### Application of the repeated quantization method to the problem of making asymptotic solutions of equations with holomorphic coefficients

#### Abstract

In this work, we derive asymptotics of solutions of ordinary differential equations with holomorphic coefficients in the neighborhood of infinity. This problem represents a particular case of the general problem of constructing asymptotics of linear differential equations with irregular singularities, namely the Poincare problem. The case of infinitely distant singular point is an example of irregular singularity and the problem of derivation of asymptotics of its solutions is reduced to the problem of constructing asymptotics of solutions in the neighborhood of zero of linear differential equations with the cusp-type singularity of the second order. If the principal symbol of differential operator has simple roots, then asymptotics of solution of equation in the neighborhood of an irregular singular point can be represented as a classic non-Fuchs asymptotics, which is a familiar fact. In the case of multiple roots, the method of repeated quantization is used. The method is based on the Laplace-Borel transform. Using repeated quantization in this paper we solve the problem of derivation of asymptotics of solutions in the neighborhood of infinity for a model problem whose singularity index has a special form. The derived asymptotics of solutions differ from the classic non-Fuchs asymptotics and represent their generalizations. The method of solution of this model problem in its essential part is transferred to the general case. Thus, this work is one of steps in solving Poincare problem.

#### Full Text:

PDF#### References

H. Poincare, “Sur les integrales irregulieres des equations lineaires,” Acta math. 1886, v. 8, p. 295-344.

Poincaré, Selected works in three volumes. Volume 3. Mathematics. Theoretical physics. Analysis of the mathematical and natural works of Henri Poincaré. “Nauka” publishing house, 1974.

J. Ecalle. “Cinq applications des fonctions résurgentes,” Prepub. Math. d'Orsay, 1984, 84T62, # 110 pp.

L. W. Thome, “Zur Theorie der linearen differentialgelichungen,” (German) 1872.

Katz D. S. “Analysis of asymptotic solutions of equations with polynomial coefficient degeneracy,” Differential equations. vol. 51, no. 12. pp. 1612–1617, 2015.

M. V. Korovina, “Repeated quantization method and its applications to the construction of asymptotics of solutions of equations with degeneration,” Differential Equations, vol. 52, no. 1, pp. 58–75, 2016.

M. V. Korovina and V. E. Shatalov, “Differential equations with degeneration and resurgent analysis,” Differential Equations, vol. 46, no. 9, pp. 1267–1286, 2010.

B. Sternin, V. Shatalov, Borel-Laplace Transform and Asymptotic Theory. Introduction to Resurgent Analysis. CRC Press, 1996.

M. V. Korovina and V. E. Shatalov, “Differential equations with degeneracies,” Doklady Mathematics, vol. 437, no. 1, pp. 16–19, 2011.

M. V. Korovina, “Existence of resurgent solutions for equations with higher-order degeneration,” Differential Equations, vol. 47, no. 3, pp. 346–354, 2011.

M. V. Korovina, “Asymptotics solutions of equations with higher-order degeneracies,” Doklady Mathematics, vol. 83, no. 2, pp. 182–184, 2011.

### Refbacks

- There are currently no refbacks.

Abava Absolutech Fruct 2020

ISSN: 2307-8162