Canonical structure of Bäcklund transformations

Volume 78A, number 1

PHYSICS LETTERS

7 July 1980

CANONICAL STRUCTURE OF BACKLUND TRANSFORMATIONS R. SASAKI The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen

0. Denmark

Received 27 March 1980

Most of the integrable nonlinear evolution equations in 1+1 dimensions are known to be Hamiltonian systems. We show that the “Generalized Bäcklund Transformations” of Calogero and Degasperis form the group of canonical transformations that keeps these Hamiltonians invariant. In other words, they form a group of symmetry transtormations for these Hamiltoalan systems.

Recent developments in the theory of nonlinear evolution equations (NEE’s) in 1 + 1 dimensions [1—4] have shown that most NEE’s that can be solved in terms of the 2 X 2 inverse spectral transform are Hamiltonian systems [3,4]. Then a natural question arises; what is the group of canonical transformations that keeps these Hamiltonians invariant, or, what is the group of symmetry transformations for these Hamiltonian systems? On the other hand, from the study of the scattering data Calogero and Degasperis [2] have found the “Generalized Backlund Transformation” (GBT) which maps a solution of an NEE to another solution of the same equation. It has been shown [2,5,6] that the GBT contains all the usual BT’s [7] for the sine-Gordon, the Korteweg de Vries (KdV), the Modified KdV (MKdV) etc. The GBT of Calogero and Degasperis takes a very simple form m the scattering data space and its group structure has recently been clarified by Konopelchenko [81. In this note we show that the GBT is a canonical transformation and demonstrate directly, i.e. m terms of the “potentials” r, q, that the “Generalized BT” keeps all of the infinite numbers of conversed quantities invariant. It has already been noted [9,10] that the usual BT’s for the s.G, the KdV etc. are canonical transformations. Kodama and Wadati [10] showed that the usual BT’s for the above equations keep their own Hamiltonians invariant. Notice that our discussion deals with two independent “potentials” r and q in con-

tradistinction to the reduced cases such as the s-G and the KdV etc., and that our conclusion is much stronger; the canonical transformation corresponding to the GBT keeps all the Hamiltonians in this scheme invariant, i.e., GBT’s form a group of symmetry transformations for these Hamiltonians. The inverse scattering method of Zakharov and Shabat [11] and AKNS [1] associates the following 2 X 2 scattering problem with the NEE under study 1)i \

a

—

ax

(~‘

~2

\

) ~~ ¶ ) r ~

,

(1)

1~

where ~ is a parameter playing the role of the eigenvalue of the scattering problem and the dependent variables (“potentials”) r(x, t) and q(x, t) are assumed to obey the boundary conditions q r

X’

q

r

X’ XX’

q

XX’

...-~0 as IxI-~oo.

(2)

The following fairly general class of equations is known to be soluble in terms of the above scattering problem, a 3U~+~2(L,t)u=0u=, ~qjI,

(3)

where 03 is the Pauli matrix, and = ‘~‘

‘

E~ n0

~

(4 )‘~

is an entire function of ~ determining the linearized dispersion relation of eq. (3). The integro-differential 7

Volume 78A, number 1

PHYSICS LETTERS

operator L is defined by 1 (ax_2rI~

2i

—2qIq

With the above definition of {i~}one can show the iden-

2rIr \ _a~+2qIr)

(5)

in which I fX,,,dy is an integration operator. Newell [3] has shown that eq. (3) can be written as a Hamilto. nian equation [12], u~= J ~—H,

(6)

where J is a constant 2 X 2 matrixJ = —202, &/~udenotes the functional derivative and H is a Hamiltonian given by H= ~ a~(t)l~+1(r, q),

(7)

{I~}(n = 1,2,3,...) is the infinite number of polynomial conserved quantities associated with the scattering problem (1) and these are in involution to each other, namely, their Poisson brackets vanish [12] ~

~

=

f dx (~i~)

tity [3,12] a3lPu =

~~‘n+1

= 0,

(8)

[H,I~]= 0. (9) The explicit forms of the {I~}are obtained from the following set of Riccati2,equations [13—15] (10) 0 Y~ r 2i~y+ qy and —

(14)

,

which leads from eq. (3) to the Hamiltonian eq. (6). The “Generalized BT” (GBT) of Calogero and Degasperis is stated as follows: if two pairs of potentials u = (~)and u’ = (~‘)satisfying the boundary conditions (2), are related by f(A)a3(u’— u) + g(A)(u’+ u) = 0, (15) then the corresponding scattering data a~(~) and o’ t(~) are related by a remarkably simple formula

f(s) ~g(~) ± ~‘±(~) f(~)~g(~)a (~).

(16)

In eqs. (15) and (16) f(s) and g(~)are arbitrary entire functions of ~ and the integro-differential operator A is defined by 1 a~ r’Iq’ rlq r’Ir + rlr’ A=-( 2i \ q’Iq qlq’ —a~+ q’Ir’ + qlr)’ (17) in which I = f~dy is an integration operator. Since —

—

~ (i’m)

which implies, in particular,

—

7 July 1980

—

—

the transformed scattering data (16) have the same tdependence as the old ones, the transformed “potential” u’ obeys the same equation as u. Konopelchenko studied the group structure of the GBT in the scattering data space and showed that the group ofthe GBT a tensor productBT” of two groups, oneconsists is calledofthe “continuous and the other the “soliton BT”. The “continuous BT” is an infinite parametrical Abelian group whose effect on the scattering data is

(11) Namely, the solutions of eqs. (10) and (11) in terms of the asymptotic expansions

ct’~~

y =y(r, q) =

where {w~}are continuous parameters. It does not add any poles to zeroes to the scattering data. The corre-

0z~q+2i~z+rz2.

~

Y~(r,q), (12)

z

z(r, q) = ~

t~Z~(r, q),

= exp

(±

~(~)

~

,

(18)

n0

sponding infinitesimal transformation in the “potential” space is

n=1

~

givetheI~ I~(r,q) 8

fdxqY~(r, q) =

—f dxrZ~(r,q).

(13)

n0

e~a

1u,

(19)

3L’

where {e~}are infinitesimal and L is defined in eq. (5). The “soliton BT” (B(d)) is an Abelian discrete free gmup

Volume 78A, number I

PHYSICS LETTERS

consisting of elements B~.2and B~V2,i.e., N_

The “elementary BT” (22) reads explicitly,

N (r~— 2ir

2ii~r’ r’I(q’r’—qr) ~~q~+2iq’+2i’qq+qI(q’r’—qr)

1.

flB,~2fl

B(d)=

7 July 1980

~

(20) 2is called an “elementary BT” and its The element effect on the B~’ scatteringdata is to add a pole at i~to &,

—

—

—

Taking we get the scalar product of (27) with a vector

B~2~(~)-+a’)=(~~~)~ (21)

0F(~)

q’r’— qr-~(qr’)~,

In terms of the “potentials” it is expressed by eq. (15) with

which removes the integration operator from eq. (27);

f(A)A—o+1, g(A)=A—~—1. (22) 112is the inverse of B~/2. TheWe element B~the “potential” space that the two kinds show in of BT’s (19) and (22) are canonical transformations that map the Hamiltonian equation (6) to itself. First observe that we have an infinite number of conserved quantities (motion invariants) {I~}in involution to each other. Therefore we have infinite dimensional infinitesimal canonical transformations generated by ~1n~[16], namely

r~—2ir—2iflr —-~qr

u

U’

—

~

‘n+l =

—

~ [u,I,~

1],

(23)

which is identical to eq. (19) due to the identity (14). Because of eq. (9), it is obvious that the transformed Hamiltonian has the same form as the old one, i.e., H’(r’,q’)H(r,q), = H(r’, q’) +

[H, ‘n+l]

+

(28a)

qX + 2iq’ + 2inq +4~r’q2= 0.

(28b)

Here the boundary conditions (2) are used. The situation is similar to the one met in the expression L’1u, in which the integration operator does not appear in the final expression. In order to prove that the transformation (28) is canonical, we have to show that two one-forms

o

=

dx r~x)dq~x), O’=fdxr’~x)dq’@)~(29)

+ ~ir’q2dr’]

0

(24)

The finite transformation is obtained by a formal exponentiation u’=e1’~ue~ =u+[k,uj +-J~j-[K, [k,u]]+...,

7

differ only by an exact one-form dØ [16]. By operating d on eq. (28b) and integrating over x with the multiplication of r’, we get 2i0’

O(e2)

H(r’, q’) + 0(e2).

‘2

0,

see eq.(31) 6p~

fl

~

‘—f dx[(—~+2i~r’+ir’2q)dq —

Here we performed a partial integration on the term r’dq~and used the boundary conditions (2). Then substitution of eq. (28a) gives ~‘—~

=dØ,

fdxq2(x)r’2(x).

~

(25)

(30)

—~

Since ~ has no explicit time dependence, the two where k has the same form as the Hamiltonian (7) k

—

26

n =0

~0nmn+1

( )

and the commutators should be considered as Poisson brackets.

Hamiltonians are numerically equal: H’(r’, q’) = H(r, q) .

(31)

Next we show that the transformed Hamiltonian H’ has the same functional dependence on r’ and q’ as that ofH on r, q, i.e., 9

Volume 78A, number 1

PHYSICS LETTERS

7 July 1980

this scheme, is provided by the Hamiltonian equation (6). H’(r’, q’)

E cx~(t)I~~1(r’, q’)

(32)

.

n0

We observe that the transformations (28a) and (28b) are again Riccati equations [141 of the same form as eqs. (10) and (11),

( ~) (+~)~ 1

0= 0=

~

)

r

—

— q’+

—

2i~(_~~~r +q

/i / 2i~~-~-q) + r ,

/

higher (3 X 3 or NXN) scattering problems [18] will

, \2

be published elsewhere.

~

2

(33b)

.

Their solutions, satisfying the boundary conditions (2), are given by — ~2

1Y,~(r,q),

r’ = n1 ~ t~-’

Before closing this note, let us remark that the transformation (28) maps the “vacuum” r q = 0 to itself. This property is markedly different from that of usual BT’s for the reduced scattering problems such as the r = —1, r = ±q,r = ±q*cases. The discussion of the GBT for the above reduced problems and for the

! q = ~ ~—nZ (r’, q’), 2 ~=i (34)

The author is very grateful to R.K. Bullough for calling his attention to this subject. He thanks P. Oleson and P. Scharbach for careful reading of the manuscript. Thanks are due to the Danish Research Council for financial support. References

provided the series converge. Multiplying q and r’ and integrating over x, we get by noting eq. (13), —

~-fdxqr’

=

fdxqY~(r, q) =E~~z~(r,q)

~

n0~1

n =1

514.

and

[5] F. Calogero, Nuovo Cimento Lett. 14 (1975) 537. [6] R.K. Dodd and K.R. Bullougli, Phys. Lett. 62A (1977)

00

~J

dx qr =

—00

—

~

n1

70.

n~f dxr’Z~(r’,q’)

[7] 11.-H. Chen, Phys. Rev. Lett. 33(1974) 925; K. Konno and M. Wadati, Progr. Theor. Phys. 53 (1975)

--00

~~I~(r

1652.

q),

n1

which implies the desired relation I~(r’,q’) ‘I~(r,q)

.

(35)

As has already been remarked in ref. [10], this type of canonical transformation forms an Abelian discrete free group, i.e. (20). Therefore, we have proved that the “Generalized BT”s form a group of canonical transformations that keeps the Hamiltonian invariant (a symmetry group). The present discussion shows that the so-called one half of the Backlund transformation is a canonical transformation [9J*.The other half, in *

Steudel’s generalized BIcklund Transformations [17] for the s-G and the KdV are these canonical transformations.

10

[1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Phys. Rev. Lett. 31(1973) 125, Stud. in Appi. Math. 53 (1974) 249. [2] F. Calogero and A. Degasperis, Nuovo Cimento 32B (1976) 201; 39B (1977) 1. [3] AC. Newell, Proc. Roy. Soc. 365A (1979) 283. [4] R.K. Dodd and R.K. Bullough, Phys. Scripta 20 (1979)

[8] B.G. Konopeichenko, Phys. Lett. 74A (1979) 189. [9] H. Flaschka and D.W. Mc~ughlin,in “Bãcklund Transformations”, R.M. Miura, ed. Lecture Note in Mathematics 515 (Springer, Berlin 1976). [10] Y. Kodama and M. Wadati, Progr. Theor. Phys. 56 (1976) 342, 1740; Y. Kodama, Progr. Theor. Phys. 57(1977)1900.

[111V.E. Zakharov and A.B. Shabat, Soy. Phys. JETP ~

(1972) 62. [12] J.M. Alberty, T. Koikawa and R. Sasaki, Niels Bohr Institute preprint. [13] M. Wadati, H. Sanuki and K. Konno, Progr. Theor. Phys. 53 (1975) 419. [14] R. Sasaki, Nucl. Phys. B154 (1979) 343. [15] R. Sasaki, to be published in Proc. Roy. Soc. [16] H. Goldstein, Classical mechanics (Addison-Wesley, Reading, MA, 1950); H. Flanders, Differential forms (Academic Press, New York, 1963). [17] H. Steudel, Ann. Physik 32 (1975) 205, 445. [18] B.G. Konopelchenko, Phys. Lett. 75A (1980) 447.

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