Bäcklund Transformation and the Hénon-Heiles System

John Weiss

Center for Studies of Nonlinear Dynamics, La Jolla Institute,
8950 Villa La Jolla Drive, Suite 2150, La Jolla, Ca 92037, USA
and Institute for Pure and Applied Physical Science,
University of California, San Diego, La Jolla, CA 92093, USA

Abstract:

A Bäcklund transformation and linearization for an instance of the Hénon-Heiles system is examined. This provides a special form of solution depending on the three parameters. In addition, a direct formulation in terms of the schwarzian derivative is defined for the Hénon-Heiles system and second Painlevé transcendent. This provides (1) a classification of the Hénon-Heiles system as equations of ``Novikov" type, and; (2) a simple method for deriving the Bäcklund transformations and special solutions of the second Painlevé transcendent. As equations of Novikov type the integrable occurrences of the Hénon-Heiles system can be completely integrated by known methods.

In ref. [1] Bäcklund transformations and the consequent linearizations of the Hénon-Heiles system were found using the methods of refs. [2], [3]. It can be shown that the solution found by this method is ``special" in that it will depend on three (not four) arbitrary parameters. To find the complete solution a direct formulation in terms of the schwarzian derivative is presented. Herein we present this result and in addition, show how a ``direct" formulation can provide a different form of Bäcklund transformation. Bäcklund transforms are derived for the second Painlevé transcendent, and it is found that the three integrable instances of the Hénon-Heiles system can be transformed into a ``canonical" class of ``Novikov" equations [4], [5], [6] considered in ref. [3].

We consider the Hénon-Heiles system

$$\ddot X=-AX-2dXY\ ,\qquad\ddot Y=-BY+cY^2-dX^2\ ,\eqno (1)$$

with hamiltonian

$$H=\textstyle{1\over 2}(\dot X^2+\dot Y^2+AX^2+BY^2)+dX^2Y-\textstyle{1\over 3}cY^3\ . \eqno (2)$$

This system is known to be integrable [7] when:

(i) $d/c=-1\ ,\qquad B=A\ ,$

(ii) $d/c=-{1\over 6}\ ,$

(iii) $d/c=-{1\over 16}\ ,\qquad B=16A\ .$ (3)

In ref. [1] we found the Bäcklund transformations for cases (ii) and (iii). Case (i) is separable [7]. The BT of ref. [1] for case (ii) may be ``improved" to the extent that the BT will depend on an additional arbitrary parameter. Therefore we first present this result. For case (ii),

$$d/c=-\textstyle{1\over 6}\ ,\eqno (4)$$

the solutions $(X,Y)$ of eqs. (1) have (meromorphic) expansions of the form:

$$X=\varphi^{-1}\sum_{j=0}^{\infty} X_j\varphi^j\ ,\qquad Y=\varphi^{-2} \sum_{j=0}^{\infty}Y_j\varphi^j\ .\eqno (5)$$

As explained in ref. [1], to define the BT we let

$$X=X_0\varphi^{-1}+X_1\ ,\qquad Y=Y_0\varphi^{-2}+Y_1\varphi^{-1}+Y_2\ , \eqno (6)$$

where $(\varphi,X_0,X,Y_0,Y_1,Y_2)$ are functions of $t$.

Their results, after evaluation:

$$Y_0 =-\varphi_t^2\ ,\qquad Y_1=\varphi_{tt}\ ,$$

$$Y_2 =\textstyle{1\over 12}(4\lambda-B-3V-3\varphi^2_{tt}/ \varphi^2_t)\ ,$$ (7)

$$X_0^2 =\varphi_t^2V\ ,\qquad X_1=-\textstyle{1\over 2}(V_t/V+\varphi_{tt}/ \varphi_t)V^{1/2}\ ,$$

where $\lambda$ is arbitrary and

$$V=\{\varphi;t\}+\lambda\ ,\eqno (8)$$

$$V_{tt}+\textstyle{3\over 2}V^2+2\bigl(\textstyle{1\over 3}B-2... ...tstyle{1\over 6}B^2-\textstyle{2\over 3}\lambda^2=0\ .\eqno (9)$$

$$\textstyle{1\over 2}V_t^2+\textstyle{1\over 2}V^3+\bigl(\text... ...ver 6}B^2-\textstyle{2\over 3} \lambda^2\bigr)V=0\ .\eqno (10)$$

The BT (6) is well defined and the integration of eqs. (1) is reduced to eq. (10), which defines $V$ as a Weierstrass elliptic function, depending on two parameters (compare ref. [1]). Eq. (8) then determines $\varphi$ and by (6), (7) $(X,X_1)$, $(Y,Y_2)$. If

$$\varphi=U_1/U_2\ ,\eqno (11)$$

and $(U_1,U_2)$ are linearly independent solutions of

$$U_{tt}=-\textstyle{1\over 2}(V+\lambda)U\ ,\eqno (12)$$

then $\varphi$ satisfies (8). Since $V$ depends on two arbitrary parameters $(\lambda,V_0)$ and two parameters can be introduced through the Moebius group (see ref. [1]), a ``general" four parameter solution should be found by the above procedure. However due to the homogeneous dependence of $(X_1,Y_2)$ on $\varphi$ the Moebius transformation introduces only one arbitrary parameter and the solution obtained is ``special" (depends on three parameters). A similar conclusion applies to case (iii) as well.

With this in mind it is interesting to compare the above results with a direct formulation of the Hénon-Heiles system in terms of the schwarzian derivative. From eqs. (1):

$$Y=-(2d)^{-1}(\ddot X\mid X+A)\eqno (13)$$

and letting

$$X=\varphi_t^{-1/2}\ ,\eqno (14)$$

find

$$Y=(4d)^{-1}\bigl(\{\varphi;t\}-2A\bigr)\ .\eqno (15)$$

By eqs. (1), (14) and (15) with

$$S=\{\varphi;t\}\ , \eqno (16)$$

the Hénon-Heiles system is

$$\ddot S=(c/4d)S^2+(B+cA/d)S=A(2B+cA/d)-4d^2\varphi_t\ .\eqno (17)$$

For the integrable cases (3):

(i) $\ddot S+\textstyle{1\over 4}S^2=A^2-4d^2/ \varphi_t\ ,$(18)

(ii) $\ddot S+\textstyle{3\over 2}S^2+(B-6A)S=2A(B-3A) -4d^2/\varphi_t\ ,$(19)

(iii) $\ddot S+4S^2=16A^2-4d^2/\varphi_t\ .$ (20)

Remarkably, eqs. (18),(19) and (20) are precisely those equations, (17), in the class examined in ref. [3]. That is, (19) is a form of higher order $KdV$ equation and (18), (20) are (related to) Caudrey-Dodd-Gibbon/Kuperschmidt equations. Therefore, with

$$V=\varphi_{tt}/\varphi_t\ ,\eqno (21)$$

applying

$$\partial/\partial t+V\ ,\eqno (22)$$

obtains the ``modified" equations of ``Novikov" type:

(i) $V_{tttt}-\textstyle{5\over 2}V_tV_{tt}-\textstyle{5\over 4} V^2V_{tt}-\textstyle{5\over 4}VV_t^2+\textstyle{1\over 16}V^5=A^2V\ ,$(23)

(ii) $V_{tttt}-\textstyle{5\over 2}V^2V_{tt}-\textstyle{5\over 2} VV_t^2+\textstyle{3\over 8}V^5+(B-6A)(V_{tt}-\textstyle{1\over 2}V^3)$

$=2A(B-3A)V\ ,$(24)

(iii) $V_{tttt}+5V_tV_{tt}-5V^2V_{tt}-5VV_t^2+V^5= 16A^2V\ .$(25)

Eqs. (23), (24) and (25) are equivalent to integrable hamiltonian systems (Novikov equations) and may be integrated by the methods of refs. [4], [5], [6]. Note that when

$$V\rightarrow 2V\ ,\qquad A\rightarrow 4A\ ,\eqno (26)$$

then eqs. (i) $\rightarrow$ (ii) and, since (i) $(d/c=-1)$ is ``separable" [7], (iii)  $\bigl(d/c=-{1\over 16}\bigr)$ [dual to (i)] is ``indirectly" separable. Case (ii)  $\bigl(d/c=-{1\over 16}\bigr)$ has recently been shown to be separable as well [8].

The previous results indicate that integrable hamiltonian ordinary differential equations might be ``canonically" classified as occurrences of equations of ``Novikov" type. For these (Novikov) equations the integrals can be found by algorithmic methods [4].

Finally, we consider the direct schwarzian formulation of the second Painlevé transcendent

$$V_{xx}-\textstyle{1\over 2}V^3-\textstyle{1\over 3}xV=\mu\ .\eqno (27)$$

Following ref. [3], let

$$V=\varphi_{xx}/\varphi_x\ ,\eqno (28)$$

and find that

$$\{\varphi;x\}+\textstyle{1\over 3}(\varphi/\varphi_x-x)=\mu\varphi/\varphi_x \ ,\eqno (29)$$

where

$$\{\varphi;x\}=(\partial/\partial x)(\varphi_{xx}/\varphi_x)- \textstyle{1\over 2}(\varphi_{xx}/\varphi_x)^2\ .\eqno (30)$$

Eq. (29) is the formulation of eq. (27) in terms of the schwarzian derivative, (30). Applying the operator

$$\partial/\partial x+\varphi_{xx}/\varphi_x\eqno (31)$$

to eq. (29) obtains (27). Although eq. (29) is not invariant under the full Moebius group (see ref. [3]), it does possess certain partial invariances that define Bäcklund transformations for eqs. (27), (29). That is, if

$$\psi=1/\varphi\ ,\eqno (32)$$

then $\psi$ satisfies eq. (29) with

$$\mu\rightarrow\textstyle{2\over 3}-\mu\ .\eqno (33)$$

On the other hand, letting

$$\psi_x=\varphi_x^{-1}\ ,\eqno (34)$$

we claim that

$$\{\psi;x\}+\textstyle{1\over 3}(\psi/\psi_x-x)=-\mu\psi/\psi_x\ ,\eqno (35)$$

i.e., $\mu\rightarrow -\mu$. Now, substituting (34) into (29), and using (35), obtains

$$\bigl(\textstyle{1\over 3}-\mu\bigr)\varphi/\varphi_x+\bigl(\... ...\textstyle{2\over 3}x+(\varphi_{xx}/\varphi_x)^2\ . \eqno (36)$$

Eqs. (24) and (36) imply that $(\varphi,\psi)$ satisfy eqs. (29), (35), respectively and thus uniquely define a Bäcklund transformation between these equations. If a solution of eq. (29) is known for a particular parameter value $\mu$, then recursive application of (32) and (36) obtains a solution for each parameter in the sequence

$$\mu_0\rightarrow\textstyle{2\over 3}-\mu_0\rightarrow\mu_0-\t... ...u_0\rightarrow\mu_0-\textstyle{4\over 3}\ \cdots\ . \eqno (37)$$

When $\mu_0=0$, a particular solution is

$$\varphi=x\ ,\eqno (38)$$

and when $\mu={1\over 3}$ eq. (29) is integrated by use of the Airy functions. In this way the known exact solutions of the second Painlevé transcendent [9] are found nearly by inspection.