Newton representation of nonlinear ordinary differential equations

Physica A 197 (1993) 191-203 North-Holland SDZ: 0378_4371(93)EOO41-c

Newton representation differential equations

of nonlinear

ordinary

Maciej Blaszakl Departamento

de FGca,

Universidade da Beira Interior, 6200 CovilhG, Porhqa12

Stefan Rauch-Wojciechowski Department

of Mathematics, Linkiiping

University, 581 83 Linkijping,

Sweden

Received 5 January 1993

A new parametrization for higher order ODE’s is introduced. It turns them into a set of Newton equations. When applied to stationary flows of soliton equations, this parametrization leads to new integrable mechanical systems. In the KdV case we find a new Poisson bracket of 3rd order.

1. Introduction

We consider here 2nth order ordinary differential equations (ODE’s) of the form Uanx +

Y”+lbl= c >

c = conk

,

(1.1)

where uZnx means the 2nth derivative and y,,+,[u] = ~,,+~(u, u,, . . . , u~~,_~~~) is a differential polynomial. We assume that eq. (1.1) is homogeneous with respect to the scaling transformation u+Au,

x+

h-1’2X,

c--f

A”+lc.

For these equations we introduce a new parametrization

(1.2) which turns them into

1 Partially supported by KBN Research Grant No. 2 2442 92 03 and a stipendy AdTlMLH ref. 304131 from the Swedish Institute. ’ On leave of absence from the Physics Department of A. Mickiewin University, 60-769 Poznan, Poland. 0378-4371193/$06.00 @ 1993 - Elsevier Science Publishers B.V. All rights reserved

192

M. Biaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

a set of Newton equations of the form ‘k,,

=

‘k+l

+fk+lkl,

‘.

.

9

‘k)

k=l,...,n-1,

7

(1.3) r “xx = c +fn+1(rr,

* . . 9 m) 7

where each variable rk scales as rk + Akrk and functions fk+lare homogeneous polynomials of the order k + 1 with respect to this scaling transformation. Notice that in (1.3) we use the notation rkx, which means only the 2nd derivative of rk. It should not lead to confusion since rk always mean Newton variables. If eqs. (1.3) are Lagrangian then its Lagrangian function 9 has an indefinite kinetic energy term and scales as 55’4 h”+*d;P. We apply parametrization (1.3) to higher order stationary flows of soliton equations and derive this way new completely integrable systems of Newton equations. In section 2 we consider the case of 4th order ODE’s and its Newton representation. As particular examples we present a new representation for stationary flows of the Korteweg-de Vries (KdV), of the SawadaKotera (SK) and of the Kaup-Kupersmidt (KK) soliton equations. In section 3 we consider the case of 6th order ODE’s and provide for them a Newton representation. This is illustrated by the 7th order KdV stationary flow. Finally, in section 4 we discuss the general case of 2nth order ODE’s of homogeneity An+’ and formulate a conjecture that they all admit the Newton representation (1.3).

2. 4th order equations A general 4th order equation which is homogeneous to the scaling (1.2), has the form U& + &.Ul, + but + du’ = c )

of order A3 with respect

fi, 6, d = const.

(2.1)

We seek for a Newton representation

rLX= r2 +

iirf ,

rL

=c+b;r,+&i,

- d” = const. a”,b,

If rl = u then r2 = u,, - a”u2and eqs. (2.2) are equivalent a=-2a”-b”, Thus every equation

L5=-2z,

(2.2)

to (2.1) if

2Z;g-d”.

(2.1) admits the Newton representation

(2.3) (2.2), where

193

M. Biaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

If we assume that eq. (1.1) is Lagrangian with 3?[U] = $US, + auuf + bu4 - cu , (homogeneous

a, b = const. ,

(2.5)

of the order h4) then its Euler-Lagrange

equation

W?

0=-=u4x-2auu,,-au~+4bu3-c

(2.6)

6l.4

is equivalent

to

rlXX= r2 + $ari ,

T2, = c + ar1r2 + ($a” - 4b)rT ,

(2.7)

which also follows from the Lagrangian _T?[r]= ri,~~, + iarfr,

+ (ka’ - b)r: + $rz + crl .

(2.8)

Notice that 5?[r] = -Z[ u ] modulo total derivative. Hamiltonian with the Hamiltonian function

System

(2.7)

X(r, s, c) = sls2 - tarfr, + (b - ta2)r: - $rt - crl , where‘ s1 = r2, and s2 = rl, are conjugate

is also

(2.9)

momenta.

Examples The equation

[u4* + (8A - 2B)uu,,

- 2(A + B)u; - yABu3],

is known [1,2] to be equivalent %X = Bq: - Aq; , under the substitution three cases (a) A=$,B=-3

= 0

to the Henon-Heiles

qz,, = -2&q,

+ ‘y/q: >

(HH) system a = const. ,

u = q1 and i u,, - 4 Bu2 = - 4Aqi.

(b) A=$,B=-$

(2.10)

(c) A=

(2.11)

It is integrable

in

a, B=-4, (2.12)

which correspond to the KdV, SK and KK hierarchies, respectively. tion of (2.2) provides for (2.10) a new physical representation dynamically equivalent to (2.11).

Applicawhich is

194

M. Blaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

(i) The 5th order Korteweg-de Wes stationary equation This is case (a), so the once integrated equation (2.10) reads Udx

lOUU,, + 5u; + 1ou3 = c *

+

It is Lagrangian L?[u] =

with

$& - 5uz.4~ + $u” -

The Newton representation

rLx= r2

(2.13)

5

2

- Zr,

,

cu.

(2.14)

for (2.2) is

r2,x = c - 5r,r, + $r: ,

(2.15)

where rl = u and r2 = u,, + 3~‘. Eqs. (2.15) follow from the Lagrangian

z[rl = r1,r2,

- grfr,

+ $rf + ar: + cr,

and have the standard Hamiltonian r2 , s2 = rl and x .z Wr, s) =slsz+

$rfr,-

ir,“-

formulation

(2.16) in term of variables rl , r2, s1 =

jjrf-cr,.

(2.17)

A bi-Hamiltonian formulation for (2.15) has been found in [3] by using its (non-canonical) equivalence with the HH system (2.11), case (a). The map connecting variables (r, s, c) and (q, p, cx) has the form

r1 =

41

7

r2=-1

242

hi>

s1= -41P1-

42P2

7

sz=p19

(2.18)

c-pf-p;_

2q:- WI; - ‘y/q:>

and it yields r1

s2

r2

Sl c - Sr,r, + $rf

ill I S1

=

c

where

5 2 zrl

r2 -

s2

x

0

= n, vslp, = n, vxO )

(2.19)

195

M. Biaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

i-L1 1 0 0 0000, -10 0 0

0

0

0 -1 0 0

II,=

0 1

0 0

0 0

0 0

2s2

i Ill =

00

0 0 -2r,

-r1

\ -2s, 1

-2s, -ri

2r2 r1

-1 r1

0

s2

lOr,r, --s25rt - 2c

- 2r, 0 + 5rf

2%

1

-lOr,rz + 5rf + 2c 2r2 - 5r: 0

.

(2.20)

The vectorfield

(2.19) belongs to the bi-Hamiltonian

ladder

(2.21) l7, VR2 = l7, vslp, ,

0=4vslr;,

where

Xl =

sls2

+

$rir,

- $rf -

ir;

-

crl ,

(2.22)

%f2= - isi - rlsls2 - 2r,r: + $r: + c(r2 + $r:) . The Poisson operator n, naturally leads to a whole family of integrable potentials if the last column in n, is substituted with Q, VS? for X = s1s2 + W(r) - crl. Potentials W(r) then satisfy a linear, 2nd order partial differential equation which follows from the Jacobi identity for II, [3]. (ii) The Sawada-Kotera stationary equation This is the case (2.10), (2.12b) which, after single integration, equation uqx+5uu2x+~u3=c. Its Newton representation 5,

= r2 ,

leads to the

(2.23) has the form

r2.u =c-5r,r,-

fri,

where rl = u and r2 = ux,, and has not any obvious Lagrangian

(2.24) formulation.

M. Btaszak, S. Rauch-Wojciechowski ! Newton representation of ODE’s

196

However system (2.24) admits a non-canonical the form

i.H I s2 s1 r2 r1 c

x =

c -

Hamiltonian

representation

of

5r,r,82 12 Sl 0 - 5r3 3 1

0

0

rl

-1

2s2

0

0

2r2

r1

23,

-s2

0

2:22

-

1 = 1 -2s, -rl x WC)

-r1 -2s -2r,

1

10r 12r +‘Or3-2c 0 31

-2r, s2

0r

2c-10r

1

@r3 3 I

(2.25)

7

where s, = r2 and s2 = rI are conjugate momenta. It follows from the standard Hamilton forxmulation of’the Henon-Heiles system and the map

r1 Sl

=

u

=

=

u3x

q1 ) =

-41P1-

r2 =

u,,

=

2

-$41-

Cl2P2 7

1 2

zq2

9

s2 = u,=p17

(2.26)

which connects the (q, p, a) and the (r, s, c) variables. (iii)

The Kaup-Kupershmidt

stationary equation

Our last example is the case of the system (2.10), (2.12~) which, after single integration, leads to the system t.L&

+

This system also does not have any obvious Newton representation has the form

rL

=

(2.27)

lOUU,, + $uf + Yu’ = c .

1.5 2 rz - arI ,

Lagrangian

formulation.

65 3 r2,X= c - $r,r2 + arl ,

where rl = u and r2 = u,, + yu”, formulation

and admits a non-canonical

Its

(2.28)

Hamiltonian

M. Btaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

=

L 0

0

45

-1

0

0

Y2

91

-i lr 1

-r2

0

92

1

-$Y,

-z2's

-s2

-s*

-c+

s2 Sl c-

0

$r,r, -

z;~Y, 6.53

197

$y,y, + Y2-

__I 2 +I?,2 4

1

BYI 65 3 I vc,

yy;

0 (2.29)

which follows from the Henon-Heiles rl=u= s1=

U&

r2=u,,+ti

41 9 +

c = u4x +

yuu,

=

4u

-jqlpl

-

case (2.12~) via the map

2=_1 4q1

$q2p2

= +p;

lOuu,, + ?&; + $”

2-1

,

2

4q2,

(2.30)

s2=ux=p1,

- $p”, - $9:

-

:qlq;

- ffl2q2,.

3. 6th order equations A most general equation

(1.1) of 6th order is

U& + Ciul.$, + Ll,u3x + El& + &4u”,+ eu2u, -I-$4” = c , where C, 6, C; 6, F, f are arbitrary sentation reads

yL = Y2f

constants.

(3.1)

The admissible Newton repre-

fzr: y%x = )

r3 + b;,y, + i?yi ,

(3.2) y3xx

= c + iiy,y, + e"r;r, +$':

+ iy; ,

where -2 r2 = u2x - au

TI=U,

,

(3.3) y3 =

u4x -

Equivalence

2&d; - (2a” + iT)uu,

+ (iii - qu3 .

of (3.2) with (3.1) requires that

a=-(2b+b”+&

6=-(8;+2b”),

E=-(6a”+b”+g”),

198

hf.

This is an choice of arbitrarily example, equations

undetermined system of equations, so we have some freedom in the the Newton representation (3.2). We can fix one tilde coefficient and then solve (3.4) with respect to the other tilde coefficients. For if we choose a solution with tixed a”, then we obtain the Newton (3.2), parametrized by a”,

BIaszak, S. Rauch-Wojciechowski I Newton representationof ODE’s

Let us consider the subclass.of Lagrangians systems. The most general Lagrangian for (3.1) (up to a total x-derivative) reads 9[u]

=

;u:,+au&

+ bu2ut + du5 + cu ,

(3.6)

where a, b and c are some suitable constants, and its Euler-Lagrange iU’l6u = 0 yields

equation

+ 2buuf - 5du4 = c .

(3.7)

uhx - 2auu4, - 4au,u3,

- 3au& + 2bu2u,

Then, in terms of Newton variables, we obtain the following (Gparameterized) equations:

rLX=

r2

+

&f,

r2xx= r2 + (2a - 4a”)r,r, + (- $6 + 2aa”r3xx

=

c +

2Grr,r3 + (-b

+ (a - 2a”)rg,

with the Lagrangian

yg2)r:

,

(3.8)

+ 6aa”- 10a”2)rfr2 + (5d - $ba” + 5a2i2 - ya”‘)r’:

199

M. Blaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

S(r, r,, c) =

ir;I + r1,r3, + r2r3 + a”;r, + (a - 2qr,r;

f (- $b + 2aC-

$+:r, (3.9) \ ,

+ (d - fba” + a*;* - $z3)r:

+ rlc

and the Hamiltonian X(r, s, c) =

is;+

s1s3 -

ya”*)r;r,

r2r3-iir~r3+(~b-2aLz+

+ (-d + fba”- a*a”*+ $T3)r; - rlc,

+ (2a”- a)&

where s1 = r3 , s2 = r2 and s3 = ri X. Notice modulo complete de&ative .

that

here

again ZJr]

(3.10) = Z3[ U]

Example. Newton representation for 7th order stationary flow of the KdV hierarchy and its bi-Hamiltonian formulation

We shall consider the 7th order stationary flow of the KdV hierarchy which corresponds to a = d = -7 and b = 25 in the Lagrangian (3.7). Its Newton representation (3.8) admits the Hamiltonian formulation (3.10). It has also the well known Ostrogradsky Hamiltonian formulation with 41=u,

u

cl2 =

q3

XT

=

uxx

7

p1 = usx + 14Uu3x + 14u,u** + 7ou*u, , p* = -u&

- 14uu,

)

P3 =

U3*

(3.11)

*

These two sets of canonical variables for the stationary flow are connected the transformation N(C) , rl = 41 7 r3 =

r2=q3-a41

“2

s2

=

9

-p2 - 2&j: + 2a”q,q3 + $(35 - 2i2)4:

s1 = p1 - 2a”q,q, + 2zq1p3 P3 -

2a”q,q2

7

by

- (35 +

s3 = q2 9

9

(3.12) =*)q:q,

c=c

7

9

which is canonical. As calculated in [4], the Ostrogradsky bi-Hamiltonian structure

formulation

has the

4

0 P c

x

=n,vx=n,v(s,

(3.13)

M. Biaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

200

with

n, = 0 1 i -2q,

4q1 0 -2q,

loq, -lOq, -2p,

-A -B -C

0 -D -E

D

E

0

where A = -2p,

+ 14Oq; + 704,” ,

B = -4P, - 14Oq,q, ,

c = 14oq,q; - 14q: -.7oq: + 2c , E = 14Oq;q, - 2p, ,

F = -28q,q,

D=4q,-7Oq:, - 2p,

and

x = ;p3

+ q*p1+ qjpz + 7q,q: + 7q:q: - 35q: - q1c )

%=

$C.

The Poisson operator II, is derived in a standard way with the use of the Miura map. Now we can use the map N(C) in order to find a bi-Hamiltonian formulation for the Newton representation (3.8). We fix a”= 1 a = - $ and the relevant Poisson operators are

fi, = N’fllNrT

r2 - 14rf

-2r, + 14r:

0

3%

-s3

A

00 1

4r, 4% 1 0

-3r, -4s, 00

-r2 Zr,-3s, -$3 + 14ri 14rf

jr,

-4r, 3r, -1 00

3r, -1 00

2r, 27, 2% 2s, 2% -- 7rf 7ri

-2s,

-2s,

-2s,

-3r,

i

=

where A = -14r,r,

-A

-2r, +7r:

-2r,+7r:

0

I (3.14)

I

- 21rfr, + y ri + 2c, N = N(- $) and N’ is the Frechet

201

M. Blarrak, S. Ranch-Wojciechowski I Newton representationof ODE’s

derivative of N. The above bi-Hamiltonian bi-Hamiltonian chain:

fiovxo=o, frov2z1= lil

formulation

l?()wf2

Vslf( ,

=

generates the following

n1 v2q , (3.15)

o=171v%3,

ii0 vz; = fil VXz , with x0=

$c,

x, = is’, + s1s3 X2

=

rls1s3

+

- FrT

-

r2r3

3r,s,s,

+

Zr2r + zr3r - ar5 212 41 2 13

cr

1 9

- 3r,si - r3si - s,s2 - r,r2r3 - r: + $ri + ?r;r2

- $r:c + r2c,

X3 = $sf + $risi + 2r:s,s,

(3.16)

+ 14r&s,

+ 3r,s,s,

-4r,r,si

- jr$z

- r2s,s3 - 2r,s,s,

- yrtr,

- 7r:ri

+ 3r,r:

+ 2r,r:

- 9r,r,s,s,

+ gr:s:

+ 4s,s: + 3ri + Sr$-,r,

+ rzr3 + 4rTc - 3r,r,c

+ ir:r,

- r3c + 2S:C ,

which are commuting and functionally independent integrals of motion. Notice that the chain starts with the Casimir of the first Poisson structure and terminates with the Casimir of the second Poisson structure. The Poisson operator fil has a similar structure as n, (2.20). It consists of the main part n being a 6 X 6 matrix and of the last column (and row). This column is equal to the dynamic vectorfield fiO VSre,. The same question (as in the 5th order KdV case) about the family of potentials W(r) such that fi, VX (X = 4s’; + s1s3 + W(r) - crl) yields a Poisson bracket, has a surprising answer. All Jacobi identities determine potential W(r) = -r2r3 + $r:r, + fr:r, ?t-i uniquely. There is no such family as in the case of separable biHamiltonian structures [5].

4. 2nth order equations In the general case, the Newton representation (1.3) remains a conjecture. There is strong evidence from the low-order cases that it is valid. It is clear that all relationships (1.3) can be recursively solved for rk: r*=t4,

‘k

=

u2(k-l)x

+

ykl”l

7

c =

U2nx

+

3/n+l[Ul

9

M. Blaszak, S. Rauch-Wojciechowski I Newton representation of ODE’s

202

where fk(rl, . . . , T~_~) and yJu] = yk(u, . . . , uz~k_l~x) are polynomials of homogeneity k with respect to the scaling (1.2). They are connected bv the recursion formula

However, the proof that arbitrary yk+i [u] can be produced through the recursion (4.1) is quite nontrivial since the relationship between the coefficients offk+l,k=l,..., n and the coefficients of yk+r[u] is in the form of a set of higher order algebraic equations. It is not clear that these equations can always be solved for the coefficients of the polynomials fk+r, k = 1, . . . , n. In the Lagrangian case of (1.1) it is necessary to prove that the Newton representation (1.3) is Lagrangian and that Z[r] = Z[u] modulo complete derivative. If it is the case, the following procedure can be applied in order to derive the Lagrangian-Hamiltonian structure of (1.1). Let us separate the linear part of the dynamics (1.3), r xx - i;k(r, c) - F,,(r) = 0 ,

(4.2)

where r2 r3

Fe(r, c) = Ill

ii C

represents the linear part of F(r, c) and F,,(r) = F(r, c) - F,(r, c) the respective nonlinear part. It is not difficult to find the Lagrangian of the linear part of the dynamics (4.2), _r%k(r,r,, c) = $(ri),~i’(ri)x

- W, (4.3)

where 1 r

w, = -

J(F,(Ar, 4, cLr)dAeF,(r,

c) = --(CLVW, .

(4.4)

0

V= (alar,, ( . . . ) . . . > is a scalar product, and the metric k is of the form

. . . , dldr,)T

is a gradient operator,

hf. Biasrak, S. Rauch-Wojciechowski I Newton representation of ODE’s

203

(4.5)

But having found the metric p we can immediately the nonlinear part of (4.2),

construct the potential of

1

W,,,W=

I V’n,(W, w) dAe-(pV)W&)

= F,,,.

(4.6)

w=w,+w,,.

(4.7)

0

Thus, our Lagrangian

reads

s(rY rxP c,= t(ri)*Pi'(rj), A suitable Hamiltonian

- w(r)

9

takes a form

sqr,s, c) = ~Si/.PSj + W(r)

(4.8)

)

where si = G’/a(r,), are the conjugate momenta to ri and the dynamics (1.3) can be put into a canonical Hamiltonian form,

(:),=(_; +w,c).

(4.9)

The above procedure was applied in the derivation of our examples of stationary KdV flows in previous sections. The general situation, described above, is not satisfactory and the relevant proofs has to be found soon. However, the existence of the representation (1.3) for homogeneous differential equations seems to be a very remarkable phenomenon with far reaching consequences in soliton theory. Most of the stationary flows of soliton equations are such homogeneous sets of ODE’s. Through the representation (1.3) they acquire the meaning of Newton equations, which makes them much more interesting objects to study.

References A.P. Fordy, Physica D 52 (1991) 204. M. Antonowicz, S. Rauch-Wojciechowski, Phys. Lett. A 163 (1992) 167. S. Rauch-Wojciechowski, Phys. Lett. A 170 (1992) 91. M. B&zak, Miura map and bi-Hamiltonian formulation for restricted hierarchy, J. Phys. A. in press. S. Rauch-Wojciechowski, Phys. Lett. A 160 (1991) 149.

flows

of the KdV