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On the Rational Solutions of the Second Painlevé Equation
A. P. VOROB'EV
The question of rational solutions of the second Painlevé equation
\[ w'' = 2w^3 + zw + a \tag{1} \]
(\(a\) is a parameter) has its own history. In the monograph [1], in reviewing the results on the study of the behavior of solutions of equation (1), it was asserted that all solutions of this equation are transcendental functions. Later, in the paper [2], rational solutions of the equation under consideration were indicated, \(w=0\) and \(w=\mp \dfrac{1}{z}\), corresponding to the values of the parameter \(a=0\) and \(a=\pm 1\). In the same paper it was asserted that equation (1) has no other rational solutions. However, in the proof of this assertion the author made an error, which was discovered in [3]. Let us formulate the results of [3] that we shall need below:
a) In order that equation (1) have a rational solution \(w_a\) for a given \(a\), it is necessary and sufficient that the system
\[ P''Q - 2P'Q' + PQ'' = 0, \tag{\(S_a\)} \]
\[ P'''Q - 3P''Q' + 3P'Q'' - PQ''' - z(P'Q - PQ') - aPQ = 0 \]
for the same \(a\) have a polynomial solution \(P_a, Q_a\) \((P_aQ_a \ne 0)\), and then the formula
\[ w_a = \frac{P'_a}{P_a} - \frac{Q'_a}{Q_a} \tag{2} \]
holds.
b) Equation (1) can have a rational solution \(w_a\) only for integral \(a\), and if such a solution exists, then it is unique.
c) For \(a=\pm2, \pm3, \pm4, \pm5\), the rational solutions of equation (1) are written out explicitly.
It should be noted that in [3] the search for rational solutions was carried out by direct substitution of polynomials into the system \((S_a)\). This method involves considerable difficulties, since one has to solve a system of \(a^2\) equations, whose solutions must additionally satisfy another \(a^2-2\) equalities.
Therefore the question whether equation (1) has rational solutions for \(|a| \ge 6\) remained open.
Here we give formulas for the polynomial solutions of the system \((S_a)\) for all integral values of the parameter \(a\), and thereby completely solve the question of rational solutions of equation (1).
Lemma 1. If the functions \(P_\alpha(z)\) and \(Q_\alpha(z)\) \((P_\alpha Q_\alpha \ne 0)\) are a solution of the system \((S_\alpha)\), then the functions \(P_{\alpha+1}(z)\) and \(Q_{\alpha+1}(z)\), defined by the formulas
\[ P_{\alpha+1}=Q_\alpha,\quad Q_{\alpha+1}=(zQ_\alpha^2+4Q_\alpha^{\prime\,2}-4Q_\alpha Q_\alpha'')/P_\alpha, \tag{3} \]
are a solution of the system \((S_{\alpha+1})\).
Proof. From the system \((S_\alpha)\) one can obtain the relation
\[ (zQQ' + 2Q'Q'' - 2QQ''')P = \alpha PQ^2 + (zQ^2 + 4Q^{\prime\,2} - 4QQ'')P'. \tag{4} \]
Since the functions \(P_\alpha\) and \(Q_\alpha\), by assumption, satisfy the system \((S_\alpha)\) and, consequently, relation (4), for the derivatives of \(Q_{\alpha+1}\) we obtain the following expressions:
\[ Q_{\alpha+1}'=[(2\alpha+1)Q_\alpha^2+Q_{\alpha+1}P_\alpha']/P_\alpha, \]
\[ Q_{\alpha+1}''=[2(2\alpha+1)Q_\alpha Q_\alpha' + Q_{\alpha+1}P_\alpha'']/P_\alpha, \tag{5} \]
\[ Q_{\alpha+1}'''=[2(2\alpha+1)Q_\alpha^{\prime\,2} +(2\alpha+1)Q_\alpha Q_\alpha''+Q_{\alpha+1}P_\alpha''']/P_\alpha . \]
The validity of the assertion of the lemma can now be easily verified by substituting into the system \((S_{\alpha+1})\), in place of \(P\), \(Q_\alpha\), and in place of \(Q\) and its derivatives, the expressions (3) and (5).
Lemma 2. If \(P_\alpha(z)\), \(Q_\alpha(z)\) \((P_\alpha Q_\alpha \ne 0)\) is a polynomial solution of the system \((S_\alpha)\), then the function \(Q_{\alpha+1}(z)\), defined by formula (3), is also a polynomial.
Proof. It is known [1] that solutions of equation (1) can have only poles with residues equal to \(\pm 1\). Hence it follows that the polynomial \(P_\alpha(z)\) cannot have multiple roots. Using this fact, we easily obtain the assertion of the lemma from relation (4), which the polynomials \(P_\alpha\) and \(Q_\alpha\) satisfy.
From these lemmas it follows:
Theorem. If equation (1), for some integer \(\alpha\), has a rational solution (2), where \(P_\alpha\) and \(Q_\alpha\) are polynomials, then the equation
\[ w''=2w^3+zw+\alpha+1 \]
has the rational solution
\[ w_{\alpha+1}=\frac{P_{\alpha+1}'}{P_{\alpha+1}}-\frac{Q_{\alpha+1}'}{Q_{\alpha+1}}, \]
where the polynomials \(P_{\alpha+1}\) and \(Q_{\alpha+1}\) are defined by formulas (3).
Corollary. Equation (1) has a rational solution for every integer \(\alpha\).
The proposition formulated gives a convenient algorithm for finding rational solutions of equation (1). Thus, using the polynomials
\[ P_5=z^{10}+60z^7+11200z, \]
\[ Q_5=z^{15}+140z^{12}+2800z^9+78400z^6-3136000z^3-6272000, \]
corresponding to \(\alpha=5\), by formulas (3) we find
\[ P_6=Q_5, \tag{6} \]
\[ Q_6=z^{21}+280z^{18}+18480z^{15}+627200z^{12}-17248000z^9+1448832000z^6+ \]
\[ +19317760000z^3-38635520000. \]
Knowing the polynomials \(P_6\) and \(Q_6\), we find
\[ P_7 = Q_6, \tag{7} \]
\[ \begin{aligned} Q_7={}&z^{28}+504z^{25}+75600z^{22}+5174400z^{19}+62092800z^{16} +13039488000z^{13}\\ &{}-828731904000z^{10}-49723914240000z^7-3093932441600000z. \end{aligned} \]
Thus, the rational solutions of equation (1) for \(\alpha=6\) and \(\alpha=7\), respectively, have the form
\[ w_6=\frac{P_6'}{P_6}-\frac{Q_6'}{Q_6} \quad\text{and}\quad w_7=\frac{P_7'}{P_7}-\frac{Q_7'}{Q_7}, \]
where the polynomials \(P_6\) and \(Q_6\) are defined by formulas (6), and the polynomials \(P_7\) and \(Q_7\) by formulas (7).
References
- Golubev V. V. Lectures on the analytic theory of differential equations. GITTL, Moscow–Leningrad, 1950.
- Schubart H. Zur Wertverteilung der Painleveschen Transcendenten, Arch. Math., 7, 1956.
- Yablonskii A. I. Proceedings of the Academy of Sciences of the BSSR, Series of Physical and Technical Sciences, No. 3, 1959.
Received by the editors
October 30, 1964
Institute of Mathematics
and Computer Technology, Academy of Sciences of the BSSR