The local Poincaré problem for irreducible branches

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare problem on constructing the asymptotics of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhoods of irregular singular points. In this study we consider such equations for which the principal symbol of the differential operator has multiple roots. The asymptotics of a solution for the case of equations with simple roots of the principal symbol were constructed earlier.

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On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2019.63 ◽

2020 ◽

Vol 150 (6) ◽

pp. 3231-3251 ◽

Cited By ~ 1

Author(s):

Maria V. Demina ◽

Claudia Valls

Keyword(s):

Differential Equations ◽

Algebraic Curves ◽

First Integral ◽

Complete Classification ◽

Invariant Algebraic Curves ◽

Liouvillian Integrability ◽

Poincaré Problem

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

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UNIPOTENT REDUCTION AND THE POINCARÉ PROBLEM

International Journal of Mathematics ◽

10.1142/s0129167x0600376x ◽

2006 ◽

Vol 17 (08) ◽

pp. 949-962

Author(s):

ALEXIS G. ZAMORA

Keyword(s):

Linear Part ◽

General Fiber ◽

Poincaré Problem

Given a fibration f : S → ℙ1, and the associated foliation [Formula: see text], the problem of bounding the genus of the general fiber of f in terms of the sheaf [Formula: see text] is studied. Using unipotent reduction of f, several bounds are obtained, under positivity assumptions on [Formula: see text]. In Sec.4, the Poincaré problem is solved, for non-degenerate [Formula: see text], assuming that all the eigenvalues of the linear part of [Formula: see text] near singularities are greater than 3.

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Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauchy–Riemann equation

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210507000108 ◽

2009 ◽

Vol 139 (1) ◽

pp. 157-181 ◽

Cited By ~ 6

Author(s):

Alip Mohammed ◽

M. W. Wong

Keyword(s):

Boundary Condition ◽

Linear Equations ◽

Robin Boundary Condition ◽

Robin Problem ◽

System Of Linear Equations ◽

Riemann Equation ◽

Cauchy Riemann ◽

Robin Boundary ◽

Poincaré Problem ◽

Well Posed

The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

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On the Poincaré problem for a compressible medium

Journal of Fluid Mechanics ◽

10.1017/s0022112074000863 ◽

1974 ◽

Vol 62 (4) ◽

pp. 657-675 ◽

Cited By ~ 9

Author(s):

Roger F. Gans

Keyword(s):

Rotation Frequency ◽

Compressible Medium ◽

Zeroth Order ◽

Frequency Shifts ◽

First Order ◽

Specific Heats ◽

Confluent Hypergeometric Functions ◽

Negative Case ◽

Poincaré Problem ◽

Order Set

By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.

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