30 S. Lou, J. Lin /Physws Letters A 185 (1994) 29-34

[0.~C~ ), 0 . , ~ ) ] = _~ 0.4 (.~,~ -./;l.~ ),

[0.~Cr, ), ~ ) l =cr~C~ Ii - . t ; . g ) ,

where the Lie product [A, B] is defined as

0 [A,B]--- ~ [ A ( F + e B ) - B ( F + e A ) ] ,

and

[0.~ Cr, ), 0.~(]i ) ] = 4 <. (?l.~ - f , ) i ) .

[ 0.~ U~ ), 0.~ Cr~ } l = } ~4 ( 3?, .1; -.t;.); } .

[ 0., C/; ), 0.~ U i ) ] = } ~ , { 3`);.L~ - 21;.~ ) .

(7)

( 8 )

{9)

lO)

I1)

12)

t f l t 1 • ~,

0.0 = - I F f ( t ) dt, 0.t = - ay.tT, 0.2 = a x j F - 4 × 3 X 2 ! . ~ , ~ F , 13)

0"3 = ~2 x Y :f F - 1 1 4 × 3 2 × 3 ! ]['y3F, 0.4(/) = f ( t ) f , . 4 X 3 3 X 4 ! (108x2.)'- 36xvZY+y4/")F . 14)

0.5 ( f ) = - 2 f ( t ) F~, + l~£VF~ 1 4 × 34X 5! ( 540X2) ' )"-- 60XV 3)'q-Y 5f ( 4 ) ) F , 15 )

or6 (/) = - f ( t l F , - 2 f .vFv- ~ ( x f - ~)'-,f ) k ,

1 + 4 X 35 X 6 ! (3240x3"~--1620x2y{'~'-f-90xy4f ~ 4 ' . - .v°F(5))F, 16)

satisfy the symmetry definition equation

(D{ + DxD, + 3 D 2 ) F . a = 0 . 17 )

The notat ions used here and those of Ref. [ 1 ] are different but equivalent. In this short Letter we would like to give the higher order symmetr ies for the BKPE ( 1 ). In order to get higher order symmetr ies of an integrable model one usually uses the mas te rsymmetry approach [2] or the recursion operator method [3]. However, both the mas te rsymmetry approach and the recursion operator method are quite difficult to apply in (2 + 1 )- dimensional cases• Recently, one of the authors (Lou) developed a simple method to get the generalized sym- metries for the usual KPE [4 ], the integrable dispersive long wave equations ( IDLWE) [ 5 ] and the Nizhn ik- Novikov-Vese lov equation ( N N V E ) [ 6 ]. Now using the method of Refs. [ 4 -6 ], we look for the symmetries of the BKPE which have the formal series

0.~(13= ~, f~"- l -k)0 .n[k] , 18) k=O

where 0.~ [k] ( k = 0, 1 .... ) are functions of x, y, u and its derivatives but are not l ime-dependent explicitly. The explicit t ime dependence of 0.n (f) has been separated out in f ~"- l k) Substituting (18) into (17) yields

f ~n-k) (Dx+DxD,+3D2)F .an[k]+4 ~. f ~ -k ) (Fan~[k] _0.~[k]F,.) = 0 . (19) k= 1 k=O

Becausef i s an arbitrary function of t, Eq. ( 1 9 ) should hold at any order of derivative o f f

Fa~x[0] - 0 . n [ 0 ] F v = 0 , (20)

F 0 . , x [ k ] - 0 . , [ k ] F x = - ( D ~ + D x D , + 3 D ~ ) F . 0 . , , [ k - I ] ( k = l , 2 .... ) . (21)

S. Lou, J. Lin / Physics Letters A 185 (1994) 29-34 31

Solving Eqs. (20) and (21) we have

an[Ol /F=gn(y) , (22)

an[k] = [ -FOx~ F-E(D4x + DxDt + 3D2)F" ]an[k- 1 ]

= [ - F O x ' F - 2 ( O 4 + DxDt + 3D2)F • ]kgn(y)r , (23)

with gn(Y) being an arbitrary function ofy. Now starting from any given function gn(Y), we can get a formal series symmetry,

~" f C n - l - k ) [ _ _ F O x l - z 4 2 an( f ) = F (Dx+DxDt+3Dy)V']kgn(y)F. (24) k=O

From the recursion relation (23), we know that if for a given integer k, say k=M,,, an[Mn] itself is a time independent symmetry, i.e.,

(D4x+DxDt+3D2)F'an[Mn] = 0 , (25)

then the formal series symmetry (24) becomes a truncated one. Now the question is what type of function gn (Y) will result in the truncated symmetries? In other words, we should fix gn(Y) and Mn such that

an [Mn] = [ - F O Z ~ F - 2 ( D 4 + D x D t + 3 D ~ ) F • ]Mngn(y)F (26)

satisfies Eq. ( 25 ). It is quite easy to see that ao ( f ) -a6 (f) shown by Eqs. ( 13 ) - ( 16 ) are just the first six nontriv- ial truncated symmetries with

1 gn(y) = 4 × 3 , _ t × n ! y n (n=0, 1, 2,..., 6 ) , (27)

Mo=0, M n = n - 1 ( n = l , 2 ..... 6 ) . (28)

Similar to the discussions for the usual KPE, IDLWE and NNVE cases [4-6 ], we know that if the next higher order truncated symmetry can be obtained than we can get infinitely many truncated symmetries by means of the mastersymmetry approach. Fortunately, further calculations show us that substituting gn(Y) given by Eq. (27) for n = 7 into Eq. (24) yields a further truncated symmetry with M7 = 7 - 1 = 6. The result reads

a7 (f) = ~FO x ' [ O yF - ' ( Ft - 2Fxxx) + 6F --2( -FxxFxy + FxFxxy) ] + !3je( 2xFy - yE t ) - ~ y(y2Fy + xyFx)

1 1 1 +3-~-4f'(xayF+4y3Fx) 1 6 2 × 4 ! f ( 4 ) x 2 y 3 F + 1 6 2 × 6 ! f ( 5 ) y S x F - 4×36×7!f{6)y7F. (29)

It is known that [ 7 ] if al and a2 are symmetries, then the commutator, [ a~, a2 ], defined by Eq. (12) is also a symmetry for the same model. Now using the symmetry a7 (f) and ao ( 1 ), we can get a set of infinitely many truncated symmetries recursively,

3 an+,(f) = ~ [aT(f~), an ( l ) ] (f-=A, n > 0 ) . (30)

However, to get the concrete expressions of an (J) for large n from Eq. (30) is still a difficult job because one has to fix some integrating functions directly from the symmetry definition equation ( 17 ). Actually, as in the cases of Refs. [ 4-6 ], we can conclude that the symmetries defined by Eq. (30) coincide with those of Eq. (24) for gn (Y) given by Eq. (27 ) for arbitrary positive integer n and Mn = n - 1. Finally, the generalized truncated sym- metries of the BKPE have the form

3 2 S. Lou, J. Lin / Physics Letters.4 185 (1994) 29 -34

t

r ~ 0 = - ~ F f f ( t ) dr.

1 n - I

Y J~"-~-k)[ - Ia 'O~II : -2(D4+D, Dt+3D2)F']ky"F ( n = l , 2 .... ) . (31) 4X 3"-1X n! k~o

In fact, if we require the symmetry to be analytic in )', then it is enough to consider the symmetries with g,,(y) given by Eq. (27) because an arbitrary Function can be expanded as a Taylor series. Though the arbitrary Func- tion g,(y) may be taken as a nonanalytic Function in the Formal series symmetry (24), we do not discuss the nonanalytic symmetries for g~(y) in this Letter because the symmetries are not truncated in that case.

In Eq. (31) (or Eq. (24) with (27)) , all the integral constants for 0; -~ should be taken as zero to get the linearly independent symmetries because the integral Function g,, (y) of Eq. (24) has been fixed as shown in Eq. (27) for all n> 0. Otherwise these symmetries will be linearly dependent.

After detailed calculations, we can see that the generalized symmetries of the BKPE shown by Eq. ( 31 ) con- stitute an infinite dimensional Lie algebra,

- - ! ( 7 . . . . , . . . . [d rm(~) ,~( f2) ] - -3 m+, ,_6[ (n -3 ) l ;12 - (m-3) f l f ' 2 ] re, n=0 ,1 ~

dr,,(/) = 0 ( r n< 0 ) . (32)

To see the correctness of (31 ) and (32) we discuss the relations between symmetries and algebras of the BKPE and those of the KPE instead of the concrete calculations. It is known that the BKPE 1 ) and the KPE

U,x= (6uux-Uxx.,.).,.- 3u,:,. (33)

are related by the transformation

u = - 2 ( l n F)~, . (34)

Then according to the transformation theorem [ 7 ] For the symmetries of two evolution equations related by a transformation, say, Eq. (34), the symmetries of the BKPE and those of the KPE are related by

crY= - 2 0 2 F - l r y F , (35)

where a ~ and O " r a r e symmetries of the BKPE and KPE, respectively. Substituting a,, (/) shown in Eq. (31 ) into ( 35 ), we reobtain the symmetries of the KPE ( 33 ) given in Ref. [ 4 ],

1 n + l

Kin( f ) - 2n!X3,+, ~ f '~"+~-k)(-O3,+6O,.u-3O~O 2 Ot)kv"=--2O~F ' " I ' -- . (Trl ] l ' = e x p ( 0~2u,/2) k = O

( m = n - 4 , Km(f)=O ( m < 0 ) ) . (36)

Correspondingly the algebra of K,~ (2') can also be reobtained from (32) by using the transformation theorem [7],

[Km(f~), K. (f2)] = [ -2Oe~F- 'o . , , ( / i ), - 2 O { F - ' G , , ( / 2 ) ] I,.=¢~, o.'~,,s.~,

2 2 - -1 = - ~ O x F G~,+~,_6[(n ' - -3)~.~-- (m' - -3)f l . f2] IF=~×p(_O~2~/2,

= l K , , + , _ 2 [ ( n + l ) A f 2 - - ( m + l ) A : f 2 ] ( m ' = m + 4 , n ' = n + 4 ) . (37)

From Eqs. (35 ) - (37) we know that when the symmetries and algebras of the BKPE are known, the correspond- ing symmetries and Lie algebras of the KPE will be determined uniquely by using transformation (35). How- ever, to get the symmetries and Lie algebras of the BKPE from those of the KPE is not a trivial problem. From Eq. (35 ) (or Eq. (36) ), one can see that four nontrivial symmetries of the BKPE, % (fl), a~ ( f ) , o2 (f) and a4 (/)

S. Lou, J. Lin /Physics Letters A 185 (1994) 29-34 33

are all correspondent to one trivial symmetry of the KPE. Generally the inverse transformation of (35) contains two arbitrary functions o f y and t,

t7 ~= -- I F[O x2tTu+ fl (y, t )X+ fE(y, t) ] . (38)

In order to fix two arbitrary functions f~ and f2 one has to substitute Eq. (38) into the symmetry definition equation (17) directly which is still a very difficult job for all the generalized symmetries of the BKPE. From the result of this paper given in Eq. (31 ) we know that

1 f ( n - l ) y n (39) f 2 = 2X 3n_ lxn! - -

1 fl = 2 x 3 n _ ~ x n ! f ( n - 2 ) [ - F O x ~ F - 2 ( D 4 + D x D t + 3 D 2 ) F ' ] Y '~

1 = -- 2X 3n_2 X ( n _ 2 ) ! f ( n - Z ) y n-2 . (40)

Finally we list some special subalgebras of (32 ) in addition to the known ones shown in ( 4 ) - ( 1 0 ) [ 1 ]. (i) The commuting algebra,

[¢rm(1), trn(1) ] = 0 . (41)

(ii) The Virasoro algebra I,

[ t rm( t ) ,an( t ) ]=~(m-n) t r ,~+n_6( t ) , m, n = 0 , 1,2 . . . . . (42)

(iii) The Virasoro algebra II,

[tr6(ts) , tr6(tr)]=~(s-r) tr6(tr+s-~) , r,s=O, +_1, _+2,.... (43)

(iv) w~ type algebra,

[trm( tS), an( tr) ]=~ [s(n-- 3 ) - - r ( m - - 3 ) ]am+n_6( U +~-l) ,

m , n = 0 , 1 , 2 .... , r , s = O , + l , + 2 . . . . . (44)

The woo algebra is found in various physical fields [8 ] such as in the sl (o9) Toda theory [ 9 ], w string and w gravity theories [ l 0 ], membrane theory [ 11 ] and integrable models [ 12,4-6 ].

Now a natural interesting problem which is worthy of further study appears: Are there generalized algebras, like those that appeared in the BKPE, KPE, IDLWE and NNVE, in other physical fields?

This work was supported by the Natural Science Foundat ion of Zhejiang Province and the National Natural Science Foundat ion of China. The author would like to thank Professors G. Ni, G. Huang, X. Hu, Q. Liu, W. Chen, H. Ruan, J. Zhang and X. Xu for their helpful discussions.

R e f e r e n c e s

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A.S. Fokas and B. Fuchssteiner, Phys. Len. A 86 ( 1981 ) 341. [ 3 ] P.M. Santini and A.S. Fokas, Commun. Math. Phys. 115 (1988) 375;

A.S. Fokas and P.M. Santini, Commun. Math. Phys. 116 (1988) 449; J. Math. Phys. 29 (1988) 1606. [4] S. Lou, Symmetries of the Kadomtsev-Petviashvili equation, to be published in J. Phys. A ( 1993 ). [ 5 ] S. Lou, Symmetries and algebras of the integrable dispersive long wave equation in (2 + 1 )-dimensional spaces, to be published in

J. Phys. A (1993).

34 S. Lou, J, Lin / Physics Letters A 185 (1994) 29-34

[ 6 ] S. Lou, Symmetry algebras of the potential Nizhnik-Novikov-Veselov model, to be published in J. Math. Phys. A ( 1993 ). [ 7 ] C.H. Gu, B. Guo, Y. Li, C. Cao, C. Tian, G. Tu, H. Hu, B. Guo and M. Ge, Soliton theory and its applications (Zhejiang Publishing

House of Science and Technology, Hangzhou, 1990) pp. 216-267. [8] E. Sezgin, preprint, IC/91/206, CTP TAMU-9/91. [9] Q.H. Park, Nucl. Phys. B 333 (1990) 267.

[ 10 ] E.S. Fradkin and M. Vasiliev, Ann. Phys. 177 ( 1987 ) 63. [ 11 ] J. Hoppe, MIT Ph.D. Thesis ( 1982 ); in: Proc. Int. Workshop on Constraints theory and relativistic dynamics, eds. G. Longhi and

L. Lusanna (World Scientific, Singapore, 1987). [ 12] Y. Yamagishi, Phys. Lett. B 259 ( 1991 ) 436.