Integrable coupled KdV systems

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 39, NUMBER 4

APRIL 1998

Integrable coupled KdV systems Metin Gu¨rses Department of Mathematics, Faculty of Sciences, Bilkent University, 06533 Ankara, Turkey

Atalay Karasu Department of Physics, Faculty of Arts and Sciences, Middle East Technical University, 06531 Ankara, Turkey

~Received 23 September 1997; accepted for publication 5 November 1997! We give the conditions for a system of N-coupled Korteweg de Vries ~KdV! type of equations to be integrable. We find the recursion operators of each subclass and give all examples for N52. © 1998 American Institute of Physics. @S0022-2488~98!03003-5#

I. INTRODUCTION

In Ref. 1 we gave an extension of the recently proposed Svinolupov Jordan KdV2,3 systems to a class of integrable multicomponent KdV systems and gave their recursion operators. This class is known as the degenerate subclass of the KdV system. In this work we will extend it to a more general KdV type of system equations containing both the degenerate and nondegenerate cases. This is a major step towards the complete classification of KdV systems. In addition we give a new extension of such a system of equations. Let us consider a system of N nonlinear equations of the form j q it 5b ij q xxx 1s ijk q j q kx ,

~1!

where i, j,k51,2,...,N, q i are functions depending on the variables x, and t, and b ij , and S ijk are constants. The purpose of this work is to find the conditions on these constants so that the equations in ~1! are integrable. In general the existence of infinitely many conserved quantities is admitted as the definition of integrability. This implies the existence of infinitely many generalized symmetries. In this work we assume the following definition for integrability: Definition: A system of equations is said to be integrable if it admits a recursion operator. The recursion operator ~if it exists! of the system of equations given in ~1!, in general, may take a very complicated form. Let the highest powers of the operators D and D 21 be respectively defined by m5degree of R and n5order nonlocality of R. In this work we are interested in a subclass of equations admitting a recursion operator with m52 and n51. Namely, it is of the form R ij 5b ij D 2 1a ijk q k 1c ijk q kx D 21 ,

~2!

where D is the total x derivative, D 21 is the inverse operator, and a ijk and c ijk are constants with s ijk 5a ik j 1c ijk .

~3!

Before starting to the classification of ~1! we recall a few fundamental properties of the recursion operator. An operator R ij is a recursion operator if it satisfies the following equation R ij,t 5F k8 i R kj 2R ik F 8j k ,

~4!

where F 8k i is the Fre´chet derivative of the system ~1!, which is given by

s it 5F 8j i s j , 0022-2488/98/39(4)/2103/9/$15.00

2103

~5! © 1998 American Institute of Physics

Downloaded 17 May 2011 to 139.179.14.104. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

2104

M. Gu¨rses and A. Karasu

J. Math. Phys., Vol. 39, No. 4, April 1998

where s i ’s are called the symmetries of the system ~1!. The condition ~3! implies that Eq. ~1! itself is assumed to be in the family of the hierarchy of equations ~or flows! q it 5 s in , n

where for all n50,1,..., s in denotes the symmetries of the integrable KdV system ~1!. For instance, for n50,1 we have respectively the classical symmetries s i0 5q ix and s i1 5q it . Equation ~5! is called the symmetry equation of ~1! with F 8j i 5b ij D 3 1s ijk q kx 1s ik j q k D.

~6!

Recursion operators are defined as operators mapping symmetries to symmetries, i.e., R ij s j 5l s i ,

~7!

where l is an arbitrary constant. Equations ~5! and ~7! imply ~4!. It is the equation ~4! which determines the constants a ijk and c ijk in terms of b ij and s ijk ’s. The same equation ~4! brings severe constraints on b ij and s ijk . We shall obtain a classification of ~1! based on the matrix b ij , ~i! ~ii!

det (b ij )50, det (b ij )Þ0,

and also we divide the classification procedure, for each class, into two parts where s ijk 5s ik j and s ijk Þs ik j . For the system of equations admitting a recursion operator we have the following proposition. Proposition 1: Let q i (t,x) be functions of t and x satisfying the N KdV equations ~1! and admitting a recursion operator R ij in ~2!. Then the constants b ij , s ijk , a ijk , and c ijk satisfy @in addition to the ~3!# the following relations: b kl c ijk 2b ik c kjl 50,

~8!

b kl a ijk 2b ik ~ a ijl 13c kjl 2s kjl ! 50,

~9!

b ik ~ 3a kjl 13c kjl 22s kjl 2s kl j ! 50,

~10!

c ijk s klm 2s ilk c kjm 50,

~11!

c ijk s klm 1c ijk s kml 2c kjm s ikl 2c kjl s ikm 50,

~12!

a ijk s klm 2s ikm a kjl 2s ilk a kjm 2s ilk c kjm 1s kjl c ikm 1a ikl s kjm 50,

~13!

c ikm ~ s kp j 2s kj p ! 50.

~14!

Now we will discuss the problem of classifying the integrable system of equations ~1! for the two exclusive cases depending upon the matrix b ij .

II. CLASSIFICATION FOR THE CLASS det „ b ij …50

In this subclass we assume the rank of the matrix b ij as N21. Investigation of the subclasses for other ranks of matrix b can be done similarly. For this case we may take b ij 5 d ij 2k i k j where k i is a unit vector, k i k i 51. In this work we use the Einstein convention, i.e., repeated indices are summed up from 1 to N . We find the following solution of ~8!–~14! for the parameters a ijk and c ijk for all N Proposition 2: Let k i be a constant unit vector and b ij 5 d ij 2k i k j . Then the complete solutions of the equations ~8!–~14! are given by Downloaded 17 May 2011 to 139.179.14.104. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions



a il j 5 23 s ijl 1 31 @ k i ~ k j a l 22k l n j ! 1 ~ 2ak i 1b i ! k l k j # , c ijl 5 13 s il j 2 31 @ k i ~ k l a j 1k j n l ! 1 ~ 2ak i 1b i ! k l k j # 1k i k l n j

2105

~15! ,

where n j 5k k k i s ijk 2ak j ,

a j 5k k k i s ik j 2nk j ,

b i 5k l k k s ilk 2nk i ,

~16!

a5k n a n , n5k i n i and s ijk ’s are subject to satisfy the following: c ijk s klm 5c kjm s ilk ,

~17!

s ijk 2s ik j 5k i @ k j ~ a k 2n k ! 2k k ~ a j 2n j !# ,

~18!

k n s nl j 5n l k j 1k l a j ,

~19!

k n s ijn 5k i n j 1b i k j ,

~20!

k n s in j 5 ~ n2a ! k i k j 1k i a j 1b i k j ,

~21!

n @~ a i 2n i ! k j 2 ~ a j 2n j ! k i # 50,

~22!

a i 5 r n i 1ak i ,

~23!

where r is a constant. At this point we will discuss the classification procedure with respect to the symbol s ijk whether it is symmetric or nonsymmetric with respect to its lower indices. i i A. The symmetric case, s jk 5 s kj

Among the constraints listed in Proposition 2 the one given in ~22! implies that s ijk ’s are symmetric if and only if a i 2n i 5 a k i where a 5a2n. There are two distinct cases depending on whether n50 or nÞ0. We shall give these two subcases as corollaries of the previous proposition. Corollary 1: Let s ijk 5s ik j and n50. Then we have the following solution for all N: a ik j 5 23 s ijk 1 31 @ k i ~ k j a k 22k k n j ! 1k k k j b i # ,

~24!

c ijk 5 13 s ijk 2 31 @ k i ~ k j a k 22k k n j ! 1k k k j b i # ,

~24!

a l 5n l ,n l 5k i k j s il j ,b i 5k j k l s il j .

~25!

where a50, r 51, and

The vector k i and s ijk are not arbitrary; they satisfy the following constraints: s ijk s klm 2s ilk s kjm 522 ~ k j n l 2k l n j !~ 2k i n m 1b i k m ! , k n s nl j 5n l k j 1k l n j ,

k n s ijn 5k i n j 1b i k j .

~26!

As an illustration we give an example for this case.1 A particular solution of the equations ~26! for N52 is b ij 5 d ij 2y i y j 5x i x j ,

~27!

a1 3 s ijk 5 a 1 x i x j x k 1 a 2 x i y j y k 1 y i ~ y j x k 1y k x j ! , 2 2 where i, j51,2 and


2106



x i 5 d i1 ,y i 5 d i2 ,

~28!

and n i 5 12 a 1 x i ,

k i 5y i ,

b i5 a 2x i.

~29!

Constants a ijk and c ijk appearing in the recursion operator are given by a ijk 5 a 1 x i x j x k 1 a 2 x i y j y k 1

a1 i y x jy k , 2

~30!

a1 a1 c ijk 5 x i x j x k 1 y i x j y k . 2 2 Taking a 1 52 and a 2 51 ~without loss of generality! we obtain the following coupled system u t 5u xxx 13uu x 1 vv x ,

v t5~ u v !x .

~31!

The above system was first introduced by Ito4 and the multi-Hamiltonian structure studied by Antonowicz and Fordy5 and by Olver and Rosenau.6 The recursion operator of this system is given by R5

S

D 2 12u1u x D 21 v 1 v xD

21

v

0

D

~32!

.

In Ref. 1 we have another example for N53. For the case nÞ0 for all N we have the following. Corollary 2: Let s ijk 5s ik j , nÞ0, and r 50. Then the solution given in Proposition 2 reduces to a il j 5 23 s ijl 1 31 ~ a22n ! k i k j k l ,

~33!

c ijl 5 13 s il j 2 31 ~ a22n ! k i k j k l , where n j 5nk j ,

a j 5ak j ,

b i 5ak i ,

~34!

and the constraint equations s ik j s kml 2s km j s ikl 50,

k n s nl j 5 ~ a1n ! k l k j ,

k n s ijn 5 ~ a1n ! k i k j .

~35!

For this case we point out that solution of ~33! and ~35! gives decoupled systems. i i B. The nonsymmetric case, s jk Þ s kj

In this case the constraints in Proposition 1, in particular ~22!, implies that we must have n50. In this case we have the following expressions for a il j and c il j for all N: a il j 5 23 s ijl 1 31 @ k i ~ k j a l 22k l n j ! 1 ~ 2ak i 1b i ! k l k j # , c ijl 5 13 s il j 2 31 @ k i ~ k l a j 1k j n l ! 1 ~ 2ak i 1b i ! k l k j # 1k i k l n j

~36! ,

where n j 5k k k i s ijk 2ak j , a j 5k

k

k i s ik j

,

b

i

~37!

5k l k k s ilk ,




2107

and the constraint equations among the parameters are c ijk c klm 2c ilk c kjm 50,

~38!

~ a ijk 2c ijk ! s klm 1 ~ c ikm 2a ikm ! s kl j 1 ~ s imk 2s ikm ! a kjl 1 ~ s kjm 2s km j ! a ikl 50.

~39!

For N52 we will give an example. Constants a ijk and c ijk appearing in the recursion operator are given by a ijk 5 a 1 x i x j x k 1 a 2 x i y j y k 1 a 3 y i x j y k , c ijk 5

~40!

a1 i x x jx k1 a 1y ix jy k , 2

where a 1 , a 2 , and a 3 are arbitrary constants and n i5 a 3x i ,

k i 5y i ,

b i5 a 2x i,

a i5 a 1x i .

~41!

We obtain the following coupled system: u t 5u xxx 13 a 1 uu x ,

~42!

v t5 a 3u xv 1 a 1u v x ,

which is equivalent to the symmetrically coupled KdV system7 u t 5u xxx 1 v xxx 16uu x 14u v x 12u x v ,

~43!

u t 5u xxx 1 v xxx 16 vv x 14 v u x 12 v x u, and the recursion operator for this integrable system of equations ~42! is R5

S

D 2 1 a 1 ~ 2u1u x D 21 !

a 3v 1 a 1v xD

21

0 0

D

.

~44!

III. CLASSIFICATION FOR THE CASE det „ b ij …Þ0

As in the degenerate case det (b ij )50, we have two subcases, symmetric and nonsymmetric. Before these we have the following proposition. Proposition 3: Let det (b ij )Þ0. Then the solution of equations given Proposition 1 is given as follows: a ijl 5 29 ~ s il j 12s ijl ! 2 91 C kl b im K mjk ,

~45!

c il j 5 19 ~ 7s il j 24s ijl ! 1 91 C kl b im K mjk , where K il j 5s il j 2s ijl

~46!

b kl c ijk 2b ik c kjl 50,

~47!

i l m l m 5C m i K r j 2C r K l j 2C j K rl 50,

~48!

c kjm K ilk 1c kjl K imk 50,

~49!

and the constraint equations


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c ijk s klm 2c kjm s ilk 50,

~50!

K kl j c ikm 50,

~51!

~ a ijk 2c ijk ! s klm 1 ~ c ikm 2a ikm ! s kl j 1K imk a kjl 1K kjm a ikl 50.

~52!

where C ir is the inverse of b ri . i i A. The nonsymmetric case, s jk Þ s kj

Equations ~47!–~52! define an over-determined system for the components of s ijk . Any solution of this system leads to the determination of the parameters a ijl and c ijl by ~45!. As an example we give the following, for N52, coupled system v t 5a v xxx 12b vv x ,

u t 54au xxx 12bu x v 1bu v x ,

~53!

where a and b are arbitrary constants. This system, under a change of variables, is equivalent to the KdV equation with the time evolution part of its Lax equation.8 The recursion operator of the system ~53! is

R5

S

4 ~ 3aD 2 1b v ! 3

b ~ 3u12u x D 21 ! 3

0

1 ~ 3aD 2 14b v 12b v x D 21 ! 3

D

.

~54!

Hence the KdV equation coupled to time evolution part of its Lax-pair is integrable and its recursion operator is given above. This is the only new example for N52 system. i i B. The symmetric case, s jk 5 s kj

When the symbol s ijk is symmetric with respect to subindices, the parameters K ijk vanish. Then the equations ~45!–~52! reduce to a ijk 5 23 s ijk ,

c ijk 5 13 s ijk ,

~55!

where the parameters b ik and s ijk satisfy b kl s ijk 2b ik s kjl 50,

~56!

s ijk s klm 2s ilk s kjm 50.

~57!

We shall not study this class in detail, because in Ref. 1 some examples of this class are given for N52. Here we give another example which correspond to the perturbation expansion of the KdV equation. Let q i 5 d i u, where i50,1,2,...,N, and u satisfies the KdV equation u t 5u xxx 16uu x . The q i ’s satisfy a system of KdV equations which belong to this subclass: q 0t 5q 0xxx 16q 0 q 0x ,

~58!

q 1t 5q 1xxx 16 ~ q 0 q 1 ! x ,

~59!

q 2t 5q 2xxx 16 @~ q 1 ! 2 1q 0 q 2 # x ,

~60!

••• .

~61!

N

q Nt 5q Nxxx 13

( @ d i~ q 0 ! 2 # x .

i51

~62!




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IV. FOKAS–LIU EXTENSION

The classification of the KdV system given in this work with respect to the symmetries can be easily extended to the following simple modification of ~1!: j 1s ijk q j q kx 1 x ij q xj , q it 5b ij q xxx

~63!

where x ij ’s are arbitrary constants. Equation ~63! without the last term will be called the principle part of that equation. Hence the equation ~1! we have studied so far is the principle part of its modification ~63!. We assume the existence of a recursion operator corresponding to the above system in the form R ij 5b ij D 2 1a ijk q k 1c ijk q kx D 21 1w ij ,

~64!

where w ij ’s are constants. We have the following proposition corresponding to the integrability of the above system. Proposition 4: The operator given in ~64! is the recursion operator of the KdV system ~63! if in addition to the equations listed in Proposition 1 @~8!–~14!# the following constraints on the constants x ij and w ij are satisfied:

x kl a ijk 2 x ik a kjl 2 x ik c kjl 1 x kj c ikl 2w kj s ikl 1w ik s kjl 50,

~65!

x kl c ijk 2 x ik c kjl 50,

~66!

x ik w kj 2 x kj w ik 50,

~67!

~ x kj 2w kj ! b ik 2 ~ x ik 2w ik ! b kj 50,

~68!

x kj a ikl 2 x ik a kjl 1w ik s kl j 2w kj s ilk 50.

~69!

Since the constraints ~8!–~14! are enough to determine the coefficients a ijk and c ijk with some constraints on the given constants b ij and s ijk , we have the following corollary of the above proposition. Corollary 3: The KdV system in ~63! is integrable if and only if its principle part is integrable. The principle part of ~63! is obtained by ignoring the last term ~the term with x ij ). Hence the proof of this corollary follows directly by observing that the constraints on the constants x ij and w ij listed in ~65!–~69! are independent of the constraints on the constants of the principle part listed in ~8!–~14!. Before the application of this corollary let us go back to the Proposition 4 and ask the question whether the KdV system ~63! admits a recursion operator with w ij 50. Corollary 4: The KdV system ~63! admits a recursion operator of the principle part. Then the last term x ij q xj is a symmetry of the principle part. If w ij 50, the above equations ~65!–~69! reduce to

x kl a ijk 2 x ik a kjl 2 x ik c kjl 1 x kj c ikl 50,

~70!

x kl s ijk 2 x ik s kjl 50,

~71!

x kj b ik 2 x ik b kj 50,

~72!

x kl c ijk 2 x ik c kjl 50.

~73!

In order that the term x ij q xj be a symmetry of the principle part, the constants x ij are subject to satisfy the following equations:

x kl s ijk 2 x ik s kjl 50,

~74!

x kj b ik 2 x ik b kj 50,

~75!


2110



x kl K ijk 1 x kj K ilk 50.

~76!

These equations simply follow from the set of equations ~70!–~73!, and hence the quantities s i 5 x ij q xj are symmetries of the principle part. By the application of Corollary 3, the full classification of the system ~63! with the recursion operator ~64! such that q it ~i.e., the system of equations themselves! belong to the symmetries of this system is possible. To each subclass given in the previous sections there exists a Fokas–Liu extension such that w ij 5 x ij with the following constraints:

x ik c kjl 2 x kl c ijk 50,

~77!

x kl a ijk 2 x kj a ilk 2 x ik ~ a kjl 2a kl j ! 50.

~78!

The above constraints are identically satisfied for the class @ det (b ij )Þ0, symmetrical case# when x ij 5 a d ij 1 b b ij . Hence the Fokas–Liu extension of the nondegenerate symmetrical case is straightforward with this choice of x ij . Here a and b are arbitrary constants. For the degenerate case the set of equations ~77! and ~78! must solved for a given principle part, b ij and a ijk . Recently a system of integrable KdV system with N52 has been introduced by Fokas and Liu.9 This system is a nice example for the application of Corollary 3. We shall give this system in its original form first and then simplify: u t 1 v x 1 ~ 3 b 1 12 b 4 ! b 3 uu x 1 ~ 21 b 1 b 4 ! b 3 ~ u v ! x 1 b 1 b 3 vv x 1 ~ b 1 1 b 4 ! b 2 u xxx 1 ~ 11 b 1 b 4 ! b 2 v xxx 50,

~79!

v t 1u x 1 ~ 213 b 1 b 4 ! b 3 vv x 1 ~ b 1 12 b 4 ! b 3 ~ u v ! x 1 b 1 b 3 b 4 uu x

1 ~ b 1 1 b 4 ! b 2 b 4 u xxx 1 ~ 11 b 1 b 4 ! b 2 b 4 v xxx 50,

~80!

where b 1 , b 2 , b 3 , and b 4 are arbitrary constants. The recursion operator of this system is given in Ref. 8. Consider now a linear transformation u5m 1 r1n 1 s,

v 5m 2 r1n 2 s,

~81!

where m 1 ,m 2 ,n 1 , and n 2 are constants, and s and r are new dynamical variables, q i 5(s,r). Choosing these constants properly, the Fokas–Liu system reduces to a simpler form r t 5 ~ rs ! x 1 a 1 r x 1 a 2 s x ,

~82!

s t 5 g 1 s xxx 1 g 2 rr x 13ss x 1 a 3 r x 1 a 4 s x , where we are not giving the coefficients a 1 , a 2 , a 3 , and a 4 in terms of the parameters of the original equation given above, because these expressions are quite lengthy. The only condition on the parameters a i is given by a 3 5 g 2 a 2 . This guarantees the integrability of the above system ~82!. On the other hand, the transformation parameters are given by m 2 52

n 1 52

b 11 b 4 m , 11 b 1 b 4 1 1

db3

,

n 25 b 4n 1 ,

~83!

d 5 b 1 ~ 11 b 24 ! 12 b 4 .

~84!

The principle part of the Fokas–Liu system ~82! is exactly the Ito system given in ~31! and hence the recursion operator is the sum of the one given in ~32! and x ij which are given by x 11 5 a 4 , x 12 5 a 3 , x 21 5 a 2 , x 22 5 a 1 . That is, R5

S

g 1 D 2 12s1s x D 21 1 a 4 r1r x D

21

1a2

g 2 r1 a 3 a1

D

.

~85!




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Another example was given very recently10 in a very different context for N52: u t 5 12 v xxx 12u v x 1u x v ,

v t 53 vv x 12au x .

~86!

The principle part of these equations is transformable to the Fuchssteiner system given in ~42!. Taking a 1 52, a 3 54, and scaling x and t properly we obtain ~without losing any generality! r t 52r xxx 16rr x ,

s t 52s x r14sr x .

~87!

The transformation between the principle part (a50) of ~86! and ~87! is simply given by u5m 1 s1 21 r, v 52r. Then the recursion operator of the system ~86! is given by R5

S

0 2a

1 2

D 2 12u1u x D 21 2 v 1 v x D 21

D

.

~88!

V. CONCLUSION

We have given a classification of a system of KdV equations with respect to the existence of a recursion operator. This is indeed a partial classification. Although we have found all conditions for each subclass, we have not presented them explicitly. We obtained three distinct subclasses for all values of N and gave the corresponding recursion operators. We also gave an extension of such systems by adding a linear term containing the first derivative of dynamical variables. We called such systems the Fokas–Liu extensions. We proved that these extended systems of KdV equations are also integrable if and only if their principle parts are integrable. For N52, we have given all subclassess explicitly. Among these the recursion operator of the KdV coupled to the time evolution part of its Lax pair seems to be new. Here we would like to add that when N52 recursion operators, including the Fokas–Liu extensions, are hereditary.11 Our classification crucially depends on the form of the recursion operator. The recursion operators used in this work were assumed to have degree two ~highest degree of the operator D in R) and nonlocality order one ~highest degree of the operator D 21 in R). The next work in this program should be the study on the classification problems with respect to the recursion operators with higher degree and higher nonlocalities. For instance, when N52, Hirota–Satsuma, Boussinesq, and Bogoyavlenskii coupled KdV equations admit recursion operators with m54 and n51.12 Hence these equations do not belong to our classification given in this work. ACKNOWLEDGMENTS

This work is partially supported by the Scientific and Technical Research Council of Turkey ~TUBITAK!. M. G. is a member Turkish Academy of Sciences ~TUBA!. M. Gu¨rses and A. Karasu, Phys. Lett. A 214, 21 ~1996!. S. L. Svinolupov, Teor. Mat. Fiz. 87, 391 ~1991!. 3 S. I. Svinolupov, Funct. Anal. Appl. 27, 257 ~1993!. 4 M. Ito, Phys. Lett. A 91, 335 ~1982!. 5 M. Antonowicz and A. P. Fordy, Physica D 28, 345 ~1987!. 6 P. J. Olver and P. Rosenau, Phys. Rev. E 53, 1900 ~1996!. 7 B. Fuchssteiner, Prog. Theor. Phys. 65, 861 ~1981!. 8 M. Ablowitz and H. Segur, Solutions and the Inverse Scattering Transform ~SIAM, Philadelphia, 1981!. 9 A. S. Fokas and Q. M. Liu, Phys. Rev. Lett. 77, 2347 ~1996!. 10 L. Bonora, Q. P. Liu, and C. S. Xiong, Commun. Math. Phys. 175, 177 ~1996!. 11 B. Fuchssteiner and A. S. Fokas, Physica D 4, 47 ~1981!. 12 M. Gu¨rses and A. Karasu, Coupled KdV Systems: Recursion Operators of Degree Four ~submitted!. 1 2
