Non-integrability of the Painlev\'{e} IV equation in the Liouville-Arnold sense and Stokes phenomena

TL;DR

This paper proves the non-integrability of a Hamiltonian system related to the Painlevé IV equation using Galois theory and Stokes phenomena, showing it cannot be solved by meromorphic first integrals.

Contribution

It demonstrates the non-integrability of the Painlevé IV Hamiltonian system in the Liouville-Arnold sense through explicit Galois group analysis and Stokes matrices, extending previous results.

Findings

The Hamiltonian system is not integrable in the Liouville-Arnold sense.

The Galois group associated with the second variational equations is non-Abelian.

The non-integrability holds for all parameter values where Painlevé IV has rational solutions.

Abstract

In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlev\'{e} equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Sim\'{o} theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for allvalues of the parameters for which the $P_{IV}$ equation has a particular rational solution the corresponding Hamiltonian system is not integrable by meromorphic first integrals which are rational in $t$.

Equations

$$\ddot{y}=\frac{1}{2y}(\dot{y}{)}^2+\frac{3}{2}{y}^3+4t{y}^2+2(t^2-a)y+\frac{b}{y},\cdot =\frac{d}{dt},$$

$${\varphi }_{\theta }(z)=3(-i\sqrt{3}{)}^{-5/2}{\int }_0^{+\infty e^{i\theta }}\frac{e^{-\frac{\xi }{z}}}{{\left(\xi -i\sqrt{3}\right)}^{5/2}}d\xi$$

$${\phi }_{\theta }(z)=3(i\sqrt{3}{)}^{-5/2}{\int }_0^{+\infty e^{i\theta }}\frac{e^{-\frac{\xi }{z}}}{{\left(\xi +i\sqrt{3}\right)}^{5/2}}d\xi .$$

$$(\text{I})$$

$$(\text{II})$$

$$a=m,b=-\frac{2}{9}(1+3n{)}^2,$$

$$\dot{x}=X_H(x)$$

$$\dot{\xi }=\frac{\partial X_H}{\partial x}(x(t))\xi .$$

$$\dot{x}(t,z)=X_H(x(t,z)).$$

$${\dot{x}}^{(k)}(t)=X_H^{(1)}(x(t))x^{(k)}(t)+P\left(x^{(1)}(t),x^{(2)}(t),\dots ,x^{(k-1)}(t)\right).$$

$$Ly=\left(L_2\circ L_1\right)y=0,L_2\circ L_1\ne L_1\circ L_2,$$

$$L_1={\partial }^2+a_1(z)\partial +a_0(z),L_2={\partial }^2+b_1(z)\partial +b_0(z)$$

$$Y^{\prime }(z)=A(z)Y(z),{}^{\prime }=\frac{d}{dz},$$

$$\displaystyle A(z)=\left(\begin{array}[]{cc}A_{1}(z)&B\\ 0&A_{2}(z)\end{array}\right)\,.$$

$$\displaystyle A_{1}(z)=\left(\begin{array}[]{cc}0&1\\ -a_{0}(z)&-a_{1}(z)\end{array}\right),\quad A_{2}(z)=\left(\begin{array}[]{cc}0&1\\ -b_{0}(z)&-b_{1}(z)\end{array}\right),\quad B=\left(\begin{array}[]{cc}0&0\\ 1&0\end{array}\right)\,.$$

$$\displaystyle\Phi(z)=\left(\begin{array}[]{cc}\Phi_{1}(z)&\Psi(z)\\ 0&\Phi_{2}(z)\end{array}\right)\,,$$

$$\displaystyle\Psi(z)=\left(\begin{array}[]{cc}\psi_{1}(z)&\psi_{2}(z)\\ \psi^{\prime}_{1}(z)&\psi^{\prime}_{2}(z)\end{array}\right)\,,$$

$${\hat{\Phi }}_0(z)=\hat{H}(z)z^{\Lambda }\exp \left(\frac{Q}{z}\right),$$

$${\hat{\Phi }}_0(z.e^{2\pi i})={\hat{\Phi }}_0(z){\hat{M}}_0.$$

$${\hat{M}}_0=e^{2\pi i\Lambda }.$$

$$y_1(z)=z^{l_1}{e}^{\frac{q_1}{z}}{\hat{\phi }}_1(z),y_2(z)=z^{l_2}{e}^{\frac{q_2}{z}}{\hat{\phi }}_2(z),$$

$$y_3(z)=z^{l_3}{e}^{\frac{q_3}{z}}{\hat{\phi }}_3(z),y_4(z)=z^{l_4}{e}^{\frac{q_4}{z}}{\hat{\phi }}_4(z),$$

$${\Theta }_j=\{{\theta }_{ji},0\le {\theta }_{ji}<2\pi \}$$

$${\Theta }_j=\left\{{\theta }_{ji},0\le {\theta }_{ji}<2\pi ,{\theta }_{ji}=\arg (q_j-q_i),1\le i\le 4,i\ne j\right\}.$$

$${\text{St}}_{\theta }={\left({\Phi }_0^{\theta +ϵ}(z)\right)}^{-1}{\Phi }_0^{\theta -ϵ}(z).$$

$$\dot{y}=\frac{\partial H_{IV}}{\partial p},\dot{p}=-\frac{\partial H_{IV}}{\partial y}$$

$$H_{IV}=y{p}^2-y^{2}p-2typ-2(v_1-v_2)p-2(v_2-v_3)y.$$

$$a=1+3v_3,b=-2(v_1-v_2{)}^2.$$

$$\frac{dy}{ds}=\frac{\partial H}{\partial p},\frac{dt}{ds}=\frac{\partial H}{\partial F},$$

$$\frac{dp}{ds}=-\frac{\partial H}{\partial y},\frac{dF}{ds}=-\frac{\partial H}{\partial t}.$$

$$\dot{y}$$

$$\dot{p}$$

Full text

Department of Mathematics and Informatics, Sofia University,

5 J. Bourchier Blvd., Sofia 1164, Bulgaria, cveti@fmi.uni-sofia.bg

Key words: Painlevé IV equation, Non-integrability of Hamiltonian systems, Differential Galois theory, Stokes phenomenon

2010 Mathematics Subject Classification: 34M55, 37J30, 34M40

1. Introduction

In this paper we study the integrability of the fourth Painlvé equation $P_{IV}$

[TABLE]

where $t\in{\mathbb{C}}$ and $a,b$ are arbitrary complex parameters from the point of view of the Hamiltonian dynamics. We prove that for one family of the parameters $a$ and $b$ the Painlevé IV equation is not integrable by meromorphic functions which are rational in $t$.

There are many works devoted to both pure mathematical aspects of the $P_{IV}$ equation [6, 5, 7, 9, 10, 13, 16, 23, 24, 25] and to its applications to mathematical physics [14, 4, 8]. In the present paper we are concerned with the Hamiltonian system corresponding to the $P_{IV}$ equation. We prove rigorously that for one two parametric family of $P_{IV}$ equation this Hamiltonian system is not integrable, i.e. there no exist two independent meromorphic first integrals in involution. As in our previous works [15, 30, 31, 32] our approach is based on the Morales-Ramis-Simó theory which reduces the problem of integrability of a given Hamiltonian system to the problem of determination of the differential Galois group of the variational equations along a particular non-equilibrium solution. In the main part of this paper we deal with the integrability of the equation (1.1) when $a=0,b=-\frac{2}{9}$ and $y=-\frac{2}{3}\,t$. It urns out that the differential Galois group of the first variational equations is Abelian. Then according to the scheme of Morales-Ramis-Simó we have to consider the higher variational equations. The second normal variational equations $(\textrm{NVE})_{2}$ are reduced to a fourth-order linear reducible ordinary differential equations with two singular points taken at the origin and the infinity point. The origin is a non-resonant irregular point of Poincaré rank 1 while the infinity point is a resonant Fuchsian singularity. By an explicit computation we show that the local fundamental set of solutions at the origin of the $(\textrm{NVR})_{2}$ contains the functions

[TABLE]

and

[TABLE]

The analytic continuation of these functions leads to the existence of non-trivial Stokes matrices at the origin of the $(\textrm{NVE})_{2}$ and therefore implies non-commutative connected component of the unit element of the corresponding differential Galois group. Then the key theorem (Theorem 3.14 in the Section 3) of this paper states

Theorem 1.1**.**

Assume that $a=0$ and $b=-\frac{2}{9}$. Then the fourth Painlevé equation is not integrable by meromorphic functions which are rational in $t$.

From the paper of Murata [22] it follows that all the rational solutions of the $P_{IV}$ equation (1.1) can be obtained by Bäcklund transformations from the following two particular solutions

[TABLE]

Since the Bäcklund transformations are birational canonical transformations [25, 35, 36] we can extend the result of Theorem 1.1 to the main theorem of this paper

Theorem 1.2**.**

Assume that

[TABLE]

where $m,n\in{\mathbb{Z}}$ such that $\frac{1}{2}(m+n)\in{\mathbb{Z}}$. Then the fourth Painlevé equation (1.1) is not integrable in the Liouville-Arnold sense by meromorphc functions that are rational in $t$.

In [32] we have proved that when the $P_{IV}$ equation has a rational solution which is obtained by Bäcklund transformations from the solution (I) the Hamiltonian system (2.17) is not integrable in the Liouville-Arnold sense. Combining Theorem 1.2 and Theorem 1.1 in [32] we establish

Theorem 1.3**.**

The fourth Painlevé equation (1.1) is not integrable in the Louville-Arnold sense by meromorphic functions which are rational in $t$.

Noumi and Okamoto are the first authors who pose the problem of irreducibility of the fourth Painlevé equation from a strictly mathematical point of view. In [23] they prove that except some special particular solutions (rational, algebraic, Riccati solutions), any solution of the $P_{IV}$ equation is non-classical in the sense of Umemura. In [36] Zoladek and Filipuk prove non-integrability of the fourth Painlevé equation as a Hamiltonian system by algebraic first integrals. Their approach is based on the Liouville’s theory of elementary functions and some properties of elliptic integrals. In [1] Acosta-Humánez, van der Put and Top claim that when $a=0,b=-\frac{2}{9}$ and $a=0,b=-2$ the Painlevé IV equation is not integrable as a Hamiltonian system. Their approach is based on the Morales-Ramis-Simó theory. In particular, they like us study the differential Galois group of the second normal variational equations. Then their statement follows from the fact that the solution of the second variational equations contains non-trivial functions, like error function, which implies non-commutative Galois group and therefore non-integrability. We note that the authors do not build explicitly the solution of the second variational equations, as well as they do not compute generators of the corresponding differential Galois group. Our method show in details the analytic obstruction for integrability and present explicitly the generators of the differential Galois group of the $(\textrm{NVE})_{2}$.

This paper is organized as follows. In the next section we briefly review the basics of the Morales-Ramis-Simó theory of the non-integrability of the Hamiltonian systems, as well as of the theory of linear ordinary differential equations and its relation to the differential Galois theory. We also recall the Hamiltonian system corresponding to the $P_{IV}$ equation. In Section 3 we study for integrability the Hamiltonian system (2.17) when $a=0$ and $b=-2/9$. The main result of this section is Theorem 3.14 which states that when $a=0,b=-2/9$ the fourth Painlevé equation (1.1) is not integrable in the Louville-Arnold sense by meromorphic functions which are rational in $t$. In Section 4 we prove Theorem 1.2.

2. Preliminaries

2.1. Non-intagrability and differential Galos theory

In this paragraph we briefly recall the theory of Morales-Ruiz, Ramis and Simó about non-integrability of Hamiltonian systems following [20, 21].

Let $M$ be a symplectic analytical complex manifold of complex dimension $2n$. Consider on $M$ a Hamiltonian system

[TABLE]

with a Hamiltonian $H:M\rightarrow{\mathbb{C}}$. Recall that by the theorem of Liouville-Arnold [2] the Hamiltonian system (2.2) is completely integrable in the Louville-Arnold sense if there exist $n$ first integrals $f_{1}=H,f_{2},\ldots,f_{n}$ functionally independent and in involution. Let $x(t)$ be a particular solution of (2.2) which is not an equilibrium point of the Hamiltonian vector field $X_{H}$. Denote by $\Gamma$ the phase curve corresponding to this solution. Then the first variational equations along $\Gamma$ are given by

[TABLE]

Using the Hamiltonian $H$ we can reduce the variational equation (2.3) in the following sense. Consider the normal bundle of $\Gamma$ on the level variety $M_{h}=\{x\,|\,H(x)=h\}$. The projection of the variational equation (2.3) on this bundle is the so called normal variational equation $(\textrm{NVE})$. The dimension of the NVE is $2n-2$. The solutions of the NVE define a Picard-Vessiot extension $L_{1}$ of the differential field $K$ of the coefficients of NVE. This in its turn defines a differential Galois group $G_{1}=Gal(L_{1}/K)$. Then the main theorem of the Morales-Ruiz and Ramis theory states [20, 21]

Theorem 2.1**.**

$($Morales-Ruiz and Ramis$)$* If the Hamilton system (2.2) is completely integrable with meromorphic first integrals in a neighbourhood of $\Gamma$, not necessarily independent on $\Gamma$ itself, then the identity component $G_{1}^{0}$ of the differential Galois group $G_{1}=Gal(L_{1}/K)$ is Abelian.*

The opposite is not true in general, i.e. if the connected component of the unit element of the differential Galois group $Gal(L_{1}/K)$ is Abelin, it is not sure that the corresponding Hamiltonian system (2.2) is integrable. To overcome this problem Morales-Ruiz, Ramis and Simó propose to use higher order variational equations [21]. Let again $x(t)$ be a particular solution of (2.2) which is not an equilibrium point of the vector field $X_{H}$. We write the general solution as $x(t,z)$, where $z$ parametrizes it near $x(t)$ as $x(t,z_{0})=x(t)$. Then we write the system (2.2) as

[TABLE]

Denote by $x^{(1)}(t,z),x^{(2)}(t,z),\ldots$ the derivatives of $x(t,z)$ with resect to $z$ and by

$X_{H}^{(1)}(x),X_{H}^{(2)}(x),\ldots$ the derivatives of $X_{H}(x)$ with respect to $x$. By successive derivation of (2.4) with respect to $z$ and evaluation at $z_{0}$ we obtain the so called $k$-th variational equations $(\textrm{VE})_{k}$ for the function $x(t)$

[TABLE]

Here $P$ is a polynomial. The coefficients depend on $t$ through $X_{H}^{(j)}(x(t),j<k$. For every $k>1$ the linear non-homogeneous system (2.5) can be arranged as a linear homogeneous system of higher dimension. This chain of linear homogeneous systems defines a chain of successive Picard-Vessiot extensions of the main differential field $K$ of the coefficients of NVE, i.e. we have $K\subset L_{1}\subset L_{2}\subset\cdots\subset L_{k}$, where $L_{1}$ is as above, $L_{2}$ is the Picard-Vessiot extension of $K$ associated with $(\textrm{VE})_{2}$, etc. We can define the Galois groups $G_{1}=Gal(L_{1}/L),G_{2}=Gal(L_{2}/K),\ldots$. Then the main theorem in the Morales-Ramis-Simó theory states [21]

Theorem 2.2**.**

$($Morales-Ruiz, Ramis and Simó$)$* If the Hamiltonian system (2.2) is completely integrable then for every $m\in{\mathbb{N}}$ the connected component of the unit element $G_{m}^{0}$ of the Galois group $G_{m}=Gal(L_{m}/K)$ is Abelian.*

Assume that $G_{1}^{0}$ is Abelian. Then the theorem Theorem 2.2 says that if we want to obtain a non-integrable result we have to find a group $G_{m}^{0},m>1$ which is not Abelian. Note that this group will be a solvable group (see [21] for details). In this case the non-integrability in the sense of Hamiltonian dynamics corresponds to integrability in the Picard-Vessiot sense.

2.2. The fundamental matrix solution of the $(\textrm{NVE})_{2}$

The main result of this paper is based on the study of the differential Galois group of the second normal variational equations since the differential Galois group of the first variational equations is Abelian. Throughout this paper we assume that $K={\mathbb{C}}(z)$. The Cyclic Vector Theorem ensures that the homogeneous linear system associated with $(\textrm{NVE})_{2}$ can be always reduced to a higher order linear scalar equation. As the initial linear system and the obtained scalar equation belong to the same differential module the connected components $G^{0}_{2}$ of the corresponding differential Galois groups are Abelian or not-Abelian at the same time (see [29, 33] for more details). It turns out that the $(\textrm{NVE})_{2}$ are reduced to a fourth order reducible scalar equation. For this reason from here to the end of this section we consider a reducible fourth order linear ordinary differential equation in the form

[TABLE]

where $L_{j},j=1,2$ are second order differential operators

[TABLE]

with $\partial=\frac{d}{dz}$ and coefficients $a_{j}(z),b_{j}(z)\in{\mathbb{C}}(z)$. For simplicity throughout this section we call the equation (2.6) with the operators (2.7) just the $(\textrm{NVE})_{2}$.

Denote by $\{w_{1}(z),w_{2}(z)\}$ a fundamental set of solutions of the equation $L_{2}\,y=0$. Then

Theorem 2.3**.**

With the above notatons the equation (2.6) possesses a fundamental set of solutions $\{y_{1}(z),y_{2}(z),\psi_{1}(z),\psi_{2}(z)\}$ where $\{y_{1}(z),y_{2}(z)\}$ is a fundamental set of solutions of the equation $L_{1}\,y=0$. The functions $\psi_{1}(z)$ and $\psi_{2}(z)$ are the solutions of the equations $L_{2}\,y=w_{j}(z),j=1,2$, respectively.

Proof.

The proof is straightforward. ∎

Denote by $w(z)$ the solution of the equation $L_{2}\,y=0$. The next theorem associate with the equation (2.6) a fourth order linear system of differential equations.

Theorem 2.4**.**

The function $y(z)$ solves the equation (2.6) if and only if the vector $Y(z)=(y(z),y^{\prime}(z),w(z),w^{\prime}(z))^{\tau}$ solves the linear system

[TABLE]

where

[TABLE]

The matrices $A_{j}(z),j=1,2$ and $B$ are second order matrices in the form

[TABLE]

Proof.

The proof is straightforward. ∎

Theorem 2.5**.**

The equation (2.6) possesses a fundamental matrix solution $\Phi(z)$ in the form

[TABLE]

where $\Phi_{j}(z),j=1,2$ are fundamental matrix solutions of the second order linear systems $U^{\prime}(z)=A_{j}(z)\,U(z)$, respectively, with matrices $A_{j}(z),j=1,2$ defined by Theorem 2.4. The matrix $\Psi(z)$ writes

[TABLE]

where $\psi_{j}(z)j=1,2$ are introdused by Theorem 2.3.

Proof.

The proof is straightforward. ∎

Remark 2.6**.**

The matrices $\Phi_{j}(z),j=1,2$ from Theorem 2.3 appear as fundamental matrix solutions for the equations $L_{j}\,y=0,j=1,2$, respectively.

2.3. The differential Galois group of the $(\textrm{NVE})_{2}$

In this paragraph we review some definitions, facts and notation from the theory of linear ordinary differential equations, as well as, from the Picard-Vessiot theory which are required to describe the differential Galois group of one equation of the kind (2.6). Most of them can be found in [27, 29, 33]. Throughout this paragraph we assume that the equation (2.6) has only two singular points over ${\mathbb{C}}{\mathbb{P}}^{1}$: one non-resonant irregular singularity at the origin of Poincaré rank 1 and one regular singularity at $z=\infty$.

We start by describing the local differential Galois group at the origin. The restriction of the theorem of Hukuhara-Turrittin-Wasow [34] gives us

Theorem 2.7**.**

Assume that the equation (2.6) has a non-resonant singularity at the origin. Then the fundamental matrix solution $\Phi(z)$ from (2.12) is represented at the origin as

[TABLE]

where $\Lambda$ and $Q$ are constant diagonal matrices. The entries of the matrix $\hat{H}(z)$ are usually divergent power series.

The formula (2.16) is usually regarded as a formal fundamental matrix solution at the origin. To introduce the formal invariants of the equation (2.6) we consider the equation (2.6) and its formal fundamental matrix solution $\hat{\Phi}(z)$ from (2.16) over the field ${\mathbb{C}}((z))$ of formal power series in $z$.

Definition 2.8**.**

With respect to the formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ given by (2.16) we define the formal monodromy matrix $\hat{M}_{0}\in GL_{4}({\mathbb{C}})$ around the origin as

[TABLE]

In particular,

[TABLE]

Denote $\Lambda={\mathrm{diag}}(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})$ and $Q={\mathrm{diag}}(q_{1},q_{2},q_{3},q_{4})$.

Definition 2.9**.**

With respect to the formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ given by (2.16) we define the exponential torus ${\mathcal{T}}$ as the differential Galois group $Gal(E/F)$, where $F={\mathbb{C}}((z))(z^{\lambda_{1}},z^{\lambda_{2}},z^{\lambda_{3}},z^{\lambda_{4}})$ and $E=F(e^{q_{1}/z},e^{q_{2}/z},e^{q_{3}/z},e^{q_{4}/z})$.

Since for $\sigma\in{\mathcal{T}}$ we have that $\sigma(z^{\lambda})=z^{\lambda}$ and $\sigma(e^{q/z})=c\,e^{q/z},\,c\in{\mathbb{C}}^{*}$ we can consider ${\mathcal{T}}$ as a subgroup of $({\mathbb{C}}^{*})^{4}$. The formal monodromy $\hat{M}_{0}$ and the exponential torus ${\mathcal{T}}$ generate topologically the differential Galois group at the origin of the equation (2.6) over ${\mathbb{C}}((z))$ (see Theorem 1.4.9 in [29]).

To introduce the analytic invariants at the origin we consider the equation (2.6) and its solutions over the field ${\mathbb{C}}(\{z\})$ of convergent power series in $z$. A fourth order linear ordinary differential equation (2.6) with a non-resonant irregular singularity at the origin of Poincaré rank 1 admits a local fundamental set of solutions at the origin in the form

[TABLE]

where $q_{j}\in{\mathbb{C}},1\leq j\leq 4$ are as above. The numbers $l_{j}$ and $\lambda_{j}$ from above are related by $\lambda_{j}-l_{j}\in{\mathbb{Z}}$. Here $\hat{\varphi}_{j}(z),1\leq j\leq 4$ are power series in $z$ which are either convergent or divergent.

Definition 2.10**.**

Under the above notations for every divergent power series $\hat{\varphi}_{j}(z)$ we define a family $\Theta_{j}$ of admissible singular directions

[TABLE]

where $\theta_{ji}$ is the bisector of the sector $\{Re\left(\frac{q_{i}-q_{j}}{z}\right)<0\}$. In particular,

[TABLE]

The application of the summability theory to the linear ordinary equations leads to the following important theorem of Ramis [27]

Theorem 2.11**.**

In the formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ from (2.16) the entries of the matrix $\hat{H}(z)$ are 1-summable in every non-singular direction $\theta$. If we denote by $H_{\theta}(z)$ the 1-sum of the matrix $\hat{H}(z)$ along a non-singular direction $\theta$ then the matrix $\Phi_{0}^{\theta}(z)=H_{\theta}(z)\,z^{\Lambda}\,\exp\left(\frac{Q}{z}\right)$ gives an actual fundamental matrix solution of the equation (2.6) on a small sector bisected by $\theta$.

For the needed aspects of the summability theory we refer to the works of Loday-Richaud [17], as well as the works of Ramis [26, 27].

Let $\epsilon>0$ be a small number. Let $\theta-\epsilon$ and $\theta+\epsilon$ be tow non-singular neighboring directions of the singular direction $\theta\in\Theta_{j}$. Let $\Phi_{0}^{\theta+\epsilon}(z)$ and $\Phi_{0}^{\theta-\epsilon}(z)$ be the actual fundamental matrix solutions at the origin of the equations (2.6) related to the direction $\theta+\epsilon$ and $\theta-\epsilon$ in the sense of Theorem 2.11. Then

Definition 2.12**.**

With respect to the actual fundamental matrix solutions $\Phi_{0}^{\theta-\epsilon}(z)$ and $\Phi_{0}^{\theta+\epsilon}(z)$ the Stokes matrix $\textrm{St}_{\theta}\in GL_{4}({\mathbb{C}})$ related to the singular direction $\theta$ is defined as

[TABLE]

The next theorem of Ramis [28] describes the differential Galois group at the origin of the equation (2.6).

Theorem 2.13**.**

$($Ramis$)$* The differential Galois group at the origin of the equation (2.6) is the Zariski closure in $GL_{4}({\mathbb{C}})$ of the group generated by the formal monodromy $\hat{M}$, the exponential torus ${\mathcal{T}}$ and the Stokes matrices $St_{\theta}$ for all singular directions $\theta$.*

The differential Galois group at $z=\infty$ of the equation (2.6) is computed by the following theorem of Schlesinger [33]

Theorem 2.14**.**

$($Schlesinger$)$* The differential Galois group at a regular singularity is the Zariski closure of the monodromy group around this point.*

For the needed facts of the local theory around regular singular points we refer to the book of Golubev [11].

In Proposition 1.3 in [18] Mitschi describes the global differential Galois group of an arbitrary differential equation. Here we formulate the result of Mitschi for the equation (2.6).

Theorem 2.15**.**

$($Mitschi$)$* The global differential Galois group $G$ of the equation (2.6) is topologically generated in $GL_{4}({\mathbb{C}})$ by the differential Galois group at the origin and the differential Galois group at $z=\infty$ of the same equation.*

2.4. $P_{IV}$ equation as a Hamiltonian system

The fourth Painlevé equation (1.1) is equivalent to the following non-autonomous Hamiltonian system of $1+1/2$ degrees of freedom [24, 25]

[TABLE]

with the polynomial Hamiltonian [24]

[TABLE]

The parameters $v_{1},v_{2},v_{3}$ satisfy the condition $v_{1}+v_{2}+v_{3}=0$. They and the parameters $a,b$ in (1.1) are related through the formulas [24]

[TABLE]

We can extend this Hamiltonian system to an autonomous Hamiltonian system on ${\mathbb{C}}^{4}$ by introducing two new dynamical variables $t$ and $F$. The variable $F$ is conjugate to $t$. The new obtained Hamiltonian system is already autonomous of two degrees of freedom with Hamiltonian $H=H_{IV}+F$. The new Hamiltonian system becomes

[TABLE]

The symplectic form $\Omega$ is canonical in the variables $y,p,t$ and $F$, i.e. $\Omega=dp\wedge dy+dF\wedge dt$.

3. Non-integrability of the Painlevé IV for $a=0,\,b=-\frac{2}{9}$

In this section we prove that when $a=0$ and $b=-\frac{2}{9}$ the Hamiltonian system (2.17) is not-integrable in the Liouville-Arnold sense.

3.1. Hamiltonian system and the first variational equations

When $v_{1}=-v_{3}=\frac{1}{3},\,v_{2}=0$ the Hamiltonian system (2.17) becomes

[TABLE]

The non-equilibrium particular solution along which we write the variational equations is

[TABLE]

Since $\dot{t}=1$ from here on we use $t$ as an independent variable instead of $s$.

Proposition 3.1**.**

The differential Galois group of the first normal variational equations along the solution (3.19) is an Abelian group.

Proof.

The first normal variational equations $(\textrm{NVE})_{1}$ along the solution (3.19) writes

[TABLE]

The $NVE_{1}$ are solvable by quadratures and the matrix

[TABLE]

is a fundamental matrix solution. Thus the differential Galois group of $\textrm{NVE}_{1}$ is isomorphic to the multiplicative group whch is an Abelian group.

This ends the proof. ∎

3.2. The second variational equations

For the second normal variational equations $(\textrm{NVE})_{2}$ we obtain the fifth-order linear system

[TABLE]

where we have put $u=y_{1}\,p_{1},\,v=y_{1}^{2},\,w=p_{1}^{2}$. The $(\textrm{NVE})_{2}$ are reduced to the following fourth-order equation

[TABLE]

The transformation $t^{2}=1/z$ takes the equation (3.20) into the equation

[TABLE]

The transformation $t^{2}=1/z$ changes the differential Galois group $G$ of the equation (3.20) but it preserves the connected component of the unit element of $G$. The equation (3.21) has two singular points over ${\mathbb{C}}{\mathbb{P}}^{1}$ – the points $z=0$ and $z=\infty$. The origin is an irregular singularity of Poincaré rank 1, while $z=0$ is a regular singularity.

Let us firstly determine the local differential Galois group at the origin of equation (3.21). The equation (3.21) is a reducible equation and it can be present as $L\,y=(L_{2}\circ L_{1})\,y=0,\,$ where the operators $L_{2}$ and $L_{1}$ do not commute and

[TABLE]

Every equation $L_{j}\,y=0,j=1,2$ is solvable. The system of functions

[TABLE]

is a fundamental set of solutions of the equation $L_{1}\,y=0$ while the system

[TABLE]

is a fundamental set of solutions of the equation $L_{2}\,y=0$. The application of the Theorem 2.3 to the equation (3.21) gives us the following existence result.

Theorem 3.2**.**

The equation (3.21) possesses a formal fundamental set of solution at the origin in the form $\{y_{1}(z),y_{2}(z),\psi_{1}(z),\psi_{2}(z)\}$ where the functions $y_{j}(z),j=1,2$ are defines by (3.22). The functions $\psi_{j}(z),j=1,2$ are given by

[TABLE]

where

[TABLE]

The elements $\hat{\phi}(z)$ and $\hat{\varphi}(z)$ are the following diveregent power series

[TABLE]

Proof.

We have only to prove that the functions $\psi_{1}(z)$ and $\psi_{2}(z)$ have the pointed form. From Theorem 2.3 it follows that the functions $\psi_{j}(z)$ must solve the equations

[TABLE]

respectively. Looking for a solution $\psi_{1}(z)$ of the first equation in the form

[TABLE]

we find that $h_{1}(z)$ must solve the equation

[TABLE]

Now it is not difficult to show that $h_{1}(z)$ has the pointed form. Similarly, for the solution

[TABLE]

of the second equation we find that the $h_{2}(z)$ must solve the equation

[TABLE]

As before one can show that $h_{2}(z)$ has the poined form.

This ends the proof. ∎

Combining Theorem 2.3, Theorem 2.7 and Theorem 3.2 we obtain the formal fundamental matrix solution at the origin of the equation (3.21).

Theorem 3.3**.**

The equation (3.21) possesses an unique formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ at the origin in the form (2.16), where

[TABLE]

The matrix $\hat{H}(z)$ is given by

[TABLE]

where the functions $h_{j}(z),j=1,2$ are defined by Theorem 3.2.

Proof.

The proof is straightforward. ∎

Once building a formal fundamental matrix solution at the origin we can compute the formal invariants of the equation (3.21).

Proposition 3.4**.**

With respect to the formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ at the origin built by Theorem 3.3 the exponential torus ${\mathcal{T}}$ and formal monodromy $\hat{M}_{0}$ of the equation (3.21) are given by

[TABLE]

where $c\in{\mathbb{C}}^{*}$.

Now we have to compute the analytic invariants of the equation (3.21). From Definition 2.10 it follows that the only admissible singular direction $\theta$ associated to the divergent power series $\hat{\phi}(z)$ is $\theta=\frac{\pi}{2}$. Similarly, the only admissible singular direction associated to the divergent power series $\hat{\varphi}(z)$ is $\theta=\frac{3\pi}{2}$. The next lemma fixes explicitly the dependence of the pointed power series on the singular directions by providing the 1-summs of both divergent power series $\hat{\phi}(z)$ and $\hat{\varphi}(z)$.

Lemma 3.5**.**

For any direction $\theta\neq\frac{\pi}{2}$ the function

[TABLE]

defines the 1-sum of the series $\hat{\phi}(z)$ in such a direction. Similarly, for any direction $\theta\neq\frac{3\pi}{2}$ the function

[TABLE]

defines the 1-sum of the series $\hat{\varphi}(z)$ in such a direction.

The function $\phi_{\theta}(z)$ $($resp. $\varphi_{\theta}(z))$ is a holomorphic function in the open disk

[TABLE]

for every $\theta\neq\frac{\pi}{2}$ $($resp. $\theta\neq\frac{3\pi}{2})$.

Proof.

Since $n+1\leq 2^{n}$ when $n\in{\mathbb{N}}$ then

[TABLE]

Hence both of the power series $\hat{\phi}(z)$ and $\hat{\varphi}(z)$ are Gevrey-1 series with constants $C=4\,\sqrt{3},\,A=\frac{2\,\sqrt{3}}{3}$. The corresponding formal Borel transforms

[TABLE]

converge in the open disk $|\xi|<\sqrt{3}$ and there

[TABLE]

The function $\phi(\xi)$ (resp. $\varphi(\xi)$) is continued analytically along any ray $\theta\neq\frac{\pi}{2}$ (resp. $\theta\neq\frac{3\pi}{2}$) from [math] to $+\infty e^{i\theta}$. Next, since

[TABLE]

and

[TABLE]

the Laplace transform $({\mathcal{L}}_{1}\phi)(z)$ (resp. $({\mathcal{L}}_{1}\varphi(z)$) is well defined along any ray $\theta\neq\frac{\pi}{2}$ (resp. $\theta\neq\frac{3\pi}{2}$) from [math] to $+\infty e^{i\theta}$. Moreover, the estimates (3.31) and (3.34) ensure that the Laplace transforms

[TABLE]

define holomorphic functions in the open disk ${\mathcal{D}}$ from (3.28). The so built Laplace transforms define the 1-sums of the series $\hat{\phi}(z)$ and $\hat{\varphi}(z)$.

This ends the proof. ∎

Remark 3.6**.**

Let $I=\left(-\frac{3\pi}{2},\frac{\pi}{2}\right)\subset{\mathbb{R}}$ and $J=\left(-\frac{\pi}{2},\frac{3\pi}{2}\right)\subset{\mathbb{R}}$. When we move the direction $\theta\in I$ (resp. $\theta\in J$) the holomorphic functions $\phi_{\theta}(z)$ (resp. $\varphi_{\theta}(z)$) glue together analytically and define a holomorphic function $\tilde{\phi}(z)$ (resp. $\tilde{\varphi}(z)$) on a sector $\tilde{{\mathcal{D}}}$ with opening $3\pi,\,-2\pi<\arg(z)<\pi$ (resp. $-\pi<\arg(z)<2\pi$) as it is shown in Figure 1. On these sectors the functions $\tilde{\phi}(z)$ and $\tilde{\varphi}(z)$ are asymptotic to the power series $\hat{\phi}(z)$ and $\hat{\varphi}(z)$, respectively, in Gevrey 1-sense and define the 1-sums of this power series there. The restriction of $\tilde{\phi}(z)$ (resp. $\tilde{\varphi}(z)$) on ${\mathbb{C}}^{*}$ is a multivalued function. In every direction $\theta\neq\frac{\pi}{2}$ (resp. $\theta\neq\frac{3\pi}{2}$) the function $\tilde{\phi}(z)$ (resp. $\tilde{\varphi}(z)$)has only one values which coincides with the function $\phi_{\theta}(z)$ (resp. $\varphi_{\theta}(z)$) from Lemma 3.5. Near the singular direction $\theta=\frac{\pi}{2}$ (resp. $\theta=\frac{3\pi}{2}$) the function $\tilde{\phi}(z)$ (resp. $\tilde{\varphi}(z)$) has two different values : $\phi^{+}_{\frac{\pi}{2}}(z)=\phi_{\frac{\pi}{2}+\epsilon}(z)$ (resp. $\varphi^{+}_{\frac{3\pi}{2}}(z)=\varphi_{\frac{3\pi}{2}+\epsilon}(z)$) and $\phi^{-}_{\frac{\pi}{2}}(z)=\phi_{\frac{\pi}{2}-\epsilon}(z)$ (resp. $\varphi^{-}_{\frac{3\pi}{2}}(z)=\varphi_{\frac{3\pi}{2}-\epsilon}(z))$ for a small number $\epsilon>0$.

Now we already can associate to the formal fundamental matrix solution $\hat{\Phi}_{0}(z)$ from Theorem 3.3 an actual fundamental matrix solution in the sense of Theorem 2.11. Denote $F(z)=z^{\Lambda}\,\exp\left(\frac{Q}{z}\right)$.

Theorem 3.7**.**

For every direction $\theta\neq\{\frac{\pi}{2},\frac{3\pi}{2}\}$ the equation (3.21) possesses an unique actual fundamental matrix solution $\Phi_{0}^{\theta}(z)$ at the origin in the form

[TABLE]

where $F_{\theta}(z)$ is the branch of the matrix $F(z)$ for $\arg(z)=\theta$. The matrix $H_{\theta}(z)$ is given by

[TABLE]

where $H_{j}(z)=\hat{H}_{j}(z),j=1,2$. The matrix $H_{12}^{\theta}(z)$ is defined as

[TABLE]

The functions $h_{j}^{\theta}(z),j=1,2$ are given by

[TABLE]

where the functions $\phi_{\theta}(z)$ and $\varphi_{\theta}(z)$ are defined by Lemma 3.5 and extended by Remark 3.6.

Near the singular direction $\theta=\frac{\pi}{2}$ the equation (3.21) possesses two different fundamental matrix solutions at the origin

[TABLE]

where $\Phi_{0}^{\pi/2+\epsilon}(z)$ and $\Phi_{0}^{\pi/2-\epsilon}(z)$ are defined by (3.35) for a small number $\epsilon>0$.

Similarly, near the singular direction $\theta=\frac{3\pi}{2}$ the equation (3.21) possesses two different fundamental matrix solutions at the origin

[TABLE]

where $\Phi_{0}^{3\pi/2+\epsilon}(z)$ and $\Phi_{0}^{3\pi/2-\epsilon}(z)$ are defined by (3.35) for a small number $\epsilon>0$.

Now we are ready to compute explicitly the Stokes matrices at the origin of the equation (3.21).

Theorem 3.8**.**

With respect to the actual fundamental matrix solution at the origin defined by Theorem 3.7 the equation (3.21) has a Stokes matrix $St_{\pi/2}$ in the form

[TABLE]

where

[TABLE]

Similarly, respect to the actual fundamental matrix solution at the origin defined by Theorem 3.7 the equation (3.21) has a Stokes matrix $St_{3\pi/2}$ in the form

[TABLE]

where

[TABLE]

Proof.

From Definition 2.12 it follows that to find the multiplier $\mu_{1}$ we have to compare the solutions

[TABLE]

and

[TABLE]

Then

[TABLE]

where $\gamma=(\pi/2-\epsilon)-(\pi/2+\epsilon)$. Without changing the integral we can deform the path $\gamma$ into a Henkel type path going along $i\,{\mathbb{R}}^{+}$ from $i\,\infty$ to $i\,\sqrt{3}$, encircling $i\,\sqrt{3}$ in the positive sense and backing to $i\,\infty$. Then

[TABLE]

where we have used the Euler’s reflection formula $\Gamma(1-z)\,\Gamma(z)=\frac{\pi}{\sin\pi\,z}$ for $z\neq{\mathbb{Z}}$. In the same manner we find that

[TABLE]

This ends the proof. ∎

Thanks to Proposition 3.4 and Theorem 3.8 we can describe the local differential Galois group at the origin of the equation (3.21).

Theorem 3.9**.**

The connected component of the unit element of the local differential Galois group at the origin of the equation (3.21) is not Abelian.

Proof.

From Proposition 3.4 it follows that the group generated by ${\mathcal{T}}$ and $\hat{M}_{0}$ is not a connected group. However the connected component of the unit element of this group is equal to the exponential torus ${\mathcal{T}}$. From the Theorem 2.13 of Ramis it follows that the local differential Galois group at the origin of the equation (3.21) is the Zariski closure of the subgroup generated by the exponential torus ${\mathcal{T}}$ and the Stokes matrices $St_{\pi/2}$ and $St_{3\pi/2}$. Denote by $S$ the Zariski closure of the subgroup generated by the Stokes matrices $St_{\pi/2}$ and $St_{3\pi/2}$. Then the element $s_{\lambda,\nu}$ of $S$ has the form

[TABLE]

where $\lambda,\nu\in{\mathbb{C}}$. Denote by $T$ the Zariski closure of the subgroup generated by the exponential torus ${\mathcal{T}}$. Then the element $\tau_{c}$ of $T$ has the form

[TABLE]

where $c\in{\mathbb{C}}^{*}$.

When $\lambda\neq 0,\nu\neq 0$ and $c^{3}\neq 1$ the commutator between $s_{\lambda,\nu}$ and $\tau_{c}$

[TABLE]

is not identically equal to the identity matrix.

The condition $c^{3}=1$ implies that the group generated by ${\mathcal{T}}$ is a finite cyclic group. For such a group the Picard-Vessiot extension $K\left(e^{\frac{i\,\sqrt{3}}{3z}}\right)$ must be an algebraic extension of the field $K={\mathbb{C}}(z)$ which is an obvious contradiction. Thus the connected component of the unit element of the local differential Galos group at the origin of the equation (3.21) is not Abelian.

This ends the proof. ∎

Now we will determine the local differential Galois group at $z=\infty$ of the equation (3.21). The transformation $z=1/x$ takes the equation (3.21) into the equation

[TABLE]

The origin is a regular singularity for the equation (3.40) and therefore the point $z=\infty$ is a regular singularity for the equation (3.21). The characteristic equation at $x=0$ for the equation (3.40) writes

[TABLE]

Its roots are $\rho_{1}=0,\rho_{2}=1,\rho_{3}=\frac{1}{2},\rho_{4}=\frac{7}{2}$ and therefore the solution at the origin can contain logarithmic terms. Fortunately the reducibility of the equation (3.40) allows us to overcome the technical difficulty in the building of a local fundamental set of solutions at such a resonant regular singularity. In fact, as we show below the solution at the origin of the equation (3.40) does not contain logarithmic terms.

The operators $L_{1}$ and $L_{2}$ for the equation (3.40) become

[TABLE]

The origin is an ordinary point for the equation $L_{1}\,y=0$. In fact, the function $y_{1}(x)=\exp\left(\frac{i\,\sqrt{3}\,x}{3}\right)$ and $y_{2}(x)=\exp\left(-\frac{i\,\sqrt{3}\,x}{3}\right)$ form a fundamental set of solutions of the equation $L_{1}\,y=0$. The following lemma gives a local fundamental set of solutions near the origin for the equation $L_{2}\,y=0$.

Lemma 3.10**.**

The equation $L_{2}\,y=0$ possesses a local fundamental set of solutions near the origin in the form

[TABLE]

where $u_{j}(x),j=1,2$ are holomorphic functions near the origin. In particular,

[TABLE]

where the coefficients $a_{2k}$ and $b_{2k}$ satisfy the recurrence relations

[TABLE]

respectively.

Proof.

The characteristic equation at $x=0$ for the equation

[TABLE]

writes

[TABLE]

Its roots are $\rho_{1}=-\frac{3}{2}$ and $\rho_{2}=\frac{3}{2}$. The local theory of regular (Fuchsian) singularity ensures that the equation $L_{2}\,y=0$ admits a local solution $w_{1}(x)$ near the origin in the form $w_{1}(x)=x^{3/2}\,u_{1}(x)$ where $u_{1}(x)$ is a holomorphic function near the origin. The function $u_{1}(x)$ must satisfies the equation

[TABLE]

Obviously the wanted function $u_{1}(x)$ is an even function in $x$. Looking $u_{1}(x)$ as a power series

[TABLE]

we find that the coefficients $a_{2k}$ must satisfy the recurrence relation (3.41).

Since $\rho_{2}-\rho_{1}\in{\mathbb{N}}$ the second solution $w_{2}(x)$ at the origin can contain logarithmic term. It turns out that the evenness of the function $u_{1}(x)$ forbids the existence of the logarithmic term. Indeed, the solution $w_{2}(x)$ depends on the solution $w_{1}(x)$ by the formula (see [11] for details)

[TABLE]

Since the function $u_{1}(x)$ is an even function, so the function $1/u^{2}_{1}(x)$. Therefore the solution $w_{2}(x)$ does not contain logarithmic term. Thus the solution $w_{2}(x)$ has the form $w_{2}(x)=x^{-3/2}\,u_{2}(x)$ where $u_{2}(x)$ must solve the equation

[TABLE]

As above we see that the function $u_{2}(x)$ is an even function

[TABLE]

and the coefficients $b_{2k}$ must satisfy the recurrence relation (3.41).

This ends the proof. ∎

Theorem 3.11**.**

The equation (3.40) possesses a fundamental set of solutions near the origin in the form $\{y_{1}(x),y_{2}(x),\psi_{1}(x),\psi_{2}(x)\}$ where $y_{j}(x),j=1,2$ are holomorphic functions near the origin. The solutions $\psi_{1}(x)$ and $\psi_{2}(x)$ write

[TABLE]

where $v_{j}(x),j=1,2$ are holomorphic functions near the origin. In particular

[TABLE]

The coefficients $c_{2k}$ and $d_{2k}$ satisfy the recurrence relations

[TABLE]

where $a_{2k}$ and $b_{2k}$ are defined by Lemma 3.10.

Proof.

From Theorem 2.3 it follows that the solutions $\psi_{1}(x)$ and $\psi_{2}(x)$ must satisfy the equations

[TABLE]

respectively, where $u_{j}(x)$ are defined by Lemma 3.10. Looking $\psi_{j}(x),j=1,2$ as $\psi_{1}(x)=x^{3/2}\,v_{1}(x)$ and $\psi_{2}(x)=x^{-3/2}\,v_{2}(x)$ we find that $v_{j}(x),j=1,2$ must satisfy the equations

[TABLE]

respectively. As above we see that the functions $v_{1}(x)$ and $v_{2}(x)$ must be even functions. Looking for them as power series in $x^{2}$ we find the recurrence relations (3.42) for their coefficients.

This ends the proof.∎

Denote by $\Phi_{0}(x)$ a fundamental matrix solution at the origin of the equation (3.40). Then from Theorem 2.5 it follows that

Theorem 3.12**.**

The fundamental matrix solution $\Phi_{0}(x)$ of the equation (3.40) is represented in a neighborhood of the origin as

[TABLE]

where the matrix $Y(x)$ is a holomorphic matrix function there. In particular $Y(x)$ is defined as

[TABLE]

where the functions $u_{j}(x)$ and $v_{j}(x),j=1,2$ are defined by Lemma 3.10 and Theorem 3.11, respectively. The matrix $J$ is given by

[TABLE]

Now we can describe the local differential Galos group at the origin of the equation (3.40).

Theorem 3.13**.**

The connected component of the unit element of the local differential Galois group at the origin of the equation (3.40) is trivial.

Proof.

With respect to the fundamental matrix solution $\Phi_{0}(x)$ fixed by Theorem 3.12 the equation (3.40) has a monodrmy matrix $M_{0}$ around the origin in the form

[TABLE]

From Theorem 2.14 of Schlesinger it follows that the local differential Galois group of equation (3.40) is generated topologically by the monodromy matrix $M_{0}$. This group is not connected. However, the connected component of the unit element of this group is trivial.

This ends the proof. ∎

Thanks to Theorem 3.9 and Theorem 3.13 we establish the main result of this section

Theorem 3.14**.**

Assume that $a=0$ and $b=-\frac{2}{9}$. Then the fourth Painlevé equation is not integrable in the Liouville-Arnold sense by meromorphic functions that are rational in $t$.

Proof.

From Theorem 3.13 it follows that the local differential Galois group at $z=\infty$ of the equation (3.21) can be considered as a subgroup of the local differential Galois group at the origin of the same equation. Then form Theorem 3.9 and Theorem 2.15 of Mitschi it follows that the connected component of the unit element of the global differential Galois group of the equation (3.21) is not Abelian. Thus the Hamiltonian system (3.1) is not integrable.

This ends the proof.∎

4. Proof of Theorem 1.2

With the exception of the first Painlevé equation all of the Painlevé equations admits groups of symmetries called auto-Bäcklund transformations. The auto-Bäcklund transformations relate one solution of a given Painlevé equation to another solution of the same equation, possibly with different values of the parameters (see [3, 8]). Using the auto-Bäcklund transformations one can obtain various property of the Painlevè equations, including hierarchies of exact solutions [3], special integrals (see [12]). In this section applying auto-Bäcklund transformations we extend the non-integrable result obtained in the previous section.

From the works of Bassom et. al [3] and Murata [22] it follows that the hierarchy stirred from the rational solution (II) by auto-Bäcklund transformations has the form

[TABLE]

where $P_{n-1}(t)$ and $Q_{n}(t)$ are polynomials of degree $n-1$ and $n$, respectively, for the parameters

[TABLE]

Proof of the Theorem 1.2. The proof follows from the fact that the auto-Bäcklund transformations are birational canonical transformations [25, 35, 36]. ∎

Acknowledgments.  The author was partially supported by Grant KP-06-N 62/5 of the Bulgarian Found ”Scientific research”.