L]~TT]~tr AL ~UOVO CIMENTO VOL. 41, ~. 17 22 Dicembre 1984

Infinitesimal B~icklund Transformation and Conservation Laws

for Nonautonomous Systems,

G. M~_HATO and A. RoY CHOWDHURY

High Energy Physics Div i s ion Department o] Physics, Jadavpnr University - Calcq~tta 700 032, Ind ia

(ricevuto il 2 Agosto 1984)

PACS. 02.30. - Funct ion theory, analysis.

S~mmary . - We have observed that it is possible to extend the concept of infinitesimal B~cklund transformation of Steudel to the case of nonautonomous systems. We have discussed the si tuat ion with cylindrical KdV as an example. The interestisg point to note is that, though the BT for CKdV does not possess the usual property of generating one soliton in a single operation, yet it generates an infinite number of symmetries, which are in accord with those obtained through the expansion of the resolvent of the linear operator associated with the equation.

The analysis of symmetries of nonlinear equations plays a central role in exposing characteristic features of completely integrable systems (~). In an interesting com- municat ion it was shown by STEUD~L (3) that these symmetries could be obtained by defining what he calls infinitesimal Bi~cklund transformation, and Noether's theorem can be applied to the proper Lagrangian system for getting the conservation laws cor- responding to each of these symmetry generator. We here have extended the idea of Steudel to nonautonomous systems of cylindrical KdV equation to obtain the heirarchy of symmetries. But, since it is not possible to have a Noether-like theorem for non- Lagrangian systems, we had to take recourse to the alternative route of getting conserva- t ion laws via the expansion of the resolvent (3) of the linear operator in the spectral parameter. Then we observe that the symmetries generated and the conserved quan- tities are connected by the sympletic operator, verifying the correctness of the generators.

The B~cklund transformation for the cylindrical KdV equation is writ ten as

X + ~ ~ 1 (1) (Z'+ Z)x-- 6y 2 ( z ' - z)~'

(1) F. MAGRI: Nonlinear evolution equations and dynamical systems, in Proceedings of the Meeting Held at Lecce (Springer Verlag, 1980). (2) IX. STEUDEL: .Ann. Phys., 32, 205, 459 (1975). (a) I. GEL'FAND and L. A. DIKII: Usp. Mat. Nauk, 30, 67 (1975).

567

5 6 8 G. 1VfAttATO a n d A. ~ o v CHOWDttURY

w h e r e z' a n d z are t h e two so lu t ion of C K d V w r i t t e n as

q~

(2) u t + 6uu~ + u~z x + 2~ = 0

a n d we h a v e se t

qJJ= Z x .

I n eq. (1) x a n d y d e n o t e t h e space a n d t i m e va r i ab le , r e spec t ive ly . I n a n a l o g y w i t h t h e idea of S t eude l we define a n d d e n o t e t h e i n f i n i t e s ima l t r ans fo r -

m a t i o n t h r o u g h

(3) V '

w h e r e z is t h e p a r a m e t e r of t r a n s f o r m a t i o n . S u b s t i t u t i n g in eq. .(1) , we h a v e

(4) + . . . ;

e q u a t i n g coefficients of d i f fe ren t powers of ~, we get success ive ly

26 A 1 - - 2 ~ / 3 y ~ Zx, 2 V ~ A2 = - - �89 + 6yz~x,

(5) V ~ A s = - - 6 V ' ~ z ~ - 6 ~ / 3 ~ z~ + ~ / ~ z~ - - - - ,

24y�89

26

2~/@"

L e t us d e n o t e t h e B ~ e k l u n d t r a n s f o r m a t i o n (1) as

(6) z ' : B ~ z

a n d cons ide r a n o t h e r so lu t i on Z" g e n e r a t e d v i a a B T w i t h p a r a m e t e r ~ + e, that is

(7)

T h e i n v e r s e B/~eklund is g i v e n as

so t h a t we o b t a i n

(s)

z ~ B ~ + e z .

z ~ B _ c J ,

A t t h i s p o i n t i t is w o r t h m e n t i o n i n g t h a t , in b o t h t h e p a p e r s (a,~) whe re t h e B T (1) is

(4) J . J . C . 1NvIMMO a nd D. G. CRIGHTON: P h y s . Zett . _4, 82, 2 i l (1~81). (5) ],,v. C. FREEMAN, G. ttORROCKS a nd P. WILKINSON: P h y s . Left . _4, 81, 305 (1981).

I N F I N I T E S I M A L B A C K L U N D T R A N S F O l l M A T I O N E T C . 569

deduced, it has been shown that this BT does not share the usual property of generating one soliton solution in a single operation and for this it was necessary to define a double Bs transformation. But our calculation here indicates that, in spite of this un- usual feature, the BT defined by eq. (1), may be interpreted as a contact transforma- tion generating symmetries, as /ormally both z and z' satisfy the same CKdV equation. Now, rewriting eq. (1) for the case of (7), we get

1 x + (~+e)~ (9) 2 ( z " - - z ) ~ - 6y ( z " + z)~,

where z " = B~,+eB_~z = z + sOc~ differentiating (9) with respect to s and setting s = O,

(lo) (dq Cq/

which yield the following equation for 0~:

2c~ dOa (11) 6y dx -- (z ' --z)O~

We now try to solve (11) in the form

co o = F., o~(z)

equating powers of ~, we get

(12)

2 1 1~ t?k = - - ~ 1 - - - - t ? k + i - - ~ AmOk-~

o r

~n

From which we can write down the following explicite structures:

1 0 0 - ~ / @ ,

X

0~ = 2 ~ / @ ~ 2 ~ / @ '

( and so on, but 01= 0 a = 0 5 = O.

I t is interesting to compare these results with a straightforward calculation of ~he Lie-B/~eklund symmetries reported in ref. (4). Since there is no Noethers theorem [or a nonautonomous system, we consider the linear operator of eq. (2).

I t is writ ten as

t L ~ = ~ ,

~ 7 0 G. NIAHATO and i . RoY C H O W D H U R Y

where5

(13) /5 = ~x 2 +

Now the resolvent of this system is really y~ which obeys the equation

(14) /~ q~ x \ [ ~ 1 )

Let us now make an expansion of this resolvent in inverse powers of 4, that is

cc

(15) -] = X / n , ~ - n , ~ 0

which yields

( ~ (16) ~J,c+1= ~-3 \12t 6] ~ + ~- ~ /~'

which yields successively the expansion coefficients and is nothing but the conserved quantities.

I t is then an important point to observe that there is an intimate relation between the symmetry generators and conservation laws through the symplectic operators which define the hierarchy of I/amiltonian structure of the system. Lastly we mention that the operator occm'ring in eq. (14) is seen to be proportional to the hereditary operator which connects the symmetry generators obtained in eq. (12)