Integrability properties of nonlinearly perturbed Euler equations

Abstract

We study the integrability properties of nonlinearly perturbed Euler equations (linear ordinary differential equations with one regular singular point in the complex plane plus a nonlinear perturbation) near the singular point. We allow for first integrals with essential singularities and give sufficient conditions for the nonintegrability of the equations in the complex domain. We extend normal form theorems for singular equations and argue that equivalence to normal forms captures the spirit of the poly-Painlevé test and is a powerful tool for a rigorous approach to nonintegrability.

Non-existence of first integrals in a Laurent polynomial ring for general semi-quasihomogeneous systems

“…Based on his criterion, he also considered the non-integrability for general semi-quasihomogeneous systems(the definition will be given below). Some similar results related to nonexistence of polynomial integrals, rational integrals and analytic integrals can be found in [2,5,6,7,8,9,10,13].…”

Partial integrability for general nonlinear systems

“…So finding a simple test for the existence or non-existence of nontrivial first integrals(in given function spaces, such as those of polynomials, rational, or analytic functions) is an important problem in considering integrability and nonintegrability, see Costin [7], Kozlov [12] and Kruskal [13]. Some works have been done in this direction.…”

Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

“…Setting. Nonlinear perturbations of Fuchsian systems are, in the present context, differential systems of the form (1) du dx = M(x)u + g(x, u) studied for u ∈ C d small, and x in a simply connected domain D ⊂ C which includes singular points of the matrix M(x).…”

“…Such systems are analytically linearizable if a Diophantine condition is satisfied [1]: Theorem 1. Assume that the eigenvalues µ 1 , ..., µ d of the matrix L satisfy the Diophantine condition: there exist C, ν > 0 so that…”