Classification of Real Solutions of the Fourth Painlevé Equation

Abstract

Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more precisely, symmetric Painlevé IV) using a sequence of symbols describing their asymptotic behavior and singularities. We give rules for the sequences that are allowed (depending on the parameter values). We look in detail at the existence of globally nonsingular real solutions and determine the dimensions of the spaces of such solutions with different asymptotic behaviors. We show that for a generic choice of the parameters, there exists a unique finite sequence of singularities for which Painlevé IV has a two-parameter family of solutions with this singularity sequence. There also exist solutions with singly infinite and doubly infinite sequences of singularities, and we identify which such sequences are possible. We look at the singularity sequences (and asymptotics) of special solutions of Painlevé IV. We also show two other results concerning special solutions: rational solutions can be obtained as the solution of a set of polynomial equations, and other special solutions can be obtained as the solution of a first-order differential equation.

1. Introduction and Contents of This Paper

The six Painlevé equations were initially discovered in the context of the classification of second-order ordinary differential equations with the property that the only movable singularities of their solutions are poles. In this context, solutions of the Painlevé equations are naturally considered as complex functions of a complex variable. Since their initial discovery, however, many applications of the Painlevé equations have emerged (see [1,2] for a comprehensive list). In many of these applications, the relevant solutions are real functions of a real variable. It is therefore of interest to have a classification of real solutions. In a recent paper [3], we described a dynamical systems approach to the fourth Painlevé equation ($P_{\mathrm{IV}}$). In the current paper, we use this approach to develop a qualitative classification of the real solutions of $P_{\mathrm{IV}}$ for real parameter values.

Recall that $P_{\mathrm{IV}}$ is the equation

$$\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^2}=\frac{1}{2w}{\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)}^2+\frac{3}{2}{w}^3+4z{w}^2+2(z^2-\alpha )w+\frac{\beta }{w},$$

(1)

with two parameters, $\alpha$ and $\beta$. Throughout this paper, we look at the case that $\alpha$ and $\beta$ are real with $\beta <0$ and $\beta \ne -2{(1\pm \alpha )}^2$. A real solution, determined, say, by the values of $w\left(0\right)$ and $w^{\prime }\left(0\right)$, may develop singularities. These singularities are simple poles, and thus it is possible to extend a solution beyond them. See Figure 1 for some typical examples of singular solutions in the case $\alpha =-2.1$, $\beta =-8.82$. The classification of solutions that we introduce in this paper is based on the asymptotics of solutions as $z\to \pm \infty$ and the sequence of singularities and zeros that appear. For a solution with a sequence of singularities and zeros that terminates (in the relevant direction), we identify four possible asymptotic behaviors of solutions as $z\to \pm \infty$. For reasons that will become clear, we label these $B_1,B_2,B_3,C$. There are poles in a neighborhood of which $w\left(z\right)\sim \frac{1}{z-z_0}$, where $z=z_0$ is the pole location, and we label these $A_2$; there are also poles in a neighborhood of which $w\left(z\right)\sim -\frac{1}{z-z_0}$, and we label these $A_3$. In the neighborhood of a zero, we have $w\left(z\right)\sim w^{\prime }\left(z_0\right)(z-z_0)$, where $z_0$ is the location of the zero. For the right-hand side of (1) to be defined at $z_0$, we need $w^{\prime }{\left(z_0\right)}^2=-2\beta$. If $w^{\prime }\left(z_0\right)=\sqrt{-2\beta }$, we label the zero $A_1$, and if $w^{\prime }\left(z_0\right)=-\sqrt{-2\beta }$, we label it Z (the reason for this distinction will become clear later). Thus, the four solutions shown in Figure 1 are assigned the sequences

$$\begin{array}{c}\overline{Z{A}_2{A}_3{A}_1}{A}_3{A}_1{A}_3{A}_1{A}_3\overline{A_{2}Z{A}_1{A}_3}\\ CZ{A}_{2}Z{A}_{2}Z\overline{A_{2}Z{A}_1{A}_3}\\ CZ{A}_{2}Z{A}_{2}ZC\\ \overline{A_3{A}_{1}Z{A}_2}Z{A}_{2}ZC\end{array}$$

where an overline indicates a subsequence that is repeated (apparently) infinitely. In fact, it is found that for this set of parameters, a Z always appears between a C and a $A_2$, between an $A_1$ and an $A_2$ and between a pair of $A_2$s. It is thus not necessary to write the Zs (and the same is true of general parameter values, although the rules for placing Zs vary).

Thus, for the classification of real solutions of $P_{\mathrm{IV}}$, we propose using a sequence that may be

• either a doubly infinite sequence of the three symbols $A_1,A_2,A_3$,
• a singly infinite sequence terminating at one end with one of the four symbols $B_1,B_2,B_3,C$
• or a finite sequence terminating at both ends with one of these four symbols.

Among the results that we present are the following: solutions without any finite singularities are of particular interest, and we use our approach to give a complete classification of such solutions with asymptotics of type $B_1,B_2,B_3$ or C at $z\to \pm \infty$. For solutions with singularities, we identify restrictions on the possible symbol sequences that are allowed. (For example, for the set of parameter values in the example above, once the Zs are removed from the sequences, a C cannot be adjacent to an $A_1$, and there cannot be two adjacent $A_1$s or $A_3$s). It is found that for any specific, generic, value of the parameters $\alpha$ and $\beta$, there is a unique finite sequence of As describing the singularities and zeros for which a two-parameter family of solutions (i.e., a two-parameter choice of $w\left(0\right)$ and $w^{\prime }\left(0\right)$) exists. (In the example above, this is $A_2{A}_2$, giving solutions with two poles as in the $w^{\prime }\left(0\right)=-6.5$ case in Figure 1).

Our approach has a number of limitations. First, as mentioned, we take $\beta <0$ and also assume $\beta \ne -2{(1\pm \alpha )}^2$. The case $\beta >0$ should, in principle, be easier, as then solutions of $P_{\mathrm{IV}}$ do not have analytic zeros. The cases $\beta =0, -2{(1\pm \alpha )}^2$ are bifurcation values, which require more careful handling. Neither of these tasks are undertaken here. Second, our previous work [3] strongly suggests that the only possible asymptotic behaviors for real solutions of $P_{\mathrm{IV}}$ are those that we call $B_1,B_2,B_3,C$. However, this is not yet proven, and it is possible that there are others, so we cannot at this stage guarantee that the classification that we propose is complete. It is, however, consistent with all numerical evidence to date. Finally, there are also certain types of sequences that we currently have no reason to exclude, but for which we do not currently have numeric evidence, and it remains an open question as to whether solutions of the relevant types exist. Despite these various limitations, the classification that we propose seems useful in understanding the remarkable variety of real solutions of $P_{\mathrm{IV}}$.

The contents of this paper are as follows. It is in fact easier to work with $s{P}_{\mathrm{IV}}$, a symmetric version of $P_{\mathrm{IV}}$. Solutions of $s{P}_{\mathrm{IV}}$, with a restriction on one of the parameters, are in $1-1$ correspondence with solutions of $P_{\mathrm{IV}}$ with $\beta <0$. In Section 2, we we introduce $s{P}_{\mathrm{IV}}$, review the necessary results from [3] and also present the symmetry group of $s{P}_{\mathrm{IV}}$. This was not used in [3], and its use allows us to significantly extend the results of [3]. In Section 3, we study real solutions of $s{P}_{\mathrm{IV}}$ with no singularities on the entire real axis and give a result classifying solutions with no singularities and with asymptotic behavior $B_1,B_2,B_3$ or C as $z\to \pm \infty$. In Section 4, we study solutions of $s{P}_{\mathrm{IV}}$ with singularities, assuming, for finite sequences of singularities, type C (but not type B) asymptotics. Using restrictions on the possible sequences of singularities from [3], and further restrictions found using the symmetry group, we give a full description of the possible sequences of singularities of $s{P}_{\mathrm{IV}}$ (assuming type C asymptotics), which determine the possible sequences of poles and zeros of solutions of $P_{\mathrm{IV}}$. In particular, we show that for generic values of the parameters $\alpha$ and $\beta$ (for which $P_{\mathrm{IV}}$ is irreducible [4,5]), there is only one possible finite sequence that describes the poles and zeros of a two-parameter family of solutions with type C asymptotics. Section 5 discusses special solutions, which remain a focus of interest for all the Painlevé equations [6,7]. Our main intention in this section is to briefly examine the singularities of the special solutions of $s{P}_{\mathrm{IV}}$ and $P_{\mathrm{IV}}$ on the real line, but, in addition, we mention two other results that we believe are new (or, at least, generalizations of existing results). In the case of rational solutions, we show that the solutions can be obtained by solving a system of polynomial equations. For certain nongeneric values of the parameters, we show that there are special solutions obtained from the solution of a single first-order differential equation. This is a generalization of a classical result (see, for example, [8]) that, for the case $\beta =-2{(\alpha \pm 1)}^2$, there are special solutions of $P_{\mathrm{IV}}$ that can be obtained from the solution of a Riccati equation. Section 6 contains a brief summary and some concluding remarks. In Appendix A, we briefly describe the numerical methods used to integrate $s{P}_{\mathrm{IV}}$ through singularities. In Appendix B, we briefly present some numerical results on global B-to-B-type solutions, the significance of which is explained in Section 3.

2. Background

2.1. The Symmetric Version of $P_{\mathrm{IV}}$

For many purposes, rather than working directly with $P_{\mathrm{IV}}$, it is useful to look at “symmetric $P_{\mathrm{IV}}$” ($s{P}_{\mathrm{IV}}$), which is the three-dimensional system

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{f}_1}{\mathrm{d}x}& =f_1(f_2-f_3)+{\alpha }_1,\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{f}_2}{\mathrm{d}x}& =f_2(f_3-f_1)+{\alpha }_2,\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{f}_3}{\mathrm{d}x}& =f_3(f_1-f_2)+{\alpha }_3,\hfill \end{array}$$

(2c)

where the 3 parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are subject to the constraint

$${\alpha }_1+{\alpha }_2+{\alpha }_3=1,$$

(3)

and the 3 unknown functions $f_1\left(x\right),f_2\left(x\right),f_3\left(x\right)$ satisfy

$$f_1+f_2+f_3=x.$$

(4)

$s{P}_{\mathrm{IV}}$ was known to Bureau [9] but was rediscovered by Adler [10] and Noumi and Yamada [11,12], amongst others. There is a $1-1$ correspondence between solutions of $P_{\mathrm{IV}}$ with $\beta <0$ and solutions of $s{P}_{\mathrm{IV}}$ with ${\alpha }_1>0$. If $f_1\left(x\right),f_2\left(x\right),f_3\left(x\right)$ is a solution of (2)–(4) and we set $w\left(z\right)=-\sqrt{2}{f}_1\left(\sqrt{2}z\right)$, then $w\left(z\right)$ is a solution of (1) with parameter values $\alpha ={\alpha }_3-{\alpha }_2$ and $\beta =-2{\alpha }_1^2$. Conversely, given a solution $w\left(z\right)$ of (1) with $\beta <0$, we obtain a solution of (2)–(4) by taking

$$f_1\left(x\right)=-\frac{1}{\sqrt{2}}w\left(\frac{x}{\sqrt{2}}\right),f_2\left(x\right)=\frac{1}{2}\left(\frac{f_1^{\prime }\left(x\right)-{\alpha }_1}{f_1\left(x\right)}-f_1\left(x\right)+x\right),f_3\left(x\right)=x-f_1\left(x\right)-f_2\left(x\right),$$

and

$${\alpha }_1=\sqrt{-\frac{\beta }{2}},{\alpha }_2=\frac{1}{2}\left(1-{\alpha }_1-\alpha \right),{\alpha }_3=\frac{1}{2}\left(1-{\alpha }_1+\alpha \right).$$

Note that a singularity of a solution of $P_{\mathrm{IV}}$ will always give rise to a singularity of the corresponding solution of $s{P}_{\mathrm{IV}}$. In addition, a zero of a solution of $P_{\mathrm{IV}}$ with $w^{\prime }\left(z_0\right)=\sqrt{-2\beta }$ will also give rise to a singularity of the corresponding solution of $s{P}_{\mathrm{IV}}$, but a zero of a solution of $P_{\mathrm{IV}}$ with $w^{\prime }\left(z_0\right)=-\sqrt{-2\beta }$ will not. This explains the difference in the labelling of the two types of zero explained in the Introduction.

Although, for the purposes of studying $P_{\mathrm{IV}}$ with $\beta <0$, it is sufficient to study $s{P}_{\mathrm{IV}}$ with ${\alpha }_1>0$, in sequel, we will allow arbitrary non-zero values of ${\alpha }_1,{\alpha }_2,{\alpha }_3$ that sum to 1. The non-vanishing of ${\alpha }_2,{\alpha }_3$ gives rise to the condition $\beta \ne -2{(1\pm \alpha )}^2$. If we introduce $\xi ,\eta$ via

$$\begin{array}{ccc}\hfill {\alpha }_1& =& \frac{1}{3}+\xi ,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\alpha }_2& =& \frac{1}{3}-\frac{1}{2}\xi +\frac{\sqrt{3}}{2}\eta ,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\alpha }_3& =& \frac{1}{3}-\frac{1}{2}\xi -\frac{\sqrt{3}}{2}\eta ,\hfill \end{array}$$

(7)

then a set of parameters for $s{P}_{\mathrm{IV}}$ corresponds to a point on the $(\xi ,\eta )$ plane. The lines on which one of ${\alpha }_1,{\alpha }_2,{\alpha }_3$ is an integer form a triangular lattice on this plane; see Figure 2. By generic parameter values, we mean any value of the parameters for which all the ${\alpha }_i$ are non-integer. For values of the parameters corresponding to the vertices of the lattice or the centers of the cells of the lattice (marked, respectively, by black and white circles in Figure 2), there exist rational solutions; see [13]. For other non-generic values of the parameters, solutions are known in terms of special functions [13]. Section 5 of this paper will be devoted to special solutions; apart from this, the focus of this paper is on generic parameter values.

2.2. The Poincaré Compactification of $s{P}_{\mathrm{IV}}$

In [3], we described the Poincaré compactification of $s{P}_{\mathrm{IV}}$ (which is also related to the compactification of $P_{\mathrm{IV}}$ on a projective space described by Chiba [14]). The Poincaré compactification is a flow on the closed unit ball in ${\mathbb{R}}^3$. Solutions of $s{P}_{\mathrm{IV}}$ between singularities correspond to orbits of the compactification in the interior of the ball; the boundary (which we call “the sphere at infinity”) is an invariant submanifold, on which the flow can be completely solved. All orbits in the interior of the ball have $\alpha$–limit sets on the closed lower hemisphere of the sphere at infinity and $\omega$–limit sets on the closed upper hemisphere. The flow has 14 fixed points on the sphere at infinity, four (which we label $B_1^{-},B_2^{-},B_3^{-},C^{-}$) in the open lower hemisphere, four (which we label $B_1^{+},B_2^{+},B_3^{+},C^{+}$) in the open upper hemisphere and six (which we label $A_1^{-},A_2^{-},A_3^{-},A_1^{+},A_2^{+},A_3^{+}$) on the equator. The points $A_1^{+},A_2^{+},A_3^{+},C^{+}$ are asymptotically stable, in the sense that any orbit in the interior of the ball, sufficiently close to one of these points, will converge to that point as $t\to +\infty$ (t denotes the independent variable of the compactification). Similarly, the points $A_1^{-},A_2^{-},A_3^{-},C^{-}$ are asymptotically unstable (i.e., stable as $t\to -\infty$). The points $B_1^{-},B_2^{-},B_3^{-},B_1^{+},B_2^{+},B_3^{+}$ are of mixed stability: there are two-dimensional stable manifolds in the interior of the ball associated with each of the points $B_1^{+},B_2^{+},B_3^{+}$ (and one-dimensional unstable manifolds on the sphere at infinity). Similarly, there are two dimensional unstable manifolds in the interior of the ball associated with each of the points $B_1^{-},B_2^{-},B_3^{-}$.

In [3], we could not exclude the possibility of there being orbits in the interior of the ball with $\alpha -$ or $\omega -$ limit sets that are closed orbits on the sphere at infinity (and not one of the 14 fixed points). However, we have no numerical evidence for such orbits, and neither is there any suggestion in the extensive literature on $P_{\mathrm{IV}}$ of a solution with appropriate asymptotic behavior. Therefore, we proceed in this paper on the assumption that no such orbit exists. We then have two partitions of the interior of the ball. The first is into the four open sets that are the basins of attraction of each of the points $A_1^{+},A_2^{+},A_3^{+},C^{+}$ as $t\to +\infty$, separated by the three non-intersecting stable manifolds of the points $B_1^{+},B_2^{+},B_3^{+}$. The second is into the four open sets that are the “basins of repulsion” of $A_1^{-},A_2^{-},A_3^{-},C^{-}$ as $t\to -\infty$, separated by the three non-intersecting unstable manifolds of the points $B_1^{-},B_2^{-},B_3^{-}$. Note that $s{P}_{\mathrm{IV}}$ has the obvious symmetry $f\left(x\right)\to -f(-x)$, which relates the two partitions.

In addition to performing a local analysis of the fixed points, in [3], we considered the question of whether there could exist orbits connecting each of the four fixed points $A_1^{-},A_2^{-},A_3^{-},C^{-}$ to each of the four fixed points $A_1^{+},A_2^{+},A_3^{+},C^{+}$, and we gave a set of rules for the permitted transitions, deduced simply by looking at the signs of $f_1,f_2,f_3$ near the fixed points, and using the fact that the signs of the parameters ${\alpha }_1,{\alpha }_2,{\alpha }_3$ determine the changes in the sign of $f_1,f_2,f_3$, respectively, at their zeros between singularities. We reproduce the rules of permitted and forbidden transitions in Figure 3. In addition to showing forbidden transitions (indicated with an X), for permitted transitions, we give a list of numbers, showing which of the functions $f_1,f_2,f_3$ change sign in the course of the transition. An empty entry indicates that a transition is permitted and none of the functions $f_1,f_2,f_3$ change sign.

As explained above, solutions of $s{P}_{\mathrm{IV}}$ between singularities correspond to orbits of the compactification in the interior of the ball. The behavior of the solutions of $s{P}_{\mathrm{IV}}$ corresponding to orbits of the compactification approaching one of its fixed points is given in Equations (15)–(17) in [3] (for type A, C and B points, respectively). Orbits near the $B_1^{-},B_2^{-},B_3^{-},C^{-}$ points describe different possible asymptotics of a solution of $s{P}_{\mathrm{IV}}$ as $x\to -\infty$. Orbits near the $B_1^{+},B_2^{+},B_3^{+},C^{+}$ points describe different possible asymptotics of a solution of $s{P}_{\mathrm{IV}}$ as $x\to +\infty$. Orbits near the $A_1^{+},A_2^{+},A_3^{+}$ points describe the behavior of a solution of $s{P}_{\mathrm{IV}}$ approaching a pole from the left. Orbits near the $A_1^{-},A_2^{-},A_3^{-}$ points describe the behavior of a solution of $s{P}_{\mathrm{IV}}$ “leaving” a pole to the right. A solution of $s{P}_{\mathrm{IV}}$ on the full real line thus corresponds to the sequence of orbits of the compactification. The first must start at one of the points $B_1^{-},B_2^{-},B_3^{-},C^{-}$ and the last must end at one of the points $B_1^{+},B_2^{+},B_3^{+},C^{+}$. If an orbits ends at the point $A_i^{+}$, then the next one must start at the point $A_i^{-}$. Thus, we naturally describe a solution of $s{P}_{\mathrm{IV}}$ (and thus also a solution of $P_{\mathrm{IV}}$) using a sequence as described in the Introduction: either a doubly infinite sequence of the three symbols $A_1,A_2,A_3$, a singly infinite sequence terminating at one end with one of the four symbols $B_1,B_2,B_3,C$ or a finite sequence terminating at both ends with one of these four symbols. As mentioned, we cannot currently exclude the possibility of orbits of the compactification with an $\alpha -$ or $\omega -$ limit set that is a closed orbit on the sphere at infinity. Such orbits would correspond to solutions of $s{P}_{\mathrm{IV}}$ with a different type of asymptotic, involving oscillations with a diverging amplitude. No evidence of such solutions currently exists. If it did, this would necessitate adding to the classification given here, but would not alter our results.

2.3. The Symmetry Group of $s{P}_{\mathrm{IV}}$

$s{P}_{\mathrm{IV}}$ clearly has a ${\mathbb{Z}}_3$ cyclic symmetry generated by the transformation $\sigma$, where

$$\sigma \left({\alpha }_i\right)={\alpha }_{i+1\mathrm{mod}3},\sigma \left(f_i\right)=f_{i+1\mathrm{mod}3}.$$

It is straightforward to directly verify that there is a further symmetry $\tau$ given by

$$\begin{array}{ccc}\tau \left({\alpha }_1\right)=-{\alpha }_1,\hfill & \tau \left({\alpha }_2\right)={\alpha }_2+{\alpha }_1,\hfill & \tau \left({\alpha }_1\right)={\alpha }_3+{\alpha }_1,\hfill \\ \tau \left(f_1\right)=f_1,\hfill & \tau \left(f_2\right)=f_2+\frac{{\alpha }_1}{f_1},\hfill & \tau \left(f_3\right)=f_3-\frac{{\alpha }_1}{f_1}.\hfill \end{array}$$

(Other authors prefer to introduce three further symmetries ${\tau }_1=\tau$, ${\tau }_2=\sigma \tau {\sigma }^2$, ${\tau }_3={\sigma }^2\tau \sigma$). The generators $\sigma$ and $\tau$ satisfy the relations

$${\sigma }^3={\tau }^2={\left(\tau \sigma \tau {\sigma }^2\right)}^3=I.$$

The infinite group generated by $\sigma$ and $\tau$ is known as the extended affine Weyl group of type $A_2^{\left(1\right)}$ [11,12]. The action of the group on the space of parameters generates the entire parameter space from a single triangular cell in Figure 2). Thus, in principle, it suffices to know the solutions of $s{P}_{\mathrm{IV}}$ in the case, say, that all the ${\alpha }_i$ are non-negative. However, since the transformation $\tau$ can add or remove singularities, it is still important to understand the qualitative behavior of solutions for all parameter values.

3. Global Solutions of ${\mathit{sP}}_{\mathrm{IV}}$

From the above discussion concerning the Poincaré compactification of $s{P}_{\mathrm{IV}}$, it is clear that global solutions of $s{P}_{\mathrm{IV}}$, with no singularities on the entire real axis correspond to orbits of the compactification going from any of the points $B_1^{-},B_2^{-},B_3^{-},C^{-}$ to any of the points $B_1^{+},B_2^{+},B_3^{+},C^{+}$. From Figure 3, we see that transitions from $C^{-}$ to $C^{+}$ are only permitted in the case that ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are all positive. By considering the signs of the solutions near the various points (using Equations (16)–(17) in [3]), it is straightforward to determine which transitions are permitted and which are prohibited between B- and C-type points. See Figure 4.

In fact, all the transitions permitted in the table exist and we have the following.

Theorem 1. 

• If ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are all positive, then there exist the following families of solutions with no singularities on the entire real axis:

  -

  a two-parameter family going from $C^{-}$ to $C^{+}$;

  -

  one-parameter families going from $C^{-}$ to each of $B_1^{+},B_2^{+},B_3^{+}$;

  -

  one-parameter families going from each of $B_1^{-},B_2^{-},B_3^{-}$ to $C^{+}$;

  -

  isolated solutions going from $B_i^{-}$ to $B_j^{+}$ for $i\ne j\in \{1,2,3\}$.

• If one of ${\alpha }_1,{\alpha }_2,{\alpha }_3$ is negative, say ${\alpha }_k$, and the other two are positive, then there exist the following families of solutions with no singularities on the entire real axis:

  -

  a one-parameter family going from $C^{-}$ to $B_{k-1\mathrm{mod}3}^{+}$;

  -

  a one-parameter family going from $B_{k-1\mathrm{mod}3}^{-}$ to $C^{+}$;

  -

  isolated solutions going from $B_{k-1\mathrm{mod}3}^{-}$ to each of $B_k^{+},B_{k+1\mathrm{mod}3}^{+}$ and from each of $B_k^{-},B_{k+1\mathrm{mod}3}^{-}$ to $B_{k-1\mathrm{mod}3}^{+}$.

• If one of ${\alpha }_1,{\alpha }_2,{\alpha }_3$ is positive, say ${\alpha }_k$, and the other two are negative, then there exist at least two solutions with no singularities on the entire real axis, going from $B_k^{-}$ to $B_{k+1\mathrm{mod}3}^{+}$ and from $B_{k+1\mathrm{mod}3}^{-}$ to $B_k^{+}$.

Proof. 

We consider the equatorial plane of the Poincaré compactification. The stable manifolds of the points $B_1^{+},B_2^{+},B_3^{+}$ intersect the equatorial plane transversely and can only reach the boundary at one of the $A^{-}$ points. They divide the equatorial plane into regions of points on orbits tending to the four points $A_1^{+},A_2^{+},A_3^{+},C^{+}$. In Figure 5, we show this division (computed numerically) in three cases. In the $+++$ case, from Figure 3, the neighborhood of each of the $A^{-}$ points can be divided into 3 regions and no more than 3 regions, corresponding to the 3 possible “destinations” of orbits starting at any of the $A^{-}$ points. Thus, two of the three stable manifolds must meet each of the $A^{-}$ points and we have the triangular configuration shown. In the $++-$ case, there can be at most two regions in the neighborhood of one of the $A^{-}$ points, at most three at one of the others and possibly all four at the last. The configuration shown is clearly the only option. In the $+--$ case, two points can have at most two regions in their neighborhood, and thus only one stable manifold can reach these points. The other point can have up to four regions in its neighborhood, and thus one of the stable manifolds must form a loop to meet this point twice. Note that it is possible that the loop might be between the other two stable manifolds, or between one of them and the boundary, as in the third image in Figure 5.

To establish the existence of orbits going from one of the $B^{-}$ or $C^{-}$ points to one of the $B^{+}$ or $C^{+}$ points, we consider the division of the equatorial plane by both the stable manifolds of the $B^{+}$ points and the unstable manifolds of the $B^{-}$ points, the latter being obtained by a half turn from the former due to the $f\left(x\right)\to -f(-x)$ symmetry of $s{P}_{\mathrm{IV}}$. See Figure 6, Figure 7 and Figure 8 for the $+++$, $++-$ and $+--$ cases, respectively. In the $+++$ case, it is clear that the rotated copy of each of the stable manifolds of the $B^{+}$ points (i.e., the unstable manifold of the corresponding $B^{-}$ point) must intersect with the stable manifolds of the other two $B^{+}$ points. It follows, by simple topological arguments, that in the $+++$ case, there must be at least one open region in the disk corresponding to solutions going from $C^{-}$ to $C^{+}$; there must be at least one curve segment corresponding to solutions going from $C^{-}$ to any of the $B^{+}$ points and from any of the $B^{-}$ points to $C^{+}$; and there must be at least one point corresponding to an orbit going from any of the points $B_i^{-}$ to the points $B_j^{+}$ with $j\ne i$. In the $++-$ case, we obtain (at least) two curve segments corresponding to solutions going from $C^{-}$ to one of the $B^{+}$ points (as allowed by the rules in Figure 4), and from one of the $B^{-}$ points to $C^{+}$. In addition, there are at least four B to B solutions. In the $+--$ case, we obtain (at least) two B to B solutions. In short, solutions exhibiting every transition allowed by Figure 4 appear, with the expected number of parameters (2 for C to C, 1 for C to B and B to C, and isolated solutions for B to B). This is the content of the theorem. □

The only assumption that has been made on the parameters in reaching the result above is that none of the ${\alpha }_i$ vanish. There is an extensive discussion in the literature [15,16,17,18] of real solutions of ($P_{\mathrm{IV}}$) in the case $\beta =0$, which is precisely the case that we have excluded. However, there are clear similarities between the existing results in the case $\beta =0$ and our results, presumably reflecting the fact that some properties persist in an appropriate limit as one of the ${\alpha }_i$ tends to zero.

Figure 7. The equatorial plane with parameter values ${\alpha }_1=0.5,{\alpha }_2=0.7,{\alpha }_3=-0.2$ ($++-$ case). Coloring as in Figure 6.

Figure 7. The equatorial plane with parameter values ${\alpha }_1=0.5,{\alpha }_2=0.7,{\alpha }_3=-0.2$ ($++-$ case). Coloring as in Figure 6.

Figure 8. The equatorial plane with parameter values ${\alpha }_1=1.1,{\alpha }_2=-0.03,{\alpha }_3=-0.07$ ($+--$ case). Coloring as in Figure 6.

Figure 8. The equatorial plane with parameter values ${\alpha }_1=1.1,{\alpha }_2=-0.03,{\alpha }_3=-0.07$ ($+--$ case). Coloring as in Figure 6.

The B-to-B solutions are of some significance. At any of the B points, one of the components $f_1,f_2,f_3$ diverges, but the others tend to zero. Thus, one component of a $B_i^{-}$ to $B_j^{+}$ solution, with $i\ne j$, gives a solution of $P_{\mathrm{IV}}$ that is not only nonsingular on the entire real axis, but also tends to 0 as $x\to \pm \infty$. Our results give methods to search for these solutions, as they sit on the intersection of one $B^{+}$ stable manifold and one $B^{-}$ unstable manifold, thus corresponding to an initial value at which both the $x\to +\infty$ and the $x\to -\infty$ asymptotics change. We show some relevant numerical results in Appendix B. Figure A1 and Figure A2 illustrate the $+++$ case. Figure A3, Figure A4, Figure A5 and Figure A6 illustrate the two different types of intersection point that occur in the $++-$ case. Figure A7 and Figure A8 illustrate the $+--$ case. Note that, in certain cases, the B-to-B solutions have the further property of having no zeros on the entire real axis.

Finally, in this section, we show some numerical results on the shape of the $C^{-}$-to-$C^{+}$ region that exists in the $+++$ case. The numerical evidence that we have points to there being only a single open region of such solutions in the space of initial data. In Figure 9, we plot the boundary of the relevant region in the space of initial data, for a variety of parameter values. As expected, we observe that as any of the parameters ${\alpha }_i$ becomes smaller, the area of the region contracts.

4. Solutions with Poles and Allowed Pole Sequences

In this section, we consider solutions of $s{P}_{\mathrm{IV}}$ that have singularities. There clearly are four possibilities: a solution could have a finite sequence of singularities, a singly infinite sequence with no singularities for x less than a certain finite value, a singly infinite sequence with no singularities for x more than a certain finite value or a doubly infinite sequence. In the first case, we need to specify the asymptotics of the solution as $x\to \pm \infty$, in the second case as $x\to -\infty$ and in the third case as $x\to +\infty$. We restrict ourselves to the generic case of type C asymptotic behavior. The discussion can be extended to include type B asymptotic behavior as well, but we do not pursue this here.

In the obvious manner, we associate with each solution of this type a symbol sequence specifying the singularities and the asymptotic behavior. In the case of a finite sequence of singularities, the sequence begins and ends with C and has a finite sequence of the symbols $A_1,A_2,A_3$ between the two Cs. In the singly infinite case, the sequence will begin or end with C, followed or preceded by an infinite sequence of As. In the doubly infinite case, the sequence only consists of As. Figure 3 shows that certain symbols cannot follow certain other symbols, depending on the signs of the parameters ${\alpha }_i$. We recall that these forbidden transitions are obtained by considering the signs of $f_1,f_2,f_3$ near the various singularities and in the appropriate asymptotic regimes (see Equations (15) and (16) in [3]), and using the fact that the sign of ${\alpha }_i$ determines the sign of $f_i^{\prime }$ at a regular zero of $f_i$ (that is, at a zero where all components of the solution are non-singular); thus, between any two singularities, there can be at most one regular zero of each of the $f_i$, and the change in sign of $f_i$ at such a zero can only be in a specific direction. In the cases of permitted transitions, Figure 3 also shows which of the functions $f_1,f_2,f_3$ have zeros (although note that the order of these zeros is not determined).

We now prove the following result.

Theorem 2. 

In the case ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$, for generic solutions,

1.

The only permitted finite singularity sequence is $CC$;

2.

The only permitted singly infinite singularity sequences are

$$\begin{array}{c}\hfill C{A}_2{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \\ \hfill C{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \\ \hfill C{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \end{array}$$

(8)

and

$$\begin{array}{c}\dots A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_1{A}_{2}C\hfill \\ \dots A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_{1}C\hfill \\ \dots A_1{A}_2{A}_3{A}_1{A}_2{A}_{3}C\hfill \end{array}$$

(9)

3.

The only permitted doubly infinite singularity sequences are

$$\begin{array}{c}\dots .A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \\ \dots .A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_1{A}_2{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \\ \dots .A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots \end{array}$$

(10)

or doubly infinite repetitions of the subsequences $A_1{A}_2{A}_3$ or $A_3{A}_2{A}_1$.

Proof. 

From Figure 3, in the case that all the ${\alpha }_i$ are positive, a singularity of type $A_i$ cannot be followed by another singularity of type $A_i$.

To obtain further restrictions on the permitted sequence of singularities, we consider the action of the symmetry group. Note that both the symmetries $\sigma$ and $\tau$ described in Section 2.2 preserve asymptotic type C behavior. The action of $\sigma$ maps singularities of type $A_i$ to singularities of type $A_{i+1\mathrm{mod}3}$ and regular zeros of $f_i$ to regular zeros of $f_{i+1\mathrm{mod}3}$. A brief calculation shows that the action of $\tau$ does not affect type $A_2$ and type $A_3$ singularities, but eliminates type $A_1$ singularities, leaving regular zeros of $f_1$ at the points of singularity; at the same time, it creates new type $A_1$ singularities out of regular zeros of $f_1$ (and this is the only way that the action of $\tau$ can create new singularities). Recall also that $\tau ({\alpha }_1,{\alpha }_2,{\alpha }_3)=(-{\alpha }_1,{\alpha }_2+{\alpha }_1,{\alpha }_3+{\alpha }_1)$ so, after the action of $\tau$, the new value of ${\alpha }_1$ is negative, and the new values of ${\alpha }_2,{\alpha }_3$ are still positive.

Suppose that a solution with ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$ has symbol sequence $C{A}_{1}C$. From Figure 3, we see that there are zeros of $f_1$ both between the first C and the $A_1$, and between the $A_1$ and the second C. Thus, applying $\tau$ gives the symbol sequence $C{A}_1{A}_{1}C$. However, now ${\alpha }_1<0$ and ${\alpha }_2,{\alpha }_3>0$, and we see from Figure 3 that, in this situation, an $A_1$ singularity cannot follow another $A_1$ singularity. Thus, we have a contradiction, and the symbol sequence $C{A}_{1}C$ is not permitted when ${\alpha }_1,{\alpha }_2,{\alpha }_3>0$. Similar arguments eliminate any symbol sequence containing any of the subsequences $C{A}_1{A}_2$, $A_2{A}_{1}C$ or $A_2{A}_1{A}_2$. The application of $\tau$ to any of these will lead to two consecutive $A_1$ singularities, which is not allowed.

Similarly, applying ${\tau }_2={\sigma }^2\tau \sigma$ (which maps the parameters to ${\alpha }_1+{\alpha }_2,-{\alpha }_2,{\alpha }_3+{\alpha }_2$, i.e., to the $+-+$ case) eliminates the substrings $C{A}_{2}C$, $C{A}_2{A}_3$, $A_3{A}_{2}C$, $A_3{A}_2{A}_3$, and applying ${\tau }_3=\sigma \tau {\sigma }^2$ eliminates the substrings $C{A}_{3}C$, $C{A}_3{A}_1$, $A_1{A}_{3}C$, $A_1{A}_3{A}_1$.

Let us now consider a permitted sequence starting $C{A}_1$. The sequences $C{A}_{1}C$, $C{A}_1{A}_2$ and $A_1{A}_1$ are not allowed, so the sequence must in fact start with $C{A}_1{A}_3$. The sequences $A_1{A}_3{A}_1$, $A_1{A}_{3}C$ and $A_3{A}_3$ are not allowed, so the sequence must in fact start $C{A}_1{A}_3{A}_2$. Continuing, we reach the conclusion that the only permitted sequence starting $C{A}_1$ is the singly infinite sequence $C{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots$. Similarly, we deduce that the only permitted sequence starting $C{A}_2$ is $C{A}_2{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots$, and the only permitted sequence starting $C{A}_3$ is $C{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots$.

This proves the section of the theorem relating to sequences (finite or singly infinite) starting with C. The section relating to singly infinite sequences ending in C is proven similarly. For doubly infinite sequences, it is straightforward to show that the subsequences $A_3{A}_1{A}_3$, $A_1{A}_2{A}_1$, $A_2{A}_3{A}_2$ have unique doubly infinite extensions, and the only other possible doubly infinite sequences of As that are not excluded are doubly infinite repetitions of one of the subsequences $A_1{A}_2{A}_3$ or $A_3{A}_2{A}_1$. □

Numerical experiments show that, in practice, there exist solutions with no singularities, as we have already documented; there also exist solutions with all the possible singly infinite singularity sequences, and solutions with the first three types of doubly infinite singularity sequence. We have not yet found evidence of the last two possibilities, viz. doubly infinite repetitions of one the subsequences $A_1{A}_2{A}_3$ or $A_3{A}_2{A}_1$. In Figure 10, we show numerical results for one specific choice of the parameters (using the numerical method explained in Appendix A for integration through poles). For different choices of initial condition at $x=0$, we integrate up to $x=10$ and down to $x=-10$ and count the number of poles in $x>0$ and in $x<0$. For the purpose of the experiment, any number of poles exceeding 10 is considered to be infinite. We find “bands” in the plane of initial values with $0,1,2,3,\dots$ poles in $x>0$, and corresponding bands (related by the symmetry $f\left(x\right)\to -f(-x)$) with $0,1,2,3,\dots$ poles in $x<0$. The only intersection of the bands is apparently a single region giving pole-free solutions. Between the bands, we observe regions where there are presumably solutions with doubly infinite singularity sequences. In the regions corresponding to singly infinite singularity sequences, we observe cases with the “last” singularity being of all 3 possible types, as shown in (8) and (9); in the doubly infinite bands, we observe cases with the “central” singularity being of all 3 possible types given in (10). However, we do not observe cases where the singularity sequence is a doubly infinite repetition of the subsequences $A_1{A}_2{A}_3$ or $A_3{A}_2{A}_1$.

Many related plots can be found in the numerical work of Reeger and Fornberg [17,18]. Much of Reeger and Fornberg’s work relates to non-generic parameter values, but Figure 8 in [18] relates to the case ${\alpha }_1={\alpha }_2=\frac{1}{4}$, ${\alpha }_3=\frac{1}{2}$, and shows what appears to be two regions in the space of initial values for which there is a pole-free solution. This arises as $u\left(0\right)$ and $u^{\prime }\left(0\right)$ in Reeger and Fornberg’s work correspond to $f_1\left(0\right)$ and $-\frac{1}{2}+f_1\left(0\right)(f_2\left(0\right)-f_3\left(0\right))$ in our work.

The results for the case that ${\alpha }_1,{\alpha }_2,{\alpha }_3$ are all positive can be generalized using the transformation group to appropriate results for any generic choice of parameters. To see the effect of the transformation $\tau$ on a particular singularity sequence (for a particular set of parameters), we use Figure 3 to locate the zeros of $f_1$ (inserting an appropriate symbol, say $Z_1$); then, we delete the existing $A_1$s and replace the $Z_1$s with new $A_1$s. The effect of the transformation $\sigma$ is simply to cycle $A_1,A_2,A_3$. Clearly, doubly infinite sequences remain doubly infinite, singly infinite sequences remain singly infinite and finite sequences remain finite. Thus, we arrive at the result stated in the Introduction: For any generic choice of the parameters, there exists a unique finite sequence of singularities for which $s{P}_{\mathrm{IV}}$ has a two-parameter family of solutions with this singularity sequence. The (singly and doubly) infinite sequences that are permitted in general also depend on the values of the parameters. For example, by application of $\tau$ to the sequence

$$\dots A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_1{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots$$

that is permitted in the $+++$ case, we obtain the same sequence with the “central” $A_1$ removed, i.e.,

$$\dots A_1{A}_2{A}_3{A}_1{A}_2{A}_3{A}_3{A}_2{A}_1{A}_3{A}_2{A}_1\dots$$

(note that a repeated $A_3$ is allowed in the case $-++$). The two sequences consisting of doubly infinite repetitions of $A_1,A_2,A_3$ or $A_3,A_2,A_1$ seem to be allowed for arbitrary values of the parameters. They are evidently invariant under the action of $\sigma$, and in fact are also invariant under the action of $\tau$, except in the case $--+$. In this case, both are transformed to the sequence that is a doubly infinite repetition of the subsequence $A_2{A}_3$ (which seems to be allowed in the cases $+-+$ and $+--$, but we have not observed its existence).

5. Special Solutions

5.1. Rational Solutions 1

In this section, we consider the rational solutions of $s{P}_{\mathrm{IV}}$ that are equivalent to the so-called “$-\frac{2}{3}z$ hierarchy” of $P_{\mathrm{IV}}$ [19]. These occur when the parameters take values at the centers of faces of the lattice in Figure 2. The fundamental example is the solution $f_1=f_2=f_3=\frac{1}{3}x$ that is obtained when ${\alpha }_1={\alpha }_2={\alpha }_3=\frac{1}{3}$. Indeed, the general solution of this kind is obtained by application of an arbitrary element of the symmetry group on this fundamental solution [12,20]. Thus, for example, by applying $\tau {\sigma }^2\tau$ to the fundamental solution, we obtain the solution

$$f_1=\frac{x^2-3}{3x}, f_2=\frac{x(x^2+3)}{3(x^2-3)}, f_3=\frac{x^4-6x^2-9}{3x(x^2-3)},$$

for parameter values ${\alpha }_1=-\frac{2}{3},{\alpha }_2=\frac{1}{3},{\alpha }_3=\frac{4}{3}$. In the context of this paper, this result is useful as, by looking at the singularities of the rational solution, we can determine the unique finite singularity sequence of solutions for all values of the parameters in the cell whose center gives the rational solution. In the case of the example given, there are singularities at $x=0,\pm \sqrt{3}$ and the sequence is $C{A}_1{A}_2{A}_{1}C$. This sequence can be read off from the group element that generated the solution. The fundamental solution has singularity sequence $CC$; applying $\tau$ gives the sequence $C{A}_{1}C$ (and parameter values $-++$), applying ${\sigma }^2$ gives the sequence $C{A}_{2}C$ (and parameter values $+-+$) and applying $\tau$ again gives the sequence $C{A}_1{A}_2{A}_{1}C$. Note that in the context of studies of the distribution of singularities of rational solutions in the complex plane [21,22], no rules are currently known for the evaluation of the effect of the transformation $\tau$.

We also note the following fact: if the action of the group element g on the fundamental solution gives f for parameter values $\alpha$, then, using the obvious notation, we have

$${\left(g^{-1}f\right)}_1={\left(g^{-1}f\right)}_2={\left(g^{-1}f\right)}_3.$$

This gives two polynomial equations that must be satisfied by the components of the rational solution f, in addition to the constraint $f_1+f_2+f_3=x$. Thus, in the above example, it is possible to check that

$$\begin{array}{ccc}\hfill 9f_1^2{f}_2^2-9f_1^2{f}_2{f}_3+3f_1^2-18f_1{f}_2+6f_1{f}_3+8,& =& 0\hfill \\ \hfill -9f_1^3{f}_2+9f_1^2{f}_2{f}_3+6f_1{f}_2-6f_1{f}_3-4& =& 0.\hfill \end{array}$$

Although this fact is interesting from the point of view of solutions of $s{P}_{\mathrm{IV}}$, it is trivial from the point of view of rational solutions of $P_{\mathrm{IV}}$. $f_3$ can be eliminated from the pair of equations above by using $f_3=x-f_1-f_2$, and then $f_2$ can be eliminated to leave a polynomial relation between $f_1$ and x. Such relations exist for any rational function $f_1$, and, in the various cases that we have checked, the relation is found to be linear in $f_1$; this is not surprising as the group element g fully determines $f_1,f_2,f_3$. (We thank one of the referees of this paper for this note).

5.2. Rational Solutions 2

In this section, we consider the rational solutions of $s{P}_{\mathrm{IV}}$ that are equivalent to the so-called “$-2z$ hierarchy” and “$-\frac{1}{z}$ hierarchy” of $P_{\mathrm{IV}}$ [19]. These occur when the parameters take values at the vertices of the lattice in Figure 2. The simplest example is the solution $f_1=x$, $f_2=f_3=0$ that is obtained when ${\alpha }_1=1$, ${\alpha }_2={\alpha }_3=0$. A complication arises that does not occur for the first set of rational solutions: applying the transformation $\tau$ requires that $f_1$ should be nonzero, and $f_1$ can be zero in the case that ${\alpha }_1=0$. The general rational solution of the second type arises by application of a restricted set of group elements to the fundamental solution, those that avoid generating the parameter value ${\alpha }_1=0$ as an intermediate step. However, there is no shortage of group elements with this property. The resulting solutions also satisfy polynomial identities, in this case obtained from the conditions

$${\left(g^{-1}f\right)}_2={\left(g^{-1}f\right)}_3=0.$$

As an example, the application of the group element $\sigma \tau {\sigma }^2\tau {\sigma }^2\tau \sigma \tau \sigma \tau$ to the fundamental solution gives

$$f_1=\frac{2x(x^2-3)(x^2+1)}{(x^2-1)(x^4+3)}, f_2=\frac{-2x(x^2-1)(x^2+3)}{(x^2+1)(x^4+3)}, f_3=\frac{x(x^4+3)}{(x^2-1)(x^2+1)},$$

for parameter values ${\alpha }_1={\alpha }_2=2,{\alpha }_3=-3$. These functions obey the polynomial identities

$$\begin{array}{ccc}\hfill f_1^2{f}_2{f}_3^2+f_1^2{f}_3-5f_1{f}_2{f}_3-f_1{f}_3^2-3f_1+6f_2+2f_3& =& 0,\hfill \\ \hfill f_1{f}_2^2{f}_3^2+5f_1{f}_2{f}_3-f_2^2{f}_3+f_2{f}_3^2+6f_1-3f_2+2f_3& =& 0.\hfill \end{array}$$

Although we have not discussed solutions with singularities with B-type asymptotics in this paper, we mention that this solution has singularity sequence $B_3{A}_2{A}_2{B}_3$.

It is straightforward to establish that there is a rational solution of $s{P}_{\mathrm{IV}}$ with $f_1=0$ (and thus ${\alpha }_1=0$) for arbitrary integer values of ${\alpha }_2,{\alpha }_3$ with ${\alpha }_2+{\alpha }_3=1$. For positive integer values of ${\alpha }_2$, the solution has

$$f_2\left(x\right)=\frac{\sqrt{2}{\alpha }_2{\mathrm{He}}_{{\alpha }_2}\left(\frac{ix}{\sqrt{2}}\right)}{i{\mathrm{He}}_{{\alpha }_2}^{\prime }\left(\frac{ix}{\sqrt{2}}\right)},f_3=x-f_2,$$

where ${\mathrm{He}}_n$ denotes the nth Hermite polynomial. This has singularity sequence $B_2{B}_2$ for odd values of ${\alpha }_2$ and $B_2{A}_1{B}_2$ for even values of ${\alpha }_2$. For non-positive integer values of ${\alpha }_2$, the solution is

$$f_2\left(x\right)=-\frac{{\mathrm{He}}_{-{\alpha }_2}^{\prime }\left(\frac{x}{\sqrt{2}}\right)}{\sqrt{2}{\mathrm{He}}_{-{\alpha }_2}\left(\frac{x}{\sqrt{2}}\right)}, f_3=x-f_2.$$

This has singularity sequence $B_3{A}_1\dots A_1{B}_3$ with $-{\alpha }_2$ successive singularities of type $A_1$. Note that since these solutions have ${\alpha }_1=0$, the rules of Figure 3 do not apply. However, since successive $A_1$ singularities are allowed in both cases $+-+$ and $--+$, it is not surprising that we also see this on the transition between them. Note also that these solutions have all their singularities on the real axis, with no further poles in the complex plane.

5.3. Other Special Solutions

In greater generality, whenever ${\alpha }_1=0$, $s{P}_{\mathrm{IV}}$ has a one-parameter family of solutions with $f_1=0$. In this case, the $s{P}_{\mathrm{IV}}$ system reduces to the first-order Riccati equation

$$f_2^{\prime }=f_2(x-f_2)+{\alpha }_2,$$

which can be linearized. By application of suitable group elements, these solutions give rise to a one-parameter family of solutions whenever any of the ${\alpha }_i$ takes an integer value. In fact, these solutions (and the corresponding solutions of $P_{\mathrm{IV}}$) can all be obtained as the solution of a first-order differential equation. To see this, note that, by application of the inverse group element to the solution, it must satisfy the single polynomial identity

$${\left(g^{-1}f\right)}_1=0.$$

Substituting

$$f_2=\frac{1}{2}\left(x-f_1+\frac{f_1^{\prime }-{\alpha }_1}{f_1}\right),f_3=\frac{1}{2}\left(x-f_1-\frac{f_1^{\prime }-{\alpha }_1}{f_1}\right)$$

gives a first-order differential equation for $f_1$. Thus, for example, by applying the transformation ${\sigma }^2\tau {\sigma }^2\tau \sigma \tau {\sigma }^2\tau {\sigma }^2\tau \sigma$ to the solutions with $f_1=0$, we obtain the set of special solutions with ${\alpha }_1=2$, and these give rise to the special solutions of $P_{\mathrm{IV}}$—see Equation (1)—with $\beta =-8$ that satisfy the first-order differential equation

$$\begin{array}{ccc}\hfill 0& =& {\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)}^4+8{\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)}^3+\left(-2w^4-8z{w}^3-8(z^2-\alpha )w^2\right){\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)}^2\hfill \\ & & +\left(-8w^4-32z{w}^3-32(z^2-\alpha )w^2-128\right)\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)+w^8+8z{w}^7\hfill \\ & & +8(3z^2-\alpha )w^6+32z(z^2-\alpha )w^5+16\left({(z^2-\alpha )}^2+1\right)w^4+64z{w}^3-256.\hfill \end{array}$$

The fact that, for suitable values of the parameters, there are first-order equations of higher and higher degree that are consistent with $P_{\mathrm{IV}}$ was observed in [23]. This generalizes the classical observation that, for suitable values of the parameters, $P_{\mathrm{IV}}$ is consistent with a Riccati equation [8].

6. Concluding Remarks

In the course of this paper, a framework has emerged for the classification of real solutions of $s{P}_{\mathrm{IV}}$ (and thus also for $P_{\mathrm{IV}}$): a solution is classified by its asymptotic behavior as $x\to \pm \infty$ and its singularity sequence, with the asymptotic behavior being superfluous in one or both limits in the cases of singly infinite or doubly infinite singularity sequences, respectively. We have established a strong result for the existence of solutions with no singularities. For the case of nonzero parameter values, solutions exist exhibiting all the transitions allowed by Figure 4. For the case of solutions with singularities and with generic (C-type) asymptotic behavior (if needed), we have given a list of the possible singularity sequences in the $+++$ parameter case, from which a similar list can be derived for an arbitrary generic (non-integer) set of parameters. In particular, we have seen that for any generic set of parameters, there is a unique allowed finite singularity sequence for solutions with C to C asymptotics. Numerics in the $+++$ case indicate that all the permitted singularity sequences actually occur, with the possible exception of doubly infinite repetitions of the subsequences $A_1{A}_2{A}_3$ and $A_3{A}_2{A}_1$. We are hopeful that it might be possible to exclude these possibilities using techniques not considered in the current paper; it is well known that there are solutions of $P_{\mathrm{IV}}$ with elliptic function asymptotics for large argument [24], and this is a question about the connection formulae for these solutions.

Note that, in the current paper, we have not investigated the possible types solutions with singularities and B-type asymptotics.

Our work has been on the basis of an assumption concerning the dynamical system described in our previous work [3]. We are satisfied with this assumption as there is no evidence to the contrary, and it is an assumption of the simplest possible scenario (that the only possible asymptotics are the B and C behaviors that we have described). However, proving this appears difficult, as it involves the local stability properties of a periodic orbit in the case that the linearized approximation gives insufficient information.

We believe that the approach given here for $P_{\mathrm{IV}}$ should be extendable to other Painlevé equations. Relevant dynamical systems have been given in [10,25]. However, the works of Chiba [14,26] suggest that more subtle compactifications will be involved.