Meromorphic solutions of algebraic differential equations

Eremenko, A. È.

doi:10.1090/trans2/126/01

Cited by 7 publications

(15 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…2) 5 According to Umemura [22], the classical functions are obtained from rational functions by successive applications of the so-called permissible operations. The latter include: derivation, quadrature, algebraic operations, solutions to linear differential equations, solutions to first order algebraic equations F (x, ẋ) = 0 and compositions with abelian functions (like the Weierstrass P-function).…”

Section: Hamiltoniansmentioning

confidence: 99%

“…They apply advanced Differential Galois Theory to prove non-integrability of some of the Painlevé equations in the class of the so-called classical functions. 5 Of course, one should expect an 'elementary' version of Theorem 1, instead of the restricted statement in Theorem 2. We are convinced that it is true, but the rigorous proof would be highly complicated.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Painlevé equations, elliptic integrals and elementary functions

Żołądek¹,

Filipuk²

2015

Journal of Differential Equations

View full text Add to dashboard Cite

The six Painlevé equations can be written in the Hamiltonian form, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of the Painlevé equations as symplectic birational transformations in C 4 and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. We also show that the original Painlevé equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions.

show abstract

Section: Hamiltoniansmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Painlevé equations, elliptic integrals and elementary functions

Żołądek¹,

Filipuk²

2015

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…The proof of Theorem D in [8] uses Theorem C which was stated in [8] but not proved. One can also give an alternative proof of Theorem D, using Nevanlinna theory instead of Theorem C, by the arguments similar to those in [9]. Lemma 1.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Meromorphic solutions of higher order Briot–Bouquet differential equations

Erëmenko¹,

Liao²,

2009

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…Nevanlinna theory has applications and analogies in many different fields of mathematics, such as differential equations [27,34,62,40,79], difference equations [121,128,129] number theory [98,99,115,127], Brownian motion [18] and even mathematical logic [60]. Recently, there has been increasing interest in applying Nevanlinna theory to study meromorphic solutions of complex difference equations [20,21,48,58,67,80], and in particular, to detect integrability in discrete equations [1,49,50,111].…”

Section: Nevanlinna Theorymentioning

confidence: 99%

“…The treatment of the full rational case by using Nevanlinna theory is not much more complicated, see for example [79]. See also [27].…”

Section: T (R W ) = Deg W (P (Z W))t (R W) + O(log R)mentioning

confidence: 99%

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Halburd

Korhonen

2007

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z − 1) = R(z, w), where R is rational in w with meromorphic coefficients.

show abstract

Meromorphic solutions of algebraic differential equations

Cited by 7 publications

References 12 publications

Painlevé equations, elliptic integrals and elementary functions

Painlevé equations, elliptic integrals and elementary functions

Meromorphic solutions of higher order Briot–Bouquet differential equations

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

Contact Info

Product

Resources

About