Symmetries and hamiltonians of Ince's XXXVIII and XLIX equations

TL;DR

This paper analyzes the symmetries of Hamiltonians associated with Ince's I_{38} and I_{49} equations, revealing differences from Painlevé equations and enhancing understanding of their structural properties.

Contribution

It provides a detailed study of the symmetries of Hamiltonians for Ince's I_{38} and I_{49} equations, connecting them to Weyl symmetries of Painlevé equations and highlighting key differences.

Findings

Identified specific symmetry structures of I_{38} and I_{49} Hamiltonians

Compared symmetries of Ince's equations with Painlevé equations

Provided insights into the structural differences between these classes

Abstract

We discuss symmetries of Hamiltonians of I$_{38}$ and I$_{49}$ equations that appear on Ince's list of fifty second-order differential equations with Painlev\'e property. This study is informed by structure of Weyl symmetries of Painlev\'e P$_{III}$ and mixed Painlev\'e P$_{III-V}$ equations and provides insights into differences between the symmetries of Painlev\'e equations and symmetries of solvable equations on Ince's list.

Equations

$$I_{38}$$

$$I_{49}$$

$$\text{1.}{r}_1=0,J=1\text{2.}{r}_0=0,J=-1$$

$$\begin{array}{rl}z{q}_z& =q\left(q-r_1\right)\left(2p+r_{0}z\right)-\left({\alpha }_1+{\alpha }_3\right)q+{\alpha }_1{r}_1+{ϵ}_0{r}_0{z}^{-J}\\ z{p}_z& =p\left(p+r_{0}z\right)\left(r_1-2q\right)+({\alpha }_1+{\alpha }_3)p-{\alpha }_2{r}_{0}z-{ϵ}_1{r}_1{z}^{J+1}.\end{array}$$

$$\begin{array}{rl}zH& =q\left(q-r_1\right)p\left(p+r_{0}z\right)-\left({\alpha }_1+{\alpha }_3\right)qp+\left({\alpha }_1{r}_1+{ϵ}_0{r}_0{z}^{-J}\right)p\\ & +\left({\alpha }_2{r}_{0}z+{ϵ}_1{r}_1{z}^{J+1}\right)q.\end{array}$$

$$\pi (q)=-p/z,\pi (p)=(q-r_1)z,\pi ({\alpha }_i)={\alpha }_{i+1},\pi (J)=-J,\pi ({ϵ}_i)=(-1{)}^i{ϵ}_{i+1},i=0,1,$$

$$P_{III}:y_{zz}=-\frac{1}{z}{y}_z+\frac{y_z^2}{y}+\mathcal{A}\frac{y^2}{z}+\mathcal{C}{y}^3+\frac{\mathcal{B}}{z}+\frac{\mathcal{D}}{y}$$

$${\pi }_0(q)$$

$${\pi }_0({\alpha }_1+{\alpha }_3)$$

$${\pi }_2(q)$$

$${\pi }_2({\alpha }_1+{\alpha }_3)$$

$$s_2(q)$$

$$s_2({\alpha }_1+{\alpha }_3)$$

$$s_0(q)$$

$$s_0({\alpha }_1+{\alpha }_3)$$

$$s_i^2=1={\pi }_i^2,{\pi }^2{\pi }_i{\pi }^2={\pi }_{i+2},{\pi }^2{s}_i{\pi }^2=s_{i+2},i=0,2,$$

$${\pi }^2(q)=-q,{\pi }^2(p)=-p-r_{0}z,{\pi }^2({\alpha }_i)={\alpha }_{i+2},{\pi }^2({ϵ}_0)=-{ϵ}_0$$

$$s_i{s}_{i+2}=s_{i+2}{s}_i,{\pi }_i{\pi }_{i+2}={\pi }_{i+2}{\pi }_i,{\pi }_i{s}_{i+2}={\pi }_{i+2}{s}_i,,i=0,2.$$

$${\left(s_0{s}_2\right)}^2$$

$${\left({\pi }_0{s}_2\right)}^2$$

$$I_{12}:y_{xx}=\frac{y_x^2}{y}+\mathcal{A}{y}^3+\mathcal{B}{y}^2+\mathcal{C}+\frac{\mathcal{D}}{y}$$

$$x=\ln z,w=qz,f=p/z$$

$$\begin{array}{rl}{w}_x& =w^2\left(2f+r_0\right)-\left({\alpha }_1+{\alpha }_3-1\right)w+{ϵ}_0{r}_0,\\ f_x& =f\left(f+r_0\right)\left(-2w\right)+({\alpha }_1+{\alpha }_3-1)f-{\alpha }_2{r}_0,\end{array}$$

$$H_{12}=f^2{w}^2+w^{2}f{r}_0+({\alpha }_1+{\alpha }_3-1)fw+{ϵ}_0{r}_{0}f+{\alpha }_2{r}_{0}w.$$

$$w_{xx}=\frac{w_x^2}{w}+w^3{r}_0^2+w^2{r}_0({\alpha }_2-{\alpha }_0)-{ϵ}_0{r}_0({\alpha }_0+{\alpha }_2)-\frac{{ϵ}_0^2{r}_0^2}{w},$$

$$f=-\frac{r_{0}y}{y-1}$$

$$\begin{array}{rl}{q}_x& =q\left(q-r_1\right)2p-\left({\alpha }_1+{\alpha }_3\right)q+{\alpha }_1{r}_1\\ p_x& =p^2\left(r_1-2q\right)+({\alpha }_1+{\alpha }_3)p-{ϵ}_1{r}_1\end{array}$$

$${\bar{H}}_{12}=f^2{w}^2-w{f}^2{r}_1-({\alpha }_1+{\alpha }_3)fw+{ϵ}_1{r}_{1}w+{\alpha }_1{r}_{1}f,$$

$$f_{xx}=\frac{f_x^2}{f}+f^3{r}_1^2+f^2{r}_1(-{\alpha }_1+{\alpha }_3)+{ϵ}_1{r}_1({\alpha }_1+{\alpha }_3)-\frac{{ϵ}_1^2{r}_1^2}{f},$$

$$\begin{array}{rl}{w}_{xx}& =\frac{w_x^2}{2}\left(\frac{1}{w}+\frac{1}{w-r_1}\right)-2r_1{ϵ}_1{w}^2+{\alpha }_1^2{r}_1\\ & +\frac{2r_1^2{ϵ}_1{w}^2}{w-r_1}-w{r}_1\frac{{\alpha }_1^2+{\alpha }_3^2+4r_1^2{ϵ}_1}{2(w-r_1)}+\frac{r_1^3{\alpha }_1^2}{2(w-r_1)}.\end{array}$$

$$y=\frac{w}{w-r_1}\text{or}w=\frac{r_{1}y}{y-1},$$

$$H_{38}=f^2{w}^2+\alpha w^{2}f+\kappa w{f}^2+\beta fw+\gamma f+\delta w,$$

$$\begin{array}{rl}{w}_x& =2w^{2}f+\alpha w^2+\beta w+2\kappa wf+\gamma ,\\ f_x& =-2w{f}^2-2\alpha wf-\beta f-\delta -\kappa f^2.\end{array}$$

$$\begin{array}{rl}\mathcal{A}& =\frac{1}{2{\kappa }^2}({\alpha }^2{\kappa }^4+{\beta }^2{\kappa }^2-2\alpha {\kappa }^3\beta +2\alpha {\kappa }^2\gamma +{\gamma }^2-2\beta \kappa \gamma ),\mathcal{B}=-\frac{{\gamma }^2}{2{\kappa }^2}\\ \mathcal{C}& =\frac{\kappa }{2}(-4\delta +2\alpha \beta -3{\alpha }^2\kappa ),\mathcal{D}=-{\kappa }^2{\alpha }^2,\end{array}$$

$$y=\frac{w}{w+\kappa }.$$

Full text

1 Introduction

Symmetry group analysis has been of crucial importance for studies of Painlevé equations and the singular behavior of solutions of second-order differential equations on the complex plane. In addition to the celebrated $6$ Painlevé equations, there are also other $44$ ordinary second-order differential equations with solutions that have no movable critical point other than poles. These equations presented in Ince’s book [4] are solvable, meaning that their solutions are expressible in terms of known functions. Comparing with literature on Painlevé equations, the Hamiltonian structure and symmetries of solvable equations on Ince’s list attracted much less attention with notable exceptions of few recent publications [5, 6, 7]. The present study fills this gap by studying symmetries of equations I38 and I49 :

[TABLE]

from Ince’s list [4, 5, 6, 7]. For the full understanding of their symmetries it is instructive to first study how their structures emerge in the context of PIII-V model [3, 1]. Note that Ince’s equation I38 (1) with $\mathcal{D}=0$ is equivalent to Ince’s equation I49 (2) with $\mathcal{A}=1$.

These two incomplete forms of (1) (with $\mathcal{D}=0$) or (2) (with $\mathcal{A}=1$) equations emerge for two special values of the parameters :

[TABLE]

of the PIII-V model:

[TABLE]

defined here in terms of the two first-order Hamiltonian equations. These equations depend on a number of parameters $J,\epsilon_{0},\epsilon_{1},r_{0},r_{1}$ together with $\alpha_{j},j=0,1,2,3$ (with $\sum_{j=0}^{3}\alpha_{j}=1$) and can be obtained from the Hamiltonian:

[TABLE]

The above equations (4) are invariant under an automorphism $\pi$ such that

[TABLE]

together with $\pi(r_{i})=r_{i+1}$. The automorphism $\pi$ satisfies $\pi^{4}=1$.

The PIII Painlevé equation

[TABLE]

emerges from PIII-V model for either $r_{1}=0$ and $J\neq 1$ or $r_{0}=0$ and $J\neq-1$ and is invariant under the extended affine Weyl group $W[s_{0},s_{2},\pi_{0},\pi_{2},\pi^{2}]$ in the former case and by $W[s_{1},s_{3},\pi_{1},\pi_{3},\pi^{2}]$ in the latter case. The extended affine Weyl group $W[s_{0},s_{2},\pi_{0},\pi_{2},\pi^{2}]$ [3] is generated by transformations

[TABLE]

that satisfy relations :

[TABLE]

for

[TABLE]

as well as commutativity rules:

[TABLE]

Relations (12) and (13) amount to the following Coxeter group relations :

[TABLE]

In the setting of PIII model it is possible to realize $W[s_{0},s_{2},\pi_{0},\pi_{2},\pi^{2}]$ symmetry as an extended affine Weyl group $C^{(1)}_{2}$ [3].

Ince’s equation $I_{12}$ :

[TABLE]

as well as the incomplete $I_{38}$ (with $\mathcal{D}=0$) and the incomplete $I_{49}$ (with $\mathcal{A}=1$) are obtained from PIII-V for either either of two values of parameters given in (3). For these models the symmetry is still given by $W[s_{0},s_{2},\pi_{0},\pi_{2},\pi^{2}]$ (or $W[s_{1},s_{3},\pi_{1},\pi_{3},\pi^{2}]$). However for parameters in (3) actions of $\pi_{i}$ on $\alpha_{j}$ become identical to those of $s_{i}$, and connection with the affine extended Weyl group $C^{(1)}_{2}$ realization can no longer be established [2].

We will illustrate the above comments by providing a brief derivation of relevant Ince’s equations for a first choice $r_{1}=0,J=1$ of the parameters among two listed in (3). Defining

[TABLE]

we obtain from (4) equations:

[TABLE]

that also can be reproduced as Hamilton equations following from a Hamiltonian :

[TABLE]

Eliminating $f$ one obtains the second order equation for $w$ :

[TABLE]

in which we recognize equation I12 of Ince (16). The second order equation for $f$ written in terms of $y$ such that

[TABLE]

leads to Ince’s equation I38 (1) with $\mathcal{A}=\alpha_{0}^{2}/2$, $\mathcal{B}=-\alpha_{2}^{2}/{2}$, $\mathcal{C}=-2\epsilon_{0}r_{0}^{2}$ and $\mathcal{D}=0$ and thus the equation obtained in this limit is only an incomplete version of I38 equation (1).

We now turn our attention to the remaining case listed in (3): $r_{0}=0$ and $J=-1$. Inserting these values directly in (4) with $r_{0}=0$ yields (for $x=\ln z$)

[TABLE]

Let us set $q=w,p=f$ and note that the Hamiltonian that reproduces the above equations is:

[TABLE]

(note that the major difference from $H_{12}$ in (18) is the term $wf^{2}$ instead for $w^{2}f$.)

For the quantity $f=p$ we find from the above equations a second order equation:

[TABLE]

in which we again recognize the XII-th equation of Ince (16). Furthermore we derive:

[TABLE]

Defining $y$ in terms of $w$ as

[TABLE]

we obtain a special incomplete case of Ince’s equation XLIX (2) with the parameters $\mathcal{A}=1,\mathcal{B}=\alpha_{3}^{2}/2,\mathcal{C}=-\alpha_{1}^{2}/2$ and $\mathcal{D}+\mathcal{E}=2r_{1}^{2}\epsilon_{1}$.

2 Completing Hamiltonian (18) to obtain Ince’s eq. 38

To obtain a Hamiltonian that would reproduce a complete equation I38 we symmetrize Hamiltonian structures $H_{12}$ from (18) and ${\bar{H}}_{12}$ from (19) by adding a term $wf^{2}$ to the Hamiltonian $H_{12}$ :

[TABLE]

where we also replaced $r_{0}$ by $\alpha$, $-(\alpha_{1}+\alpha_{3}-1)$ by $\beta$, $\epsilon_{0}r_{0}$ by $\gamma$ and $\alpha_{2}r_{0}$ by $\delta$ to have a more general expression. The corresponding Hamilton equations are

[TABLE]

Taking a second derivative of the first equation in (21) and eliminating $f$ one obtains a second-order differential equation for $w$ which agrees with I38 in (1) with the parameters:

[TABLE]

when using the variable $y$ :

[TABLE]

Similarly obtaining a second-order differential equation for $f$ from (21) and defining $y$ through $f=-\alpha y/(y-1)$ will also yield equation (1). Due to the fact that addition of $\kappa wf^{2}$ term rendered the system (20) explicitly symmetric in $w,f$ the Hamilton equations (21) are invariant under $w,f$ rotation here referred to as an $R$ operation :

[TABLE]

Obviously $R^{2}=1$. Also the Hamilton equations (21) are invariant under

[TABLE]

which also squares to $1$. In addition the model is also invariant under modified transformations (11) and (10):

[TABLE]

and

[TABLE]

connected to each other via $s_{0}=\pi^{2}s_{2}\pi^{2}$ and both being a symmetries of the Hamilton equations (21).

We can also define transformations

[TABLE]

with an explicit action for $\mathcal{S}_{0}$ and $\mathcal{S}_{2}$ being :

[TABLE]

[TABLE]

with both transformations keeping the Hamilton equations (21) invariant. We note that $w$ remains invariant under actions of $\mathcal{S}_{0},\mathcal{S}_{2}$ while $f$ remains invariant under actions of $s_{0},s_{2}$.

In conclusion addition of an additional cubic term to Hamilton structures (18) or (19) associated with I12 yields a Hamilton structure (20) of I38 with a symmetry group no longer involving the automorphisms $\pi_{i},i=0,2$. The manifest symmetry between $f$ and $w$ variables gives raise to a new $w-f$ rotation $R$.

3 Hamiltonian for Ince’s I49 and its symmetries

We will here derive Ince’s equation I49 (2) from the Hamilton function :

[TABLE]

where we allowed for the first time a term of the 5-th power in $f,w$ : $kf^{3}w^{2}$ in addition to terms already present in $H_{38}$ (20). A term of this dimension appears in the Hamiltonian of the Painlevé VI equation. The corresponding Hamilton equations are

[TABLE]

leading to Ince’s equation I49 (2) when coefficients $k$ and $\alpha$ are related through the condition:

[TABLE]

or $\alpha=-(k+1)$. The coefficients $\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{E},\mathcal{D}$ of equation (2) are given in terms of coefficients $\kappa,\alpha,\beta,\gamma,\delta$ from (28) as follows :

[TABLE]

The parameter $\alpha$ needs to have values different from $=-1,-1/2,0,$ to avoid that $\mathcal{A}=0$ or $\mathcal{A}=1$ or $\mathcal{D}$ or $\mathcal{E}$ becoming infinite.

As we will see below the condition (30) required so that the Hamilton equations derived from (28) would reproduce equation I49 in (2) also enables several symmetries of (28) system. In the discussion below we will assume that the condition (30) holds and consider those symmetries that maintain this relation.

One symmetry transformation that keeps equations (29) invariant is

[TABLE]

with

[TABLE]

For equations (29) with condition $k=-\alpha-1$ satisfied we can also define additional symmetry $\pi_{0}$ made possible by presence of term $kf^{3}w^{2}$ in addition to $\alpha fw^{2}$ in $H_{49}$. This symmetry is defined as follows

[TABLE]

in terms of $X$ being a root of a quadratic equation:

[TABLE]

Acting with $\pi_{0}$ directly on $X=(-\kappa\pm\sqrt{\kappa^{2}+4\gamma(\alpha+1)})/2$ yields

[TABLE]

Thus there are two possible values for $\pi_{0}(X)$ :

[TABLE]

where we associated two different transformations $\pi_{0\,+}$ and $\pi_{0\,-}$ to two possible actions of $\pi_{0}$ on $X$. Both $\pi_{0\,+}$ and $\pi_{0\,-}$ act on other quantities in accordance with (33).

Both transformations $\pi_{0\,+}$ and $\pi_{0\,-}$ keep equations (29) invariant and preserve the condition $k=-(\alpha+1)$. Furthermore it holds that :

[TABLE]

with few intermediate explicit formulas being:

[TABLE]

From relations (37) and

[TABLE]

and $s_{2}\pi_{0\,-}(X)=X$ it follows that

[TABLE]

Thus both $\pi_{0\,+}$ and $\pi_{0\,-}$ satisfy the relation $(s_{2}\pi_{0})^{4}=1$ although only $\pi_{0\,+}$ squares to one! In addition $\pi_{0\,+}$ and $\pi_{0\,-}$ transformations satisfy the relation

[TABLE]

that can be rewritten equivalently as

[TABLE]

using relation (36). The last identity can also be written as

[TABLE]

Equations (29) with the condition $k=-(\alpha+1)$ are also invariant under transformations of $\pi^{2}$ defined as

[TABLE]

From

[TABLE]

it follows that

[TABLE]

and since $\pi^{2}(k)=-\pi^{2}(\alpha)-1$ then $\pi^{2}$ defined in (43) is an automorphism that squares to $1$ and leaves (29) with the condition $k=-(\alpha+1)$ invariant !

Define now

[TABLE]

which obviously squares to one. Then

[TABLE]

It follows then that

[TABLE]

For the transformation

[TABLE]

we are able to find based on (33) and (43) the transformations rules :

[TABLE]

In addition it holds

[TABLE]

Furthermore it follows that

[TABLE]

and also

[TABLE]

We also find from relations (40) and (42) that:

[TABLE]

In conclusion adding a $5$-th power term to Hamilton structure (20) of I38 yielded Hamilton structure (28) of I49 and restored $\pi_{0}$ symmetry that was absent in $H_{38}$. Remarkably, the underlying symmetry group contains transformations $s_{0},s_{2},\pi_{0},\pi_{2}$ and $\pi^{2}$ of $W[s_{0},s_{2},\pi_{0},\pi_{2},\pi^{2}]$. However the Coxeter group relations are in some cases (e.g (39) and (47)) of higher order as compared with simple Coxeter group relations (14) and (15) of symmetry transformations of PIII or I12 models.

4 Conclusions

We presented here a study of Hamiltonian structures of I12, I38 and I49 and their symmetries. The mixed PIII-V equations taken for various special values of the underlying parameters provided a useful starting point for this analysis.

The Hamiltonian structure of I12 shared its symmetry generators and underlying Coxeter relations with PIII equation although its symmetry generators acted differently on a set parameters hindering its interpretation as an extended affine Weyl group as described in details in [2]. To obtain I38 and I49 hamiltonian structures extra terms needed to be added to the Hamiltonian for I12. For I38 that resulted in additional symmetry rotation operation and a totally different content of the underlying symmetry group. For I49 model the underlying symmetry structure contains the same generators as those of PIII or I12 but the added higher dimensional term in the Hamiltonian resulted in different higher order Coxeter relations among symmetry generators.

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JFG and AHZ thank CNPq and FAPESP for financial support. VCCA thanks grant 2016/22122-9, São Paulo Research Foundation (FAPESP) for financial support.