of constructing asymptotics of solutions to higher order equations is much more complicated. To solve it, the re-quantization method is used, which was not required when solving a similar problem for 2nd order equations. Here we solve a model problem, which is an important next step towards solving the general problem formulated by Poincare, the problem of constructing asymptotics of solutions in the neighborhood of an arbitrary irregular singular point for an equation of arbitrary order. The problem of further research is to generalize the solution method outlined in the article to equations of arbitrary orders.