Physica D 52 (1991) 204-210 North-Holland

The H6non-Heiles system revisited

A l l a n P. F o r d y

Department of Applied Mathematical Studies and Centre ]'or Nonlinear Studies, Unit:ers'ity of Leeds', Leeds, LS2 9JT, UK

Received 18 September 1990 Revised manuscript received 31 January 1991 Accepted 5 February 1991 Communicated by H. Flaschka

The known integrable cases of the Hdnon-Hei l e s system are shown to be closely related to the stationary flows of the known (and only) integrable fifth-order (single component and polynomial) nonlinear evolution equations. This is further evidence that these are the only integrable cases of the H6non-Hei les system. Lax pairs are deduced for each of the integrable cases and used to construct the constants of motion. A curious Lax operator recently found by the Painlev6 method is explained.

1. I n t r o d u c t i o n

The general H d n o n - H e i l e s system and its energy are given by

ql + Clql = bq21 - aq 2,

42 + c2q2 = - 2 a q l q 2 ,

~bq 1 • , .2 d l 2 + c l q ~ + c 2 q Z ) + a q l q 2 _ , 3 E = 2 ( q l + 2

( l . l a )

( l . l b )

( l . l c )

With parameter values c I = c 2 = a = b = 1, this system is known to have irregular behaviour at high

energies [1]. On the o ther hand there are three known in tegrable cases of the H d n o n - H e i l e s system,

character ised by certain ratios of the parameters :

( i ) a/b= - 1 , c j = c 2,

(ii) a/b = - 1 / 6 , c l , c 2 arbitrary,

(iii) a / b = - 1/16 , c 2 = 16c I.

( 1 . 2 a )

(1.2b)

(1.2c)

These parameter ratios were isolated by the Painlev6 method [2-4], a l though second integrals were already known for the first two cases. There was some speculation in ref. [2] that these are not the only integrable cases, but direct Hamil tonian analysis did not isolate any fur ther ones [5, 6] and this situation has not changed to date. Fur ther developments in the Painlev~ approach have led to B~icklund

t ransformations for the known integrable cases [7, 8] and a curious spectral problem for case (ii) [9]. In the present paper, the variable q~ is shown to satisfy a four th-order ordinary differential equat ion

for all values of a and b (c 1 and c 2 are set to zero without loss of generality). The three integrable cases correspond precisely to the stationary flows of the only three integrable cases of fifth-order polynomial nonl inear evolution equat ions of scale weight 7 (respectively the Sawada -Ko te ra , fifth-order KdV and

0167-2789/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

A.P. Fordy / The Hdnon-tteiles system ret'isited 205

Kaup-Kupershmidt equations). Similarly, the variable (12/2aq2 satisfies one of the corresponding (stationary) modified equations, which are the only integrable fifth-order polynomial nonlinear evolution equations of scale weight 6.

This result further verifies that the known cases are the only integrable cases of the H6non-Hei les system.

Furthermore, the known spectral problems of these stationary equations are used to derive the two constants of motion for each of the integrable H6non-Hei les systems. An explanation is given of the curious spectral problem derived in ref. [9] for case (ii).

2. Results

To give the appearance of a stationary partial differential equation let

u = q l , u~ =c)1, etc. (2.1)

and, without loss of generality set c I = c 2 = 0. Differentiate (1.1a) twice, use (1.1b) to eliminate/J2, (1.1c) to eliminate 0 2 and (1.1a) to eliminate q22. The result is an autonomous, fourth-order equation for u:

u ........ + (8a - 2 b ) u u x x - 2(a + b ) u ~ - 3-2°abu3- = - 4 a E . (2.2)

Remark. Alternatively, by eliminating second and higher derivatives of u (and q2) from the left-hand side of (2.2) we obtain - 4 a E , with E given by (1.1c).

This equation has the following scale symmetry:

x ~ s - l x , u ~ s 2 u . (2.3a)

Fifth-order (polynomial and autonomous) nonlinear evolution equations with this scale symmetry are of the form

. . . . . . I 3 u , = u ......... + A u u ~ ' + B u x u ~ x + C u 2 u , = ( u x x x x + A u u x x + ½ ( B - A ) u 2 + ~ Cu )x" (2.3b)

There are three known integrable cases of this class of equations:

5 3 (i) u, = (u ........ + 5uux~. + ~u )x, (2.4a)

(ii) u, = (u ........ + lOuu,., + 5u 2 + 1 0 U 3 ) x , (2.4b)

15 2 20u3 ~ (2.4c) (iii) u , = ( u , x x x+ 1 0 u u x x + T u x + T Ix,

known respectively as the Sawada-Kotera equation, Lax's fifth-order KdV flow and the Kaup-Kupershmidt equation (see ref. [10] and references therein). According to refs. [11, 12] these are

206

the only

(i) (ii)

(iii)

Remark.

A.P. Fordy / The HJnon-Heiles system ret,isited

integrable cases. Comparing these with (2.2), we find

1 I a---g, b - 2, ( 2 5 a )

a = ½ , b - - - 3 , (2.5b)

a = ¼ , b - - - - 4 . (2.5c)

The integrability of (2.3b) is invariant under the scale change u ~ r u , which means that a ~ ~a, b ~ ~rb, so that only the ratios a/b are important.

Thus eq. (2.2) corresponds to the stationary flow of an integrable fifth order nonlinear evolution equation of the form (2.3b) only for the ratios (1.2), corresponding to the three integrable cases of the H6non-Hei les system (1.1).

2.1. The spectral problems

Eqs. (2.4) have Lax form:

L,= [P,L], (2.6)

where L and P are respectively (see ref. [10]):

(i) L = a 3 + u a , P=905+15ua3+15uxaZ+(5u2+lOUxx)a, (2.7a)

(ii) L = a 2 + u , P=16aS+4Oua3+6OuxaZ+(5Ouxx+3OuZ)a+15Uxxx+3Ouux, (2.7b)

(iii) L=a3+2ua+ux, P=905+30ua3+45uxa2+(20u2+35uxx)a+lOu,~x+20uu,. (2.7c)

The stationary equations have commutator form:

[P,L] = 0 . (2.8)

The corresponding H6non-Hei les system has form (2.8), but with P and L written in terms ql, q2 and their first derivatives. The important operator is P, which now plays the role of the spectral operator:

(i) P=905+15qta3+15glla2-5qZ2a, (2.9a)

(ii) P = 16 a 5 + 40q~ a 3 + 60ql a 2 - (120q~ 2 + 25q2:) a - 60q~gl~ - 15qz(t 2, (2.9b)

35 2~ (iii) P = 9 3 5 + 3 0 q , 3 3 + 4 5 c ) , a z - ( l z 0 q 2 + T q z ) a - ( 6 0 q l t ~ l + 5 q z q 2 ). (2.9c)

From these we can derive the constants of motion. The details will only be presented for case (i).

Case (i). We have

~0 °) + ql~ (1) = A~b, (2.10a)

9~0 (5) + 15q1~ (3) + 15ql~b (2) -- 5qzZ~b (1) = /x~ (2.10b)

A.P. Fordy / The Hdnon-Heiles system reHsited 2(}7

where q,~k) means the k th derivative with respect to the HEnon-Heiles time. Here, t* is the separation constant resulting from considering the stationary flow of (2.4a). From (2.10a), only 4', ~ ) and g,{2) are independent and, if we use (2.10a) and the equations of motion (1.1a), (1.1b) with (2.5a), then (2.10b) can be written as a second-order equation:

J 2 q 2 ] ~ ( l ) + 6 a q l ~ ~ . . (9A - 301)6 {2)- v(3ql + 2] = (2.11a)

Differentiating this equation twice and (each time) eliminating higher derivatives of ~ and qi, we obtain an eigen-equation for the vector q* = (g,, tO (~), ~(2))v:

Aqr = tzqt

with

A =

6Aql I 2 - 2 ( 3 q , +q2 2)

9A2 + 3Aql -3Aql --q2q2 1 2 q2 --2A(3ql +q22) 9A2- 2

so that

det A = 729A 5 - 162EA 3 + K2A,

where

I(q12q_02) .ff I 2 I 3 E = ~ 2qlq2 + 6q l ,

9A - 301 q2

2

q2q2 -- 3aql

1 2 K = 3~1q2 + ~-q2(3q, + q22).

(2.11b)

(2.11c)

(2.11d)

(2.11e)

E will be recognised as (1.1c) with (2.5a). K is the second integral.

Remark. This system separates in coordinates q~ _+ q2, these variables having individual energies E . . In terms of these, K = ½ ( E + - E ).

Case (ii)

In this case, since the spectral problem of the KdV hierarchy is only second order, only g, and g,~J~ are independent. The equation corresponding to (2.11a) is

(16A 2 + 8Aq, - q2~tbd)2] + (q202 - 4A0,)~ = Iz~/'. (2.12a)

The relevant matrix A is now only 2 × 2, but its determinant is still of fifth degree:

det A = -256A 5 - 32EA 2 + 8KA, (2.12b)

where

I / q 2 F02 I 2 E = 2 ~ 1 2 ) + q 3 + : q , q z , K = q 2 0 1 0 2 - q 1 0 2 + 1 8 q 2 4 q_ 2qlq2"l 2 2 (2 .12c)

208 A.P. Ford)'/The Hdnon-Heiles system revisited

Once again, E is just (1.1c) with (2.5b) and K is the second integral, first written down by Greene (see

refs. [2, 3])

I Remark. This K is just the flux (written in terms of q,, 0i) corresponding to the conserved density su + of

the KdV hierarchy.

Case (iii)

In this case 0 satisfies

±,,2,t,(I) (12Aql ~ . 9A0 ¢2} + 4~2"~ + - = 7q2 q2 )11/ /.t o . (2.13a)

The matrix A is now 3 × 3 and the de terminant is of fifth degree in A:

det A = 243A 5 162EA 3 + 7KA, (2.13b)

where

I " 2 " " 4 3 I 2 I 2 4 1 +, (2.13c) E = ?(q, + q ~ ) + sqi + xq lq 2, K = 3gt4 + 3qlq2dl 2 - ~" " . q2qlq2 ~qlq2 74q2,

giving the two integrals of motion [5].

Remark. K is presumably just the flux (written in terms of qi, o,) of the conserved density: u 4 - 9 2 ~, " ~uu, + %tGx of eq. (2.4c), but I have not checked this.

2.2. The spectral problem q[ Newell et al.

Using the Painlevd method the authors of ref. [9] derived a curious integro-differential Lax opera tor

for case (ii). Here 1 give the relationship of their Lax opera tor to the P of (2.7b).

Consider the fractional powers of the opera tor L of (2.7b):

= ~ _ , + I . u ~ ) ( 2 . 1 4 ) L3/e ~.13+ ~ll()q- 4 l l , q - b l ( t I 2bl ~) - + . . . . b I = ~,(U.,., + 3 ~ ,

so that (multiplying by L ) :

L5 (3u ...... + 2 u _ ) ~ ) + 3 3 (2.15a) ( L 5 / 2 ) + = ~ ¢ + 5~u33+Tu,+02 + ~ + 7( u .... + u u , + ) + b t 3 + eb l , .

where ( )+ means the differential part of the operator . Using the Hdnon-He i l e s equation of motion we

can write this in terms of qi, gli:

5 33 - . ' " + ~(5q ,g f l+qeg l2) 11+, (q~ 3 + 3 q 2 0 2 ) , ( 2 . 1 5 b ) ( L 5 / 2 ) + = 3 5 + ~q, + ' 4 q , ~ t 2 - 5 ( 5 q r + q 2 ) 3 - : 4

and this last term can be written as

(~2 + q, )q2 ~+ ' q e = q ~ + 3 q 2 0 2 . (2.16)

A.P. Fordy / The H~rton-Heiles ~system ret'isited 209

Thus we have

- ' i) 'q~) ---L[. so P = 16LL (2.17) ( L 5 / 2 ) + = ( a 2 + q l ) ( ~ ) 3 + 3 q , 3 + 3 0 , 16q2 ~ ,

-4L being the ope ra to r derived in ref. [9].

In ref. [9] it is conjec tured that /~ indicates a connect ion with the loop a lgebra of A~ I. Indeed, such an ope ra to r appea r s against /1(2)3 in table 4 of ref. [13] and is associated with a sequence of equat ions of KdV type (table 7 of ref. [13]). However , the H 6 n o n - H e i l e s system (case (ii)) is not the s ta t ionary flow of one of this" hierarchy, but of the A~ ~) hierarchy associated with L = 02 + u. The appea rance of the A (2) 3 ope ra to r seems to be a co-incidence.

2.3. The modified systems

The second of the H 6 n o n - H e i l e s equat ions can be wri t ten as

q j = -- q 2 x x / 2 a q 2 . (2.18a)

Thus, defining t, as follows, we have

I' = q 2 x / 2 a q 2 ~ u = q l = - - t ' x -- 2 a c 2 . (2.18b)

Different ia t ing the first of the H 6 n o n - H e i l e s equat ions once, leads to a four th -order equat ion for c:

t ,~ ...... . + 2 (b + 6a)c~t,x~ + 4 a ( b - 4a)(c2c~.~. + uc 2) - 16a3bc 5 = 0. (2.19)

The mapp ing (2.18b) thus takes solutions of (2.19) onto solutions of (2.2) for all values of a and b. In fact, the Miura maps (2.18b) map solut ions .of

c t = [t, ........ + 2(b + 6a)c~cx. ̀ . + 4 a ( b - 4a)(c2cx.~ + t,c 2) - 16a3bcS]x (2.20a)

onto solutions of

_ Z~abu ~l u , = [ u ........ + ( 8 a - 2 b ) u u .... - 2 ( a + b ) u 2, 7 I x " (2.20b)

Remark. Eqs. (2.20a), (2.20b) can be wri t ten in Hami l ton ian form:

t,, = -Ot3 ~ ' , f f"[ , , ] l ,2 2 a ( b 4 a ) t , 2 , x + , ,~ s 3 . (, = - ~ t x x + - ,2 ( 2 a + s b ) t ~ + x a o c , (2.21a)

u , = ( 1 t 3 + 8 a u o + n a u x ) 8 , , ~ , ~ [ u ] = - ½ u 2 - } b u 3, (2.21b)

and, denot ing (2.18b) by M:

O 3 + 8 a u O + 4 a u , = M ' ( - ~ ) ( M ' ) * [ . . . . , . 2, ,- ~, f f ~ = 2 U o M ( m o d a m O ) , (2.21c)

where M ' = - O - 4at, is the FrEchet derivative of M and (mod Im O) means " m o d u l o exact derivatives". Thus M is a Hami l ton ian Miura m a p be tween (2.20a), (2.20b) for any values of a and b. For the three integrable cases these Miura maps and Hami l ton ian s t ructures were already known [10].

210 A.P. Fordy / The HJnon-Heiles system ret,isited

For the three integrable cases, ~, then satisfies

( i ) Cxxxx+5l ,x t '~x - -5 ( t '2L '~ .~+CV~)+t '5=O,

( i i ) t.'xxx. ,. -- lO(u2t'xx + t'U2x) + 6 v 5 = O,

(iii) ~ : ~ . ~ . - 5 t ' x t , ~ x - 5(vzt'x~ + vt~'2~) + c '5 = O.

( 2 . 2 2 a )

( 2 . 2 2 b )

(2 .22c )

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