Preamble

DIFFERENTIAL EQUATIONS 1967, VOLUME III, NO. 3

ON A CLASS OF SOLUTIONS OF THE SIXTH PAINLEVÉ EQUATION

N. A. LUKASHEVICH, A. I. YABLONSKII

Consider the sixth Painlevé equation:

$$ \begin{aligned} \frac{d^2 w}{dz^2} = & \frac{1}{2} \left( \frac{1}{w} + \frac{1}{w-1} + \frac{1}{w-z} \right) \left( \frac{dw}{dz} \right)^2 - \left( \frac{1}{z} + \frac{1}{z-1} + \frac{1}{w-z} \right) \frac{dw}{dz} \\ & + \frac{w(w-1)(w-z)}{z^2(z-1)^2} \left[ \alpha + \beta \frac{z}{w^2} + \gamma \frac{z-1}{(w-1)^2} + \delta \frac{z(z-1)}{(w-z)^2} \right] \end{aligned} \tag{1} $$

where $\alpha, \beta, \gamma, \delta$ are constants.

It is well known that the sixth Painlevé equation (1) defines a new transcendental function $w(z, \alpha, \beta, \gamma, \delta)$, which is a single-valued function of $z$ except for the fixed singular points $z=0, 1, \infty$. However, for certain specific values of the parameters $\alpha, \beta, \gamma, \delta$, equation (1) may possess solutions that can be expressed in terms of elementary or classical transcendental functions.

In this paper, we investigate a class of solutions for equation (1) that can be expressed through hypergeometric functions. We demonstrate that under certain constraints on the parameters, the sixth Painlevé equation can be reduced to a simpler form or related to the solutions of linear differential equations.

Specifically, we consider the case where the parameters satisfy certain algebraic relations. By applying appropriate transformations to both the independent variable $z$ and the dependent variable $w$, we identify conditions under which the transcendental nature of the solution simplifies. This approach allows for the construction of explicit solutions for

W W — 1 w — Z w = •

2(2-1) (w — z) a, p, Riccati constants. Following the remarks in work \cite{...}, we seek an equation of the form:

$$w' = a(z)w^2 + b(z)w + c(z), \quad (2)$$

all solutions of which are also solutions to equation (1). We shall demonstrate that, under certain relations between $j$ and $S$, equation (1) possesses a one-parameter family of solutions that constitutes the general solution of a specific Riccati equation. By substituting (2) into (1) and comparing the coefficients of identical powers of $w$, we obtain:

$$z^2(z - 1)^2 a^2(z) = 2a,$$

$$[z^2(z - 1)^2 a'(z) - z(z - 1)(z + 1)a(z)] + z(z - 1)(2z - 1)a(z) + 2a(z)(z + 1) = 0,$$
$$z^2(z - 1)^2 [b'(z) + 2a(z)c(z)] - z(z - 1)(z + 1)b(z) - z(z - 1)a'(z) + 2(z + 1)b(z) + 2a(z)(z + 1) - 1[(2z - 1)b(z) + 2a(z)c(z)] = 0.$$

• 4z + 1) + 2p z + 2 T (z — 1) + 25 z (z — 1) = 0,

(z) — (z — 1) [(2z — 1) (1 — 2z — 6 (z) — 2a (z — 2 p ( z + 1 ) — 2 ( z — 1 ) —25(z —1) = 0 , (z) — 2 (z 2za (z) (z) — (z) ] (z — 1) [(1 — 2z— z 1) = 0, ( z + ( z ) 2 ( z 1) - 0,

$(z-1)^2 c^2(z) + 2\epsilon = 0$.

From the first and last equations of the system, we respectively find $f/2a$.

c(z) =

2 (2 - 1) Painlevé found the general solution of the equation in the case where $f = 6$ \cite{1}.

The equalities (3) transform the second and sixth equations of system (A) into identities. The third and fifth equations of system (A) are Riccati equations with respect to $b(z)$, specifically:
$$ \begin{aligned} 2z - 1 + 2ac - 3a^2 z + 2a' (z + 1) + z(z - 1) 2b' + b^2 + 2z - 1 + 4z + 2b(z - 1) \\ (z - 1)^2 (z - 1) (z + 1) \cdot 2zac + 3c^2 + 2(z + 1) c' - 2(z - 1) b(z) + 4z + z(z - 1) \end{aligned} $$

• = 0. (5)

The fourth equation is linear with respect to $z(z-1)$. By rearranging the terms, we obtain:
$$2za + 2(z-1)(2z-1)c - 2(z-1) \dots$$
We seek the general form of the function $\phi(z)$ that reduces equations (4)–(6) to identities. It should be noted that if $\phi(z)$ is a solution to equations (4) and (5), then $\phi(z)$ also satisfies the following linear differential equation:
$$2b(z) [(z+1)(\sqrt{2a} - f - 2ft) - 3z + 1] / \sqrt{2a} = 2(a+ft)(z+1)$$
Furthermore, it identically vanishes the quadratic trinomial:
$$+ \dots - \frac{(\sqrt{2a} - z + \alpha)(\dots)}{(z-1)(ft-a)(z+1)(z-1)\sqrt{2a}}$$
It is easily established that $\phi(z)$ will satisfy equation (5) provided that $\sqrt{2a} - (a + ft + \dots)$.

$b(z) = \dots$

$\sqrt{2a} - f - 2ft / \sqrt{2ft} - (a + ft - 5) / \sqrt{2a}$

/ 2 ^ — / ^ 2 f t — 1 = £ 0 . (9)

By direct substitution of $(9)$ into $(7)$, we verify that $b(z)$ as defined in $(9)$ is indeed a solution to equation $(7)$. Since $(9)$ was derived by eliminating $b'(z)$ from equations $(6)$ and $(7)$, the expression in $(9)$ also satisfies equation $(6)$. It remains to determine the conditions under which $(9)$ causes $(8)$ to vanish identically.

LUKASHEVICH, YABLONSKII. By substituting and equating the coefficients of like powers to zero, we obtain the following relations: $(\mu_k + f = \Gamma_2 p_i) - f_{2o} - p -$.

$2X p_i + X(f_k + K = \gamma + 1) - \gamma(f_k + 1 = 2\gamma - 1) +$

$f = \gamma - f = -\gamma f_{2a} + 2(- - 5) = 0$, $(f_{2a} + f = 2p + f \mu_{\gamma p} + a -$. It is directly evident that $+ f_k + f = 2p = 0$. By summing equations $(10)$–$(10)$, we obtain:

$(X + \mu + f_{2a} + f - = 2p)^2 = 0,$

That is, equations $(10)$–$(10)$ are dependent by virtue of the given conditions. Furthermore, the difference between the terms yields $(X + \mu + f_{2a} + f)$, which also vanishes. Substituting the values of $X$ and $\mu$ into $(10)$, we obtain after simplification: $a + 3p \approx m - (3a) + 4f - ab(-a - f - 1) + 2(a - b - 6ab - 2a + 2p - 2p5 +)$.

$2 - f_{2fS} + \sigma^* = 0, \quad (12)$

Moreover, the value taken is the one that coincides with $\sqrt{2\alpha - 2\rho}$, as the values of $\sqrt{2\alpha - 2\rho}$ are pre-selected. This condition ensures the existence of a general form for the function $b(z)$ that satisfies equations $(4)$--$(6)$. Consequently, the equation possesses families of solutions that constitute general solutions to the equations for any choice of the radical values $\sqrt{2\alpha}$ and $\sqrt{\dots}$ (taken according to $(9)$), provided that relation $(12)$ is satisfied. We now transition from these equations to linear equations. By setting $2(2W) = \dots$, we obtain:

$- \sigma = 0. \quad (15) \frac{2}{(2-1)} \frac{2}{(2-I)^2}$

The equations belong to the Fuchsian class with singular points at $z=1$ and are characterized by the Riemann P-function:

$v = P \begin{Bmatrix} 0 & 1 & \infty & \\ -f & -2\beta & 0 & z \\ 2 & \dots & \dots & \end{Bmatrix}$

By substituting $v = z^{f+2\alpha} \dots$, equation $(15)$ is reduced to the Gauss hypergeometric equation with arbitrary parameters $\alpha_1, \beta_1, \gamma_1$. Thus, the following holds:

Theorem. If $w(z)$ is a solution to any hypergeometric equation, then by selecting the parameters $a, b, \gamma, \delta$ according to equations (17), (9), and (12), the function $w(z)$ is a solution to the sixth Painlevé equation.

Remark.

1. In the transition from (13) to (15), it was assumed

In the case where $a = 0$, equation (13) yields the linear equation:
$2(2-1)/-2p + 1/-2ft + 1$.

Remark 2. In equation (9), it was assumed that the radicals in (3) were chosen such that they are non-zero. If we assume that $2ft - 1 = 0$, then for the function (2) to exist, it is necessary that the numerators 1 and $p$ also vanish, which leads to the condition:

$m + b = \dots$ (22)

In this case, equations (6) and (7) coincide. By integrating one of them, we again obtain:

$b(r) = \dots$ (23)

However, in this context, the integration constant is an arbitrary value that must be determined from the equation $f = 2ft$. All arguments regarding equations (8) and (10)–(10') remain valid if conditions (21) and (22) are taken into account, since relation (11) still holds in this case. Consequently, if (12) is replaced by (21) and (22) is determined from (10'), we obtain the solution in the form of (23). These final arguments are also applicable to the case where $2ft + 1 = 0$ in equation (20), as condition (21) holds there as well.

References

References

Ince, E. L. Ordinary Differential Equations. GNTIU, Kharkov, 1939.

Erugin, N. P. Doklady Akademii Nauk BSSR, Vol. II, No. 4, pp. 139–142, 1958.

Received by the Editorial Board: December 20, 1966.
Affiliation: V. I. Lenin Belarusian State University, Institute of Mathematics.