Non-Hamiltonian symmetries of a Boussinesq equation

Non-Hamiltonian symmetries of a Boussinesq equation

Citation for published version (APA):

Eikelder, ten, H. M. M., & Broer, L. J. F. (1986). Non-Hamiltonian symmetries of a Boussinesq equation. Journal of Mathematical Physics, 27(8), 2151-2153. https://doi.org/10.1063/1.527036

DOI:

10.1063/1.527036

Document status and date: Published: 01/01/1986 Document Version:

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Non-Hamiltonian symmetries of a Boussinesq equation

H. M. M. Ten Eikelder

Department

0/

Mathematics and Computing Science, Eindhoven University o/Technology, P. O. Box 513, Eindhoven, The Netherlands

L. J.

F. Broer

Department

0/

Physics, Eindhoven University o/Technology, P. O. Box 513, Eindhoven, The Netherlands (Received 12 March 1986; accepted for publication 16 April 1986)

For a class of Hamiltonian systems there exist infinite series of non-Hamiltonian symmetries. Some properties of these series are illustrated using a Boussinesq equation. It is shown that the recursion operators generated by these non-Hamiltonian symmetries are powers ofthe original recursion operator. A class of recursion formulas for the constants of the motion (not for the corresponding symmetries!) is given.

I. INTRODUCTION

For a certain class of Hamiltonian systems there exist so-called recursion operators for symmetries. Repeated ap-plication of such a recursion operator yields a series of sym-metries. Often it is possible to construct in this way infinite series of Hamiltonian symmetries (corresponding to con-stants of the motion) and infinite series of non-Hamiltonian symmetries. The most well-known example is the Korteweg-de Vries equation, where the Lenard operator generates an infinite series of Hamiltonian symmetries and an infinite series of non-Hamiltonian symmetries. In this pa-per we use a Boussinesq equation to illustrate some propa-per- proper-ties of these series, in particular the series of non-Hamilto-nian symmetries. Similar results can be obtained for various other equations, see Ten Eikelder.I,2 In this paper we work within the framework of differential geometry. For defini-tions of various concepts (symmetry, recursion operator for symmetries, etc.) see, for instance, Ref. 2, where also nota-tions and convennota-tions are given.

II. SYMMETRIES OF A BOUSSINESQ EQUATION

We study a Boussinesq equation of the form

v,

=w

x '

w, =

vVx

+

AV

xxx '

- 00

<x

< 00, t>O. (1)

We consider (1) as an evolution equation in a topological vector space 'lr of pairs of smooth functions (v,w), which decay, together with their x derivatives, sufficiently fast for Ixl~oo. The spaces 'lr and 'lr* are constructed such that I

Three infinite series of symmetries now can be defined by

X k

=

AkX_{O' Yk}

=

Aky_o, Zk

=

AkZo,

k

=

0,1,2, ....

It is shown by Fokas and Anderson4 _{that the Nijenhuis}

ten-sor of A vanishes (in their terminology, A is a hereditary symmetry). So for all vector fields A we have 2' AA A

their duality map (.,.) is the L2 inner product. A possible choice is 'lr

=

.Y p X.Y p and 'lr* = ~ p X ~ p' where the

function spaces .Y p and ~ p are described in Ref. 1. In terms

of u = (v,w)e'lr we can write (1) as

u

=X(u) (U(t)

=

:t

U(t»).

A Hamiltonian form of ( 1) is well known. Let the function (functional) Fo on 'lr be given by

Fo =

f'"

(..!..

v3 -

..!..

AV!

+..!..

w2)dX

-'" 6

2

2

and let the symplectic form 0 on 'lr be (represented by the linear mapping 0: 'lr ~'lr*) given by

O=(a~1 a~).

Then the vector field X can be written as X = 0 -I dF_o

(d = exterior derivative), so (1) is a Hamiltonian system. The invariance of ( 1) for translations along the t and x

axis and for a scale transformation yields the following ele-mentary symmetries: Xo=X= (

Wx

),

vVx

+

AV

xxx (2)

(

Vx )

(2V

+

xv", ) Yo = , Zo

=

3

+

2tXo·

Wx

w+xwx

A recursion operator for symmetries of ( 1 ), written in terms of the "coordinates" of a modified Boussinesq equation, has been given by Fordy and Gibbons.3 _{In terms of the "original}

coordinates" v and

w

this operator reads

L

A2' A A (2' A = Lie derivative in direction of A). This also can be verified by a straightforward computation.

LetA andBbe vector fields on Ysuch that 2' AA = aA

and .!f BA = bA for a,heR. Define Ak = AkA and Bk

=

AkB, for k

=

0,1,2, .... Using the fact that the Nijenhuis tensor of A vanishes, it is easily shown (see, for instance, Ref.

2151 J. Math. Phys. 27 (8), August 1986 0022-2488/86/082151-03$02.50 @) 1986 American Institute of Physics 2151

2) that the Lie bracket [Ak,BI ] is given by

[Ak,Bd =iaBk+l-kbAk+I+Ak+/[A,Bj. (3)

A simple computation shows that

.2" Xo A = 0, .2" Y" A = 0, .2" z" A = 3A, ( 4 )

[Xo,Yo] = 0, [Zo,xo] _{= 2Xo, [Zo,Yo]} = Yo'

Substitution in (3) yields that the only non vanishing Lie brackets between the elements of the series X k' Y_{k ,} and Z k are given by

[Zk,Xd = (3/+2)Xk + l , [Zk,Ytl = (3/+ l)Yk+ l ,

[Zk,Ztl

=

3(1- k)Zk+ I' (5)

Since the Nijenhuis tensor of A vanishes, it immediately fol-lows that

.2" X_k A = 0, .2" Y_k A = 0, .2" Zk A = 3A k + I,

k = 0,1,2, ....

(6)

The first relation corresponds to the well-known fact that A is also a recursion operator for symmetries of the "higher-order Boussinesq equations" II = X k' The second relation

shows that A is also a recursion operator for the equations

11= Yk •

Next we discuss some properties of the series of symme-tries Z k' For every non-Hamiltonian symmetry Z a

nonvan-ishing recursion operator for symmetries is given by n - 1.2" Z n. If Z is a Hamiltonian symmetry this expression

yields

°

[because .2" zn = d(nZ) = 0]. Note that the re-cursion operators obtained in this way are always the prod-uct of a canonical operator n - I (also called Hamiltonian operator or implectic operator) and a closed operator .2" Z (n) (also called symplectic operator). Most interesting

recursion operators have such a factorization, see, for in-stance, MagrV Fuchssteiner and Fokas,6 or Gel'fand and Dorfman.7 _{In Ref. 2, we computed recursion operators for}

the massive Thirring model by this method.

The symmetries Zo and ZI turn out to be non-Hamilto-nian. The corresponding recursion operators are found to be n -I .2" z" n = 31 (l = identity mapping:

rr

---+

rr),

n-l_{.2"z,n=6A. (7)}

So the recursion operator A can be reconstructed from the symmetry ZI' From (6) and (7) it is easily shown by induc-tion that

.2"~,n=3k(k+1)!nA\ k=0,1,2,.... (8)

Since the Lie derivatives and the exterior derivative com-mute, this relation yields a very simple proof of the well-known fact that all the two-forms nA k are closed. This prop-erty implies that

LZkn=dcnZk) =dcnAkZo ) = .2"zo(nAk)

= (.2" z"n)Ak + n.2" Zo (Ak) = (3k + 3)nAk #0,

k=0,1,2,.... (9)

Thus we have proved that all the symmetries Zk are non-Hamiltonian and that the corresponding recursion opera-tors are powers of A (up to a multiplicative constant).

Because Xo is a Hamiltonian symmetry .2" Xo n = 0. A simple computation shows that Yo = n -I _dGo with

2152 J. Math. Phys., Vol. 27, No.8, August 1986

Go = S': 00 vw dx, so Yo is also a Hamiltonian symmetry. A

computation similar to (9) shows that all the symmetries

Xk , Yk (k = 0,1,2, ... ) are Hamiltonian vector fields, i.e., there exist two series of constants of the motion Fk and G_k

such that

Xk =n-IdFk, Yk =n-IdGk, k=0,1,2,.... (10) The corresponding symmetries commute, so all these con-stants of the motion are in involution. The existence of the series Fk is a standard property in this case, see, for instance, Ref. 6. It follows from (10) that

nAkX = _dFk,

which can be considered as "pre-Hamiltonian" forms for

X = Xo. The original Hamiltonian form is obtained for k = 0, while formally k = - 1 with F -I = S", 00

!w

dx

yields the second Hamiltonian form of the Boussinesq equa-tion.

We now give a class of recursion formulas for the con-stants of the motion Fk and Gk • The Hamiltonian vector

field corresponding to the function .2" z,F_k on

rr

is n -I _{d.2" z,Fk} = n -1.2" z, _dFk

= .2" z, (n -I _{dFk ) -} (.2" z,n -I _)dFk = [ZI,xk] + n- I (.2" z,n)n -I _dFk

= (3k + _2)Xk+1 + (31 + 3)AIX k

= (3k

+

31

+

5)n-1 dFk + I'

where we used (5) and (9). This yields the recursion formu-las

F - 1 .2" F - 1 (d )

k + I - 3k + 31 + 5 z, k 3k + 31 + 5 Fk ,ZI .

(11 ) In a similar way we get

G - 1 .2"G

k + I - 3k

+

31

+

4 z, k _3k

+

1 ₃₁

+

₄ (dG _k' Z) _{I .}

(12) Note that in these recursion formulas it is not necessary to reconstruct a functional from its derivatives. The part of ZI

with "coefficient" t is 2XI [see (2)], so this term can be omitted in (11) and (12).

The symmetry ZI is given by

where

ZI.I = 12vw + 2wx

a

_{-IV + 2vx}

a

-Iw

+

40Awxx

+

x(4(vw)x

+

8Awxxx )' ZI.2 = 4v3

+ _2vvx

a

_{-IV + UVxxx}

a -

_{IV + 58Avv.u}

+

45Av~ + _{48,.1, 2vxxxx + 9w}2 _{+ 2wx}

a

_-Iw

+

x( 4v2

vx

+

1Uvvxxx

+

24Avxvxx

+

8,.1, 2vxxxxx

+

_{4wwx ).}

The part of ZI with coefficient

x

turns out to be YI • So ZI

= C1 + xYI + 2tX1, where C I contains (also non10cal)

terms not depending explicitly on x and t. Similar relations

H. M. M. Ten Eikelder and L. J. F. Broer 2152

turn out to hold for the other symmetries Zk' For 1 = 1 we obtain from (11) and (12) the recursion formulas

F k+1 = __ I_fco (8F k _{ZI,I +} 8Fk ZI,2)dX, 3k

+

8 -

co

8v 8w (13) I

fco

(8Gk _8Gk ₎ Gk+1

= - - -

--ZI,I +--ZI.2 dx. 3k

+

7 -

co

8v 8w (14)

Starting with Fo and Go these relations enable us to generate the series Fk and G_{k •} In fact it is also possible to begin with

F - I and G - I =

J"':

co

v dx.

A constant of the motion that depends explicitly on t is J

=

J"':

co

(xv

+

tw)dx. Constants of the motion of this type always exist if a conserved density (in this case v) has a flux that is also conserved, see Broer and Backerra.8 _The

Hamil-tonian symmetry corresponding to J is formally given by

- I

(0)

Z_I=O dJ= 1 .

It can be shown that (11) and (12) also hold for 1

= -

1 and k~O. This yields the relations

1 1

fco

8Fk

Fk _ 1

=---"?z

Fk

=---

--dx,

3k

+

2 - 1 3 k

+

2 _

co

8w

2153 J. Math. Phys., Vol. 27, No.8, August 1986

1 1

fco

8Gk

Gk _ 1

=---"?z

Gk

=---

--dx

3k

+

1 - 1 3 k

+

1 _

co

8w '

k = 0,1,2, ....

While (13) and (14) allow us to go upwards in the series of constants of the motion, these two relations allow us to go downwards.

'H. M. M. Ten Eikelder, "Symmetries for dynamical and Hamiltonian sys-tems," CWI tract 17 (Centre for Mathematics and Computer Science, Amsterdam, 1985).

2H. M. M. Ten Eikelder, J. Math. Phys. 27, 1404 (1986). 3A. P. Fordy and J. Gibbons, J. Math. Phys. 22,1170 (1981). 4A. S. Fokas and R. L. Anderson, J. Math. Phys. 23,1066 (1982).

SF. Magri, "A geometrical approach to the nonlinear solvable equations," in Nonlinear Evolution Equations and Dynamical Systems, Lecture Notes in Physics, Vol. 120 (Springer, Berlin, 1980).

6B. Fuchssteiner and A. S. Fokas, Physica D 4, 47 (1981).

71. M. Gel'fand and I. Ya. Dorfman, Funct. Anal. Appl. 13, 248 ( 1979); 14, 223 (1980).

8L. J. F. Broer and S. C. M. Backerra, Appl. Sci. Res. 32, 495 (1976).

H. M. M. Ten Eikelder and L. J. F. Broer 2153