A Systematic Analysis of the Properties of the Generalised Painlev\'e--Ince Equation

TL;DR

This paper analyzes the generalized Painleve-Ince equation, exploring its symmetry properties and singularity structure, revealing conditions for integrability and symmetry, and demonstrating its behavior under the Painleve test.

Contribution

It provides a detailed symmetry and singularity analysis of the generalized Painleve-Ince equation, identifying conditions for maximal symmetry and integrability.

Findings

Maximal symmetry when eta = lpha^2/9

Fails the Painleve test for arbitrary parameters

Becomes integrable under symmetry and singularity analysis

Abstract

We consider the generalized Painlev\'e--Ince equation, \begin{equation*} \ddot{x}+\alpha x\dot{x}+\beta x^{3}=0 \end{equation*} and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $\beta =\alpha ^{2}/9~$the given differential equation is maximally symmetric and well-known that it pass the Painlev\'{e} test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlev\'{e}--Ince equation fails at the Painlev\'{e} test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlev\'{e}--Ince equation is integrable is terms of Lie symmetries and of the Painlev\'{e} test.

Equations

$$\ddot{x}+\alpha x\dot{x}+\beta x^3=0$$

$$\ddot{x}+3x\dot{x}+x^3=0,$$

$$\ddot{x}+\alpha x\dot{x}+\beta x^3=0$$

$${\Gamma }_1$$

$${\Gamma }_2$$

$$\beta ={\alpha }^2/9.$$

$$u=x\text{mbox}\mathrm{and}$$

$$v=\dot{x}.$$

$$v{v}^{\prime 3}=0.$$

$$w=\frac{v}{u^2}\text{mbox}\mathrm{and}$$

$$z=\frac{v^{\prime }}{u}.$$

$$wz+\alpha w+q=0.$$

$$x=a{\tau }^r$$

$$\frac{2a}{{\tau }^3}-\frac{\alpha a^2}{{\tau }^3}+\frac{\beta a^3}{{\tau }^3}=0$$

$$a=0,\frac{\alpha -sqrt{\alpha }^2-8\beta }{2\beta }\text{mbox}\mathrm{and}\frac{\alpha +sqrt{\alpha }^2-8\beta }{2\beta }.$$

$$x=a{\tau }^{-1}+m{\tau }^{-1+s}$$

$$s=-1,\frac{\alpha \sqrt{{\alpha }^2-8\beta }-{\alpha }^2+8\beta }{2\beta }$$

$$s=-1,\frac{-\alpha \sqrt{{\alpha }^2-8\beta }-{\alpha }^2+8\beta }{2\beta }$$

$$x(t)=\frac{1}{y(t)}.$$

$$y\ddot{y}-2{\dot{y}}^2+\alpha \dot{y}-\beta =0$$

Full text

[TABLE]

and we perform a detailed study in terms of symmetry analysis and of the singularity analysis. When the free parameters are related as $\beta=\alpha^{2}/9~{}$the given differential equation is maximally symmetric and well-known that it pass the Painlevé test. For arbitrary parameters we find that there exists only two Lie point symmetries which can be used to reduce the differential equation into an algebraic equation. However, the generalized Painlevé–Ince equation fails at the Painlevé test, except if we apply the singularity analysis for the new second-order differential equation which follows from the change of variable $x=1/y.$ We conclude that the Painlevé–Ince equation is integrable is terms of Lie symmetries and of the Painlevé test.

1 Introduction

The Painlevé–Ince Equation,

[TABLE]

is well-known to have a number of interesting features. Firstly it has eight Lie point symmetries [16] with the algebra $sl(3,R)$ in the Mubarakzyanov Classification Scheme [17, 18, 19, 20] and so is linearisable by means of a point transformation. Secondly it passes the Painlevé Test in terms of the procedure of the ARS algorithm [1, 2, 3] although in a way which was regarded as unacceptable at the time. In 1993 Lemmer et al [15] showed that the singularity was a simple pole and that the resonances for $a=1$ were a perfectly normal 1 and the generic $-1$ whereas for $a=2$ they were $-1$ and an unexpected $-2$. This result was not acceptable to some workers in the field, but a subsequent programme initiated by Mac Feix [14] gave substance to the result and a fuller exposition is to be found in the paper of Andriopoulos et al [5] in which the whole question of positive and negative resonances was answered in terms of regions in the complex plane centred on the singularity. Thirdly the Painlevé–Ince Equation is a member – the second – of the Riccati Hierarchy [10, 11, 12, 13] based on the recursion operator, $D+x$ with $D=\frac{d\,}{dt}$, applied to $\dot{x}+x^{2}$ as initial member 111One could simply start at $x$..

The generalised Painlevé–Ince Equation is defined as

[TABLE]

in which $\alpha$ and $\beta$ are constants, by means of some ingenious manipulations. In this paper we approach the question of integrability of (2) by means of the systemic methods of analysis, namely the search for Lie point symmetries and the determination of the existence of a Laurent series about a singularity.

2 Symmetry Analysis

The Lie point symmetries of (2) are easily calculated using the Mathematica add-on, SYM [7, 8, 9, 4]. For general values of $\alpha$ and $\beta$ there are two symmetries, namely,

[TABLE]

except when the parameters are related according to

[TABLE]

Then there are the eight symmetries of the Painlevé–Ince Equation up to the effect of the rescaling by $\alpha$.

The invariants for the symmetry, $\Gamma_{1}$ are

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In the new variables (2) becomes

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In the new variables $\Gamma_{2}$ becomes $u\partial_{u}+v\partial_{v}$ when the superfluous term in $t$ is ignored. The invariants for the once extended form of this symmetry are

[TABLE]

With the invariants (9) and (10 the first-order equation (8) becomes the algebraic equation,

[TABLE]

3 Singularity Analysis

We now turn to the analysis of (2) in terms of the Painlevé-Test as summarized in the ARS algorithm. The first step is to determine whether a singularity exists and, if so, to calculate its coefficient. To this end we make the substitution

[TABLE]

into (2), where $\tau=t-t_{0}$ and $t_{0}$ is the location of the putative singularity. We find that the terms balance in exponent when $r=-1$, ie the singularity is a simple pole. Moreover all terms are dominant. When we replace $r$ with $-1$, the equation for the leading-order coefficient is

[TABLE]

which has the solutions

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The first solution must be rejected as being irrelevant. The other two can take any value as we are working in the complex time plane.

The second step of the ARS algorithm is the location of the resonances, that is the exponents at which the remaining constants of integration enter the Laurent expansion. We make the substitution

[TABLE]

into (2) (recall that all terms in (2) are dominant) and collect the terms linear in $m$ for these are the terms at which a new constant enters into the expansion. For that constant to be arbitrary the coefficient of $m$ must be zero. The coefficient is a polynomial in $s$ and the values of $s$ which render it zero are the resonances. When we do this, we obtain

[TABLE]

in the former case and

[TABLE]

in the latter case. The value $-1$ is generic.

Acceptable values for the nongeneric resonance must be real and rational with the practical limitation that fractional resonances should not have denominators which are large for means that the complex plane is divided into unworkable sections because of too many branch cuts. Note that there are many possibilities for the second (nongeneric) resonance to be negative. Consequently the ‘impossible’ result for the Painlevé–Ince Equation found by Lemmer et al could be regarded as commonplace. Naturally there are also a multitude of values for which the second (nongeneric) resonance is irrational and/or complex.

4 Conclusion

We have seen that the generalised Painlevé–Ince Equation possesses two Lie point symmetries and is reducible to an algebraic equation for all values of the parameters. From singularity analysis we further see that the singularity is always a simple pole. Various possibilities exist for the nature of the Laurent expansion about this simple pole. It can be either a Right – expansion over a disc centered on the pole – or a Left – expansion about the pole over the complex plane without a disc – Painlevé Series depending upon the values of the parameters. Alternatively it can be a mess, thereby indicating nonintergability.

An interesting feature occurs if one inverts the dependent variable by setting

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Then equation (2) takes the following form

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The two symmetries look the same under the coordinate transformation, except that the plus sign becomes a minus sign in $\Gamma_{2}$. Naturally we are not considering the particular case in which $\alpha^{2}=9\beta.$

In terms of the singularity analysis for equation (19) only the first two terms are dominant, the singularity is a simple pole and the coefficient of the leading-order term is unspecified. (A zeroth-order possibility also exists, but that is not a singularity.) The resonances are at $-1$ and zero in line with the arbitrary coefficient of the leading-order term. The Laurent expansion corresponding to the simple pole is a Right Painlevé Series. The interesting thing is that the coefficients of the higher-order terms in the expansion vanish for the special relationship between $\alpha$ and $\beta$ which give rise to equations of maximal symmetry, that is equations of Painlevé–Ince form. Hence, we can infer that the generalized Painlevé–Ince equation (2) passes the Painlevé test under the coordinate transformation (18) if and only if $\alpha^{2}=9\beta$.

That is not the first example of a differential equation which pass the singularity test under a coordinate transformation. A recent discussion on that property can be found in [21].

Acknowledgements

PGLL thanks the Durban University of Technology, the University of KwaZulu-Natal and the National Research Foundation of South Africa for support. This work was undertaken while we enjoyed the gracious hospitality of Surananee University o Technology, Thailand.