##
**Meromorphic solutions of difference Painlevé equations.**
*(English)*
Zbl 1219.39001

Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes 155. Helsinki: Suomalainen Tiedeakatemia; Joensuu: Univ. of Eastern Finland, Faculty of Science and Forestry (Diss.) (ISBN 978-951-41-1046-7/pbk; 978-951-41-1047-4/ebook). 59 p. (2010).

This dissertation concerns the open problem to establish a systematic framework that enables us to classify integrable difference equations. An ordinary differential equation is said to possess the Painlevé property if all of its solutions are single-valued about all movable singularities, and it has turned out that this property is a powerful indicator of integrability especially of a general class of second order ones. P. Painlevé and his colleagues have classified the six equations out of a general class of second order ordinary differential equations, which are now known as Painlevé equations I–VI and are indeed integrable in general. In line with the growing importance of interpretation of those rich harvests in the field of differential equations in the language of discrete mathematics, the question of finding an analogue of the Painlevé property for difference equations has been attracting many researchers to this field of study for a long time.

A great number of difference versions of these six Painlevé equations have been distinguished, and they have been found to share several varied integrability properties. The most widely used detector of integrable discrete analogues of the Painlevé equations is the so-called singularity confinement test which is found by B. Grammaticos, T. Tamizhmani, A. Ramani and K. M. Tamizhmani [“Growth and integrability in discrete systems”, J. Phys. A, Math. Gen. 34, No. 18, 3811–3821 (2001; Zbl 1006.39018)]. The test has been successfully applied to discover many important discrete equations, which are widely believed to be integrable, but it causes us some difficulties for the actual applications. On the other hand, discrete equations may be considered as delay equations in the complex plane in order to apply complex analytic methods including Nevanlinna’s value distribution theory for the purpose. M. J. Ablowitz, R. G. Halburd and B. Herbst [“On the extension of the Painlevé property to difference equations”, Nonlinearity 13, No. 3, 889–905 (2000; Zbl 0956.39003)] have proposed to regard those equations to be of Painlevé type, if they permit (sufficiently many) non-rational meromorphic solutions of finite order in Nevanlinna theoretic sense.

In this paper the author observes a class of second order difference equations of the form \(L(w)=R(z,w)\) with a two-variable function \(R(z,w)\) which is rational in \(w\) and meromorphic in \(z\) and \(L(w)\) is chosen as a difference operator in the following variations:

\[ w(z+1)+w(z-1), \quad w(z+1)w(z-1), \;\text{and} \;\bigl\{w(z)z(z+1)-1\bigr\}\bigl\{w(z)w(z-1)-1\bigr\} \]

as the families of difference Painlevé I-II, III and V equations, respectively. R. G. Halburd and R. J. Korhonen [“Finite-order meromorphic solutions and the discrete Painlevé equations”, Proc. Lond. Math. Soc. (3) 94, No. 2, 443–474 (2007; Zbl 1119.39014)] showed that if an equation in the first family has at least one admissible meromorphic solution of finite order, then it reduces into one in a short list of equations containing the known difference Painlevé I and II. In Theorem 3.2 of this paper, the author gives its slightly generalized form with the requirement of finite order by {hyper-order less than one}.

As the main result of this paper, Theorem 3.4 deals with the difference Painlevé III equations and their proofs are given in Chapter 4, while the whole Chapter 5 is devoted to the classification of the family of difference Painlevé V equations. Then, it is shown that if this equation has at least one admissible meromorphic solution \(w\) of hyper-order less than one, then either the \(w\) satisfies a difference Riccati equation, or the equation can be transformed into such a difference Painlevé or a linear equation. By those results, the author does indicate that the existence of an admissible meromorphic solution {of hyper-order less than one} to a difference equation is again a strong indicator of integrability of a class of equations of the above form. Recall that we say that \(w\) is an admissible solution of \(L(w)=R(z,w)\) when the coefficients of \(R(z,w)\) have slow growth with respect to \(w\) in the sense of Nevanlinna theory, as a generalization of a non-rational solution to an equation with the rational coefficients. The importance of the restriction on their growth to be of hyper-order less than one might be understood with Lemmas 3.11 and 3.12 where he shows that solutions with sufficiently many non-confined singularities satisfying an inequality should have hyper-order at least one.

His strategies for classifying equations \(L(w)=R(z,w)\) are as follows: First, in order to restrict possible forms of \(R(z,w)\), he applies both the so-called Valiron-Mohon’ko identity (Theorem 3.6) and a shift-invariance theorem on non-decreasing continuous functions of hyper-order less than one (Theorem 3.10). Then in Section 4.2, he makes out a complete list \(L(z_j,w)\) of some \(w(z_j)\)-points of the solution \(w\) with equidistant distribution. In Sections 4.3 and 4.4, by investigating the ingredients of the list, he can obtain possible coefficients of \(R(z,w)\) which has been factored possibly over the field of algebroid functions, small with respect to the solution \(w\). There are, indeed, so many cases to be managed concerning possible forms of equations and possible coefficients. And the requirements, of course, are extremely more in Chapter 5 for the classification of the family of difference Painlevé V equations. The procedures for this accomplishment have been done efficiently and completely without any overlooks throughout.

Finally, in Chapter 6, the author discusses certain known results in related fields such as number theory and discrete mathematics, some concrete examples of integrable equations and alternative approaches to the problem studied here in his dissertation. These give the reader an overview of a sort of esoteric methods to detect integrable difference equations and what remains open about integrability of Painlevé type equations. One could find there that the relation between the condition for a meromorphic function \(f\) to be of hyper-order less than one and the zero algebraic entropy condition posed by J. Hietarinta and C. Viallet when the singularity confinement test does not work sufficiently for integrability of the difference equation under observation. In fact, the former condition implies that the Nevanlinna characteristic \(T(r,f)\) of \(f\), which corresponds to the degree of rational functions, behaves approximately like \(\log T(r,f) \leq r^{1-\varepsilon}\) for some \(\varepsilon>0\), while the latter is given by \(\lim_{n\to\infty}\frac{\log d_n}{n}=0\) with the degree \(d_n\) of the iterates of the difference equation.

A great number of difference versions of these six Painlevé equations have been distinguished, and they have been found to share several varied integrability properties. The most widely used detector of integrable discrete analogues of the Painlevé equations is the so-called singularity confinement test which is found by B. Grammaticos, T. Tamizhmani, A. Ramani and K. M. Tamizhmani [“Growth and integrability in discrete systems”, J. Phys. A, Math. Gen. 34, No. 18, 3811–3821 (2001; Zbl 1006.39018)]. The test has been successfully applied to discover many important discrete equations, which are widely believed to be integrable, but it causes us some difficulties for the actual applications. On the other hand, discrete equations may be considered as delay equations in the complex plane in order to apply complex analytic methods including Nevanlinna’s value distribution theory for the purpose. M. J. Ablowitz, R. G. Halburd and B. Herbst [“On the extension of the Painlevé property to difference equations”, Nonlinearity 13, No. 3, 889–905 (2000; Zbl 0956.39003)] have proposed to regard those equations to be of Painlevé type, if they permit (sufficiently many) non-rational meromorphic solutions of finite order in Nevanlinna theoretic sense.

In this paper the author observes a class of second order difference equations of the form \(L(w)=R(z,w)\) with a two-variable function \(R(z,w)\) which is rational in \(w\) and meromorphic in \(z\) and \(L(w)\) is chosen as a difference operator in the following variations:

\[ w(z+1)+w(z-1), \quad w(z+1)w(z-1), \;\text{and} \;\bigl\{w(z)z(z+1)-1\bigr\}\bigl\{w(z)w(z-1)-1\bigr\} \]

as the families of difference Painlevé I-II, III and V equations, respectively. R. G. Halburd and R. J. Korhonen [“Finite-order meromorphic solutions and the discrete Painlevé equations”, Proc. Lond. Math. Soc. (3) 94, No. 2, 443–474 (2007; Zbl 1119.39014)] showed that if an equation in the first family has at least one admissible meromorphic solution of finite order, then it reduces into one in a short list of equations containing the known difference Painlevé I and II. In Theorem 3.2 of this paper, the author gives its slightly generalized form with the requirement of finite order by {hyper-order less than one}.

As the main result of this paper, Theorem 3.4 deals with the difference Painlevé III equations and their proofs are given in Chapter 4, while the whole Chapter 5 is devoted to the classification of the family of difference Painlevé V equations. Then, it is shown that if this equation has at least one admissible meromorphic solution \(w\) of hyper-order less than one, then either the \(w\) satisfies a difference Riccati equation, or the equation can be transformed into such a difference Painlevé or a linear equation. By those results, the author does indicate that the existence of an admissible meromorphic solution {of hyper-order less than one} to a difference equation is again a strong indicator of integrability of a class of equations of the above form. Recall that we say that \(w\) is an admissible solution of \(L(w)=R(z,w)\) when the coefficients of \(R(z,w)\) have slow growth with respect to \(w\) in the sense of Nevanlinna theory, as a generalization of a non-rational solution to an equation with the rational coefficients. The importance of the restriction on their growth to be of hyper-order less than one might be understood with Lemmas 3.11 and 3.12 where he shows that solutions with sufficiently many non-confined singularities satisfying an inequality should have hyper-order at least one.

His strategies for classifying equations \(L(w)=R(z,w)\) are as follows: First, in order to restrict possible forms of \(R(z,w)\), he applies both the so-called Valiron-Mohon’ko identity (Theorem 3.6) and a shift-invariance theorem on non-decreasing continuous functions of hyper-order less than one (Theorem 3.10). Then in Section 4.2, he makes out a complete list \(L(z_j,w)\) of some \(w(z_j)\)-points of the solution \(w\) with equidistant distribution. In Sections 4.3 and 4.4, by investigating the ingredients of the list, he can obtain possible coefficients of \(R(z,w)\) which has been factored possibly over the field of algebroid functions, small with respect to the solution \(w\). There are, indeed, so many cases to be managed concerning possible forms of equations and possible coefficients. And the requirements, of course, are extremely more in Chapter 5 for the classification of the family of difference Painlevé V equations. The procedures for this accomplishment have been done efficiently and completely without any overlooks throughout.

Finally, in Chapter 6, the author discusses certain known results in related fields such as number theory and discrete mathematics, some concrete examples of integrable equations and alternative approaches to the problem studied here in his dissertation. These give the reader an overview of a sort of esoteric methods to detect integrable difference equations and what remains open about integrability of Painlevé type equations. One could find there that the relation between the condition for a meromorphic function \(f\) to be of hyper-order less than one and the zero algebraic entropy condition posed by J. Hietarinta and C. Viallet when the singularity confinement test does not work sufficiently for integrability of the difference equation under observation. In fact, the former condition implies that the Nevanlinna characteristic \(T(r,f)\) of \(f\), which corresponds to the degree of rational functions, behaves approximately like \(\log T(r,f) \leq r^{1-\varepsilon}\) for some \(\varepsilon>0\), while the latter is given by \(\lim_{n\to\infty}\frac{\log d_n}{n}=0\) with the degree \(d_n\) of the iterates of the difference equation.

Reviewer: Kazuya Tohge (Kanazawa)

### MSC:

39A12 | Discrete version of topics in analysis |

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |

39A10 | Additive difference equations |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |

30D35 | Value distribution of meromorphic functions of one complex variable, Nevanlinna theory |