On construction of recursion operators from Lax representation

Transcription

1 On construction of recursion operators from Lax representation Metin Gürses, Atalay Karasu, and Vladimir V. Sokolov Citation: J. Math. Phys. 40, 6473 (1999); doi: / View online: View Table of Contents: Published by the American Institute of Physics. Related Articles Conditional stability theorem for the one dimensional Klein-Gordon equation J. Math. Phys. 52, (2011) Solitary wave evolution in a magnetized inhomogeneous plasma under the effect of ionization Phys. Plasmas 18, (2011) Modulational instability of ion acoustic waves in e-p-i plasmas with electrons and positrons following a q- nonextensive distribution Phys. Plasmas 18, (2011) On polygonal relative equilibria in the N-vortex problem J. Math. Phys. 52, (2011) Global existence and blow-up phenomena for a weakly dissipative periodic 2-component Camassa-Holm system J. Math. Phys. 52, (2011) Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors:

2 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 12 DECEMBER 1999 On construction of recursion operators from Lax representation Metin Gürses a) Department of Mathematics, Faculty of Sciences, Bilkent University, Ankara Turkey Atalay Karasu Department of Physics, Faculty of Arts and Sciences, Middle East Technical University, Ankara Turkey Vladimir V. Sokolov Landau Institute, Moscow, Russia Received 28 June 1999; accepted for publication 16 August 1999 In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdVtype systems of integrable equations American Institute of Physics. S I. INTRODUCTION It is well known that most of the integrable nonlinear partial differential equations, u t F t,x,u,u x,...,u nx, 1 admit a Lax representation, L t A,L, 2 so that the inverse scattering method is applicable. The generalized symmetries 1 of 1 have also Lax representations with the same L operator, The recursion operator R, satisfying the equation see Ref. 2 L tn A n,l, n 1. 3 R t D F,R 0, 4 where D F is the Frechét derivative of the function F, generates symmetries of 1 starting from the simplest ones. In general, R is a nonlocal operator a pseudodifferential operator. The construction of the recursion operator of a given integrable system 1 is not an easy task. Several works are devoted to this subject. Among these works, most of the authors use 4 for the construction of the recursion operator. 3 8 There are several difficulties in this direct approach. The main problems are the choices of the order of R and the structure of the nonlocal terms. This is an approach having no relation with the Lax representation 2. On the other hand, some of the authors used Lax representation for this purpose. Most of these works are related to the squared eigenfunctions of the Lax operator 9 13 and are based on finding an eigenvalue equation for the squared eigenfunctions of the Lax operator. The operator corresponding to this eigenvalue equation turns out to be the adjoint of the recursion operator. a Electronic mail: gurses@fen.bilkent.edu.tr /99/40(12)/6473/18/$ American Institute of Physics

3 6474 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov There is an alternative use of the Lax representation to construct recursion operators. This approach is based on the explicit construction of the A n operators 3. It was first used by Symes, 14 Adler 15 see also Dorfman Fokas, 16 Fokas Gel fand 17 and Antonowicz Fordy. 18,19 Although these authors use the Lax representation in different ways, their approach is basically the same. Symes and Adler use the Gel fand Dikii 20 construction of the A n operators. On the other hand, Antonowicz Fordy determines these operators from integrability condition 3 and by using an ansatz for A n. Their basic aim is to determine the Hamiltonian operators 1 and 2 21 of the equations under consideration. The recursion operator is simply given by R Their approach is based on some explicit formulas for coefficients of the A n operator. This is necessary to find the Hamiltonian operators 1 and 2, and it seems that this approach is quite effective to determine the bi-hamiltonian structure for the simple cases but it becomes more complicated when the L-operator has a sophisticated structure. If one is interested only in the determination of the recursion operator R, we shall show in this work that it is possible to succeed this without any concrete information of the coefficients of A n operators. We use only an ansatz à PA R that relates A n operators for different n. Here P is some operator that commutes with the L operator and R is the remainder. We follow this basic idea, partially used by Symes, 14 Adler. 15 Shabat and Sokolov, 22 and establish an extremely simple, effective, and algorithmic method for the construction of recursion operators when the Lax representation 2 is given. 23 In the next section we consider the case where L is a scalar operator. We first consider the case where L is a differential operator and then the case where it is a pseudodifferential operator. In each case we present our method, discuss the reductions, and give examples for illustrations. In Sec. III we consider Lax operator taking values in a Lie algebra. We give our method both for the general case and also for the reductions. We give one example for each case in the text. Several additional examples are given in the Appendices A, B, and C corresponding to all different cases. II. SCALAR LAX REPRESENTATIONS First we consider equations with the scalar Lax representations of the form L t A,L, 5 where L is, in general, a pseudodifferential operator of order m and A is a differential operator whose coefficients are functions of x and t. The different choice of operators A for a given L leads to a hierarchy of nonlinear systems 3. It is well known that one can define operators A n by the following formula: 20 A n L n/m, 6 where L n/m is a pseudodifferential series of the form L n/m n v i D i and (L n/m n ) i 0 v i D i. Here v i are some concrete functions depending on the coefficients of L and D is the total derivative with respect to x. In Refs. 25 and 26 the relationships between the Kac Moody algebras and special types of scalar differential and pseudodifferential operators L were established. All corresponding integrable systems are Hamiltonian ones. For most of them a second Hamiltonian structure is not known up to now. In this section and Appendices A, B, and C we consider the simplest systems from Refs. 25 and 26 as examples and find their recursion operators. In the sequel these examples will be referred to as Drinfeld Sokolov DS systems. It is interesting to note that in all these examples the order of the recursion operator is equal to the Coexter number of the corresponding Kac Moody algebra.

4 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from A. Gel fand Dikii systems In this section we shall consider the case where L is a differential operator, L D m u m 2 D m 2 u 0, 7 where u i, i 0,1,...,m 2 are functions of x, t. In the framework of Ref. 25, this corresponds to the (1) Kac Moody algebras of the type A m 1. To show that 3 is equivalent to a system of (m 1) evolution equations with respect to u i one can use the following standard reasoning. Set L n/m L n/m L n/m, 8 where (L n/m ) is the differential part of the series L n/m and (L n/m ) is a series of order 1. Since L,L n/m 0 we have L n/m,l L, L n/m. 9 The left-hand side of 9 is a differential operator, but the right side is a series of order n 2. Thus, both sides of 3 are differential operators of order n 2 and it is equivalent to a system of evolution equations for the dependent variables u i, i 0,1,...,m 2. This system can be obtained by comparing the coefficients of D i, where 0,...,m 2 in 3. Since L (n m)/m LL n/m, then we have A m n LL n/m L L n/m L L n/m, 10 which leads directly to L tn m A n m,l LL tn L L n/m,l. 11 The above equation 11 has been given also by Symes 14 see also Adler s paper 15. In his work Symes expressed the coefficients of the both parts of 11, in a rather complicated way, in terms of some finite set of coefficients of the resolvent for an L operator. That allows him to express L tn m in terms of L tn. This relation gives directly the recursion operator. He gave explicit formulas for the cases m 2 and m 3. In this section we shall show that in order to construct the recursion operator it suffices to know only that L tn m LL tn R n,l. 12 Obviously, it follows from the following. Proposition 1: For any n, A n m LA n R n, 13 where R n is a differential operator of order m 1. Proof: The relation 13 coincides with 10 if we put R n L L n/m. 14 Since (L n/m ) is a series of order 1, then ord(r n ) m 1. Remark 1: It follows from the formula A n m L n/m L L n/m L L n/m L, 15 that

5 6476 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov A n m A n L R n, 16 and L tn m L tn L L,R n, 17 where R n is a differential operator of order m 1. To find the recursion operator we can simply equate the coefficients of different powers of D in 12. It is easy to see that in this comparison of the coefficients of D i, i 2m 2,...,m 1 we determine R n in terms of the coefficients of operators L and L tn. It is important that the resulting formulas turn out to be linear in the coefficients of L tn. The remaining coefficients of D i, i m 2,...,0 in 12 give us the relation u 0 u 0, 18 R u m 2 t n m u m 2 t n where R is a recursion operator. Instead of 12 one can use 17. The corresponding recursion operators coincide. Example 1. KdV equation: The KdV equation, u t 1 4 u 3x 6uu x, 19 has a Lax representation with L D 2 u, A L 3/2. 20 Since in this case L tn 2 u tn 2 u n 2 and L tn u tn u n, the main relation 12 takes the form u n 2 D 2 u u n R n,l, with R n a n D b n. Now if we equate successively to zero the coefficients of D 2, D, and D 0 equation, we obtain 21 in the above a n 1 2D 1 u n, b n 3 4u n, and u n 2 1 4D 2 u 1 2u x D 1 u n, that gives the standard recursion operator for the KdV equation, R 1 4D 2 u 1 2u x D In the same way one can find a recursion operator for the Boussinesq equation see Appendix A. B. Symmetric and skew-symmetric reductions of a differential Lax operator The standard reductions of the Gel fand Dikii systems are given by the conditions L* L or L* L. Here * denotes the adjoint operation defined as follows. Let L be a differential operator,

6 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from L a i D i. Its adjoint L* is given by L* ( D) i a i. It is easy to see that if L* L then m ord(l) must be an even integer. For the case L* L, it must be an odd integer. It is well known that for both reductions all possible A n are defined by 6, where n takes odd integer values. This condition provides that (A n )* A n that is necessary for 3 to be compatible. If L* L, the formula A n m (LL n/m ) (L (n m)/m ) gives a correct A n operator since n m is an odd integer. Thus, in this case Proposition 1 remains valid and the recursion operator can be found from 12 or 17. On the other hand, if L* L then both integers m and n are odd and hence their sum m n is an even integer. This means that (L (n m)/m ) cannot be taken as an A n operator. In this skew adjoint case we must take A n 2m L n 2m /m L 2 L n/m, to find the recursion operator. Following the proof of Proposition 1 we obtain Proposition 2. Proposition 2: If L* L then A n 2m L 2 A n R n, 23 where ord(r n ) 2 ord(l). It follows from 23 that L tn 2m L 2 L tn R n,l. 24 Remark 2: Instead of 23 we can use the ansatz A n 2m LA n L R n, 25 or A n 2m A n L 2 R n. 26 The recursion operators obtained by the utility of 23, 25, and 26 all coincide. In the works, 25,26 more general reductions L L were also considered. Here L KL*K 1, where K is a given differential operator, such that LK 1 is a differential operator. In this general reductions, as well, possible A n operators are given by 6, with n being an odd integer. Propositions 1 and 2 are valid for this general symmetric and skew-symmetric cases and hence one can use Eqs. 12, 24 accordingly to obtain the recursion operators. Example 2. Kupershmidt equation: This equation, u t u 5x 10uu 3x 25u x u 2x 20u 2 u x, 27 has the Lax pair In this case L* L; therefore we use Eq. 24 with L D 3 2uD u x, A L 5/3. 28 R n a n D 5 b n D 4 c n D 3 d n D 2 e n D f n. 29 By equating the coefficients of powers of D in 24, we obtain a n 2 3D 1 u n, b n 11 3 u n, c n uD 1 u n 73u n,x, d n u x D 1 u n 22uu n 27u n,2x,

7 6478 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov e n u 2x D 1 u n 2D 1 u 2x u n 40u 2 D 1 u n 8D 1 u 2 u n 134u n,3x 212uu n,x 184u x u n, f n,x u 4x D 1 u n 74u 3x u n 126u 2x u n,x 40uu 2x D 1 u n 40u x 2 D 1 u n 136u x u n,2x 27uu x u n 28u n,5x 64uu n,3x 16u 2 u n,x, and the recursion operator for the Kupershmidt equation: R D 6 12uD 4 36u x D 3 49u 2x 36u 2 D 2 5 7u 3x 24uu x D 13u 4x 82uu 2x 69u x 2 32u 3 2u x D 1 u 2x 4u 2 2 u 5x 10uu 3x 25u x u 2x 20u 2 u x D C. Pseudodifferential Lax operator In this section we generalize our scheme to the case of pseudodifferential Lax operators. The only difference is that in formulas like 13 and 23 the R n operator also becomes a pseudodifferential operator. It follows from these formulas that the structure of the nonlocal terms in R n is, in general, similar to the nonlocal terms in L since A n m and A n are differential operators. For skew-symmetric case, A n may be defined by either 23 or 25, or 26. In the pseudodifferential case they are not equivalent, in the sense that the nonlocal part of R n depends on which ansatz we choose. For illustration, let us consider the case L MD 1, where M is a differential operator. The following lemma shows that if L L or L L, where L DL*D 1, 31 then the formulas 13 and 25 are much suitable then 16, 23, and 26. Lemma: Let L L, where 1. Then where R n is defined by 13, and R n D m 1 a 0, for 1, 32 R n D 2m 1 a 1 D 1, for 1, 33 where R n is defined by 25. Proof: If L MD 1 then L L implies M* M. It is easy to show that (L 1/m ) L 1/m. Hence (L n/m ) L n/m for an odd integer n. Define now a series K n by L n/m DK n. It is easy to prove that K n * K n. Since K n (K n ) (K n ) and (K n )* K n, we have K n * K n, K n * K n. From the last formula it follows that ord(k n ) 2, which leads to an important result, This implies that A n L n/m D K n. LA n M K n 34 is a differential operator. Now using 34 in 13 and 25 for the cases 1 and 1, respectively, we find the ansatz for A n given by 32 and 33.

8 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from Example 3 ( 1): It is known that the KdV equation has, besides the standard Lax representation, the following Lax pair: L D 2 u D 1, A L The L operator satisfies the reduction L L. According to the formula 33 we have It follows from 25 that R n a n D b n c n D 1. a n D 1 u n, b n u n, c n u n,x ud 1 u n. The remaining equation in 25 gives the recursion operator R D 2 4u 2u x D Example 4 ( 1). DSIII system: The DSIII system 25,26 is given by u t u 3x 6uu x 6v x, v t 2v 3x 6uv x. 37 The nonlocal Lax representation for this system is L D 5 2uD 3 2D 3 u 2Dw 2wD D 1, A L 3/4, 38 where w v u 2x. Since L L we can use 32, which gives us R n a n D 3 b n D 2 c n D d n. 39 By equating the coefficients of the powers of D in 25, we obtain a n D 1 u n, b n 4u n, c n 1 2 6uD 1 u n 11u n,x 2D 1 uu n 2D 1 v n, d n,x 1 2 6u 2x D 1 u n 10u x u n 5u n,3x 4uu n,x 6v n,x. The recursion operator of the DSIII is found as with R R 0 0 R 0 1 R 1 0 R 1 1, 40 R 0 0 D 4 8uD 2 12u x D 8u 2x 16u 2 16v 2u 3x 12uu x 12v x D 1 4u x D 1 u, R D 2 8u 4u x D 1, R v x D 12v 2x 4v 3x 12uv x D 1 4v x D 1 u, 41 R 1 1 4D 4 16uD 2 8u x D 16v 4v x D 1. This recursion operator has recently been given in Ref. 6.

9 6480 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov III. MATRIX L OPERATOR OF THE FIRST ORDER In this section we demonstrate how our approach, given in the previous sections, can be generalized to the case where L is a matrix operator of the form A. General case L D x a q x,t. 42 Let us consider the Lax operator 42, where q and a belong to a Lie algebra g and is the spectral parameter. The constant element a is supposed to be such that g Ker ad a Im ad a. 43 First, let us recall the procedure 25 of constructing the A operators for the Lax operator 42. Proposition 3: There exist unique series, such that u u 1 1 u 2 2, u i Im ad a, 44 h h 0 h 1 1 h 2 2, h i Ker ad a, 45 e ad u L L u,l 1 2 u, u,l D x a h. 46 Let b be a constant element of g such that b,ker(ad a ) 0. It follows from 45 that b n,d x a h 0. Hence b,n,l 0, where b,n e ad u b n. 47 Then the corresponding A operator of the form A b,n b n a n 1 n 1 a 0, 48 is defined by the formula A b,n b,n, 49 where n i i n 0 i i. 50 According to 47, b,n 1 b,n. 51 Hence A b,n 1 b,n b,n b,n. 52 The last formula shows that A b,n 1 A b,n R n, R n g, 53 where R n does not depend on. Substituting 53 into the Lax equation L tn 1 A b,n 1,L, we get L tn 1 L tn R n,l. 54

10 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from Using the ansatz 54, one can easily find the corresponding recursion operator. Example 5: The system u t 1 2u xx u 2 v, v t 1 2v xx v 2 u, 55 is equivalent to the nonlinear Schrödinger equation, has a Lax operator L D u. 56 v 0 The Lie algebra g in this example coincides with sl(2). Using 54 with we find that R n a n b n c n a n, a n 1 2D 1 vu n uv n, b n 1 2u n, c n 1 2v n, and the recursion operator of the system 55 is given by R 2D ud 1 1 v ud 1 u u. vd 1 1 v 2D vd 1 B. Reductions in matrix case 57 In the general case considered in the previous section the A n operators belong to the Lie algebra, a i 0 a i i, a i g, Z, 58 that is a subalgebra of the Lie algebra, a a i i, i g, Z. 59 A standard reduction is defined by any automorphism of the Lie algebra g of finite order. Because Id, the eigenvalues of are i,i 0,..., 1, where is a primitive root of unity. Let g i be an eigenspace corresponding to eigenvalue i. Then the following reduction a j g i, where i j(mod ) in 58 and 59 is compatible with Eqs. 3. Note that according to this definition a g 1, and the potential q(x,t) in 42 belongs to g 0 or, the same, satisfies (q) q. It is easy to see that, to satisfy such a reduction, we must use the ansatz A b,n A b,n R n, 60 where R n r 1 1 r 0, r i g i. 61 Further generalizations are associated with modifications of sign in 50, which corresponds to the simplest decomposition of algebra a into the direct sum of two subalgebras,

11 6482 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov a a a, 62 where a is given by 58 and a 1 a i i, a i g. 63 The sign in 50 is the projection of onto a parallel to a. If we have a different decomposition 62, then the construction from Proposition 3 is also valid, but we have the following condition: R n a a, 64 instead of R n g. If we also have the reduction, we must use the most general ansatz 60, where R n a a. 65 Example 6: Let us consider the following equation: u t 1 4u xxx 3 8u xx u 3 8uu xx 3 8uu x u, 66 where u is a square matrix of arbitrary size, or more generally, u belongs to an arbitrary associative algebra K. This equation has a Lax representation with L D u Here 1 is the unity of K. The reduction 67 can be described as follows see Ref. 27. The Lie algebra g is the algebra of all 2 2 matrices with entries belonging to K. The automorphism is defined by X TXT 1, 68 where T Obviously 2 Id and eigenvalues of are 1 and 1. The corresponding eigenspaces are g 0 * 0, g 0 * 1 0 *, 69 * 0 and therefore the coefficients a i in 59 have the following structure: a 2 j * 0 0 *, a 2 j 1 0 * * The subalgebra a is given by 58, where the coefficients have the structure 70 and, additionally, The subalgebra a has the following form: a 0 *

12 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from a a i i, 71 where a 0 is of the form a 0 0, K. 0 The A operator for 66 is given by formula A ( a,3 ) see 49, where a , and means the projection onto a parallel to a. According to 65, R n is of the form It follows from R n a n 0 0 a n 2 0 b n c n 0 d n L tn 2 2 L tn R n,l, 73 that u n a n,x a n,u b n c n 0 c n b n a n,x 0, d n b n,x ub n 0, d n c n,x c n u 0, u n 2 d n,x d x,u. Finding a n, b n, c n, and d n from this system, we obtain the following recursion operator: R D ad u D R u 2D ad u 1 D L u D 2D ad u 1, 74 where R u and L u are the operators of right and left multiplications by u, respectively. Note that in the commutative case 66 coincides with the modified KdV equation. It is easy to verify that 74 becomes the standard recursion operator of a modified KdV equation. All factors in 74 have to be regarded as operators acting on a noncommutative polynomial depending on u,u x,u xx,.... IV. CONCLUSION In this work we devoted ourselves in the construction of recursion operators when the Lax representation is given. We have shown that our approach can be easily generalized to all cases where the L operator is a polynomial of. It would be interesting to generalize it for the cases of more complicated dependence of L as well as for the cases of 2 1-dimensional equations, Toda-type lattices, and ordinary differential equations. ACKNOWLEDGMENTS We would like to thank Dr. Jing Ping Wang for reading the manuscript and pointing out some misprints. This work is partially supported by the Scientific and Technical Research Council of Turkey TUBITAK. M. G. is a member of the Turkish Academy of Sciences TUBA. V.S. is supported by Russian Foundation of Basic Researches RFBR Grant No and INTAS.

13 6484 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov APPENDIX A: EXAMPLE TO SEC. II A The Boussinesq equation, u tt 1 3 u 4x 2 u 2 2x, A1 can be expressed in the form of a pair of first-order evolution equations, u t v x, v t 1 3 u 3x 8uu x. A2 This system has a Lax pair, L D 3 2uD u x v, A L 2/3. A3 To construct the recursion operator for this system, we use Eq. 12 with the differential operator, R n a n D 2 b n D c n. By equating the coefficients of the powers of D in 12, wefind a n 2 3D 1 u n, b n 1 3 5u n D 1 v n, c n 1 9 6v n 8uD 1 u n 10u n,x, and after that we obtain the recursion operator of the form 40 for A2 with R 0 0 3v 2v x D 1, R 1 0 D 2 2u u x D 1, R D ud 2 5u x D 3u 2x 16 3 u 2 2 3u 3x 16 3 uu x D 1, A4 R 1 1 3v v x D 1. APPENDIX B: EXAMPLES TO SEC. II B 1. Sawada Kotera equation The Lax pair for the Sawada Kotera equation, 28 u t u 5x 5uu 3x 5u x u 2x 5u 2 u x, B1 is given by L D 3 ud, A L 5/3. B2 In this example, L L, where L D 1 L*D and L is skew-symmetric, then we use 24. The operator R n has the same form as 29, with the coefficients given by a n 1 3D 1 u n, b n 5 3u n, c n 1 9 5uD 1 u n 29u n,x, d n 1 9 5u x D 1 u n 14uu n 26u n,2x,

14 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from e n u 2x D 1 u n 2D 1 u 2x u n D 1 u 2 u n 5u 2 D 1 u n 28u n,3x 32uu n,x 32u x u n, f n 0. The recursion operator is given as R D 6 6uD 4 9u x D 3 9u 2 11u 2x D 2 10u 3x 21uu x D 5u 4x 16uu 2x 6u x 2 4u 3 u 5x 5uu 3x 5u x u 2x 5u 2 u x D 1 u x D 1 u 2 2u 2x. B3 2. DSI system The DSI system, 25,26 u t 3vv x, v t 2v 3x 2uv x vu x, B4 has a Lax representation with L D 3 u v D 1 2 u v x D 3 u v D 1 2 u v x, A L 1/2. B5 Here R n is a differential operator of order 5, and since L is symmetric we again use Eq. 12. The expressions for the coefficients of the operator R n are very long and complicated. Hence we do not display them here. We find that the recursion operator R of this system is of the form 40, where R 0 0 4D 6 24uD 4 27u x D u 2x 18u 2 42v 2 D u 3x 12uu x 30vv x D 26u 4x 82uu 2x 69u x 2 222vv x 141v x 2 16u 3 48v 2 u 2 2u 5x 10uu 3x 25u x u 2x 10u 2 u x 15v 2 u x 30vv 3x 45v x v 2x 30uvv x D 1 2u x D 1 3v 2 2u 2 u 2x, R vD 4 204vD v 2x 32uv D vu x 7v 3x 22uv x D 6 13vu 2x 10u x v x v 4x 5uv 2x 4vu 2 12v 3 108vv x D 1 v 2u x D 1 6uv 9v 2x, B6 R vD 4 268v x D v 2x 32uv D vu x 219v 3x 106uv x D 2 27vu 2x 92u x v x 99v ax 99uv 2x 4vu 2 12v vu 3x 35u 2x v x 45u x v 2x 10uvu x 18v 5x 30uv 3x 10u 2 v x 15v 2 v x D 1 2v x D 1 3v 2 2u 2 u 2x, R D 6 216uD 4 432u x D u 2x 18u 2 22v 2 D u 3x 36uu x 70vv x D 3 18u 4x 18uu 2x 9u x 2 98vv 2x 67v x 2 32uv v 3x 2v x u vu x D 1 v 2v x D 1 6uv 9v 2x. 3. DSII system The DSII system, 25,26

15 6486 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov u t 3v x, v t 2 v 3x uv x vu x, B7 has a Lax representation with L D 5 ud 3 D 3 u v 1 2u 2 D D v 1 2u 2 D, A L 1/2. B8 Since L is symmetric we again use Eq. 12. In this case the operator R n is given as follows: R n a n D 5 b n D 4 c n D 3 d n D 2 e n D, B9 where a n 1 3D 1 u n, b n 5 3u n, c n 1 9 5uD 1 u n 3D 1 v n 29u n,x, d n 1 9 5u x D 1 u n 26u n,2x 14uu n 12v n, e n u 2x u 2 3v D 1 u n 3D 1 vu n uv n 9uD 1 v n 2D 1 u 2x u n 1 2u 2 u n 54u x u n 28u n,3x 32 uu n,x u n u x 42v n,x. The recursion operator 40 for the system can be found as 29 R 0 0 D 6 6uD 4 9u x D 3 11u 2x 9u 2 42v D 2 10u 3x 21uu x 30v x D 5u 4x 16uu 2x 6u x 2 60v 2x 4u 3 24vu u 5x 5uu 3x 5u x u 2x 5u 2 u x 15vu x 15v 3x 15uv x D 1 u x D 1 2u 2x u 2 3v, R D 4 48uD 2 87u x D 6 7u 2x u 2 6v 27v x D 1 3u x D 1 u, B10 R vD 4 106v x D 3 165v 2x 32uv D 2 54vu x 132v 3x 74v x u D 30vu 2x 79u x v x 54v 4x 57uv 2x 4u 2 v 24v 2 10vu 3x 25v x u 2x 30u x v 2x 10uvu x 9v 5x 15uv 3x 5u 2 v x 15vv x D 1 v x D 1 3v u 2 2u 2x, R D 6 54uD 4 135u x D u 2x 9u 2 22v D u 3x 27uu x 28v x D 3 9u 4x 9uu 2x 9u x 2 21v 2x 16vu 18 v 3x u x v v x u D 1 3v x D 1 u. 4. DSIV system The DSIV system, 25,26 which is also known as the Hirota Satsuma system, 30,31 u t 1 2u 3x 3uu x 6vv x, v t v 3x 3uv x, B11 has Lax representation with L D 2 u v D 2 u v, A L 3/4. B12

16 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from Since the operator L is symmetric we use Eq. 12. In this case the operator R n has the same form as 39, with coefficients given by a n 1 2D 1 u n, b n 7 4u n 1 2v n, c n 1 8 6uD 1 u n 2D 1 uu n 4D 1 vv n 17u n,x 12v n,x, d n,x u 2x D 1 u n 12v 2x D 1 u n 30u x u n 8u x v n 24uu n,x 15u n,3x 12v x v n 8uv n,x 20vv n,x 28v n,3x. The recursion operator 40 for the given system is R D 4 2uD 2 3u x D 2u 2x 4 u 2 v 2 3uu x 6vv x 1 2u 3x D 1 u x D 1 u, R 1 0 5vD 2 4v x D v 2x 4uv 2u x D 1 v, R v x D 3v 2x v 3x 3uv x D 1 v x D 1 u, B13 R 1 1 D 4 4uD 2 2u x D 4v 2 2v x D 1 v. 5. N 3 Hirota Satsuma system This system is given by 29 u t 1 4u 3x 3uu x 3 v 2 w x, v t 1 2v 3x 3uv x, B14 w t 1 2w 3x 3uw x. This is an example for the N 3 system that covers some other N 2 systems as special cases. For instance, letting w 0, we get DSIV and letting v 0 we get DSIII systems. The corresponding Lax pair is L D 2 2u 2v D 2 2u 2v 4w, A L 3/4. B15 In this case the operator L is symmetric and hence R n has the same form as 39, with the coefficients a n D 1 u n, b n 7 2u n v n, c n uD 1 u n 4D 1 uu n w n 2vv n 17u n,x 12v n,x, d n,x u 2x D 1 u n 24v 2x D 1 u n 60u x u n 16u x v n 15u n,3x 48uu n,x 24v x u n 40v x v n 20v n,3x 16vv n,x 20w n,x. The recursion operator is given by R R R 0 R 0 2 R 1 0 R 1 1 R 1 2 R 2 0 R 2 1 R 2 2, B16 where

17 6488 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov R 0 0 4D 1 4 4uD 2 6u x D 4 u 2x 4u 2 4v 2 4w 4 4u 1 3x 3uu x 6vv x 3w x D 1 4u x D 1 u, R vD 2 4v x D v 2x 8uv 4u x D 1 v, R 0 2 5D 2 8u 4u x D 1, R 1 0 5u x D 6v 2x 2 v 3x 6v x u D 1 4v x D 1 u, R 1 1 D 4 8uD 2 4u x D 8 8w 2v 2 8v x D 1 v 8D 1 w x, R v x D 1 2D 1 v x, R 2 0 5w x D 6w 2x 2 v 3x 6w x u D 1 4w x D 1 u. R vD 1 w x 8w x D 1 v, R 2 2 D 4 8uD 2 4u x D 16 w v 2 4w x D 1 16vD 1 v x. B17 APPENDIX C: EXAMPLES TO SEC. III 1. Non-Abelian Schrödinger equation This is the system given by u t 1 2u xx uvu, v t 1 2v xx vuv, C1 where u and v belong to K see Example 6 for the notations. The Lax operator of C1 is given by L D u. C2 v 0 The corresponding formula 54 reduces to 0 u n 1 v n 1 0 u n 0 v n R 0 n,l, C3 where R n a n b n c n a n. C4 The formula C3 gives us both a n,b n,c n and the recursion operator R. They are given by a n 1 2D 1 u n v uv n, b n 2u 1 n, c n 1 2u n, C5 R 1 2 D R ud 1 R v L u D 1 L v R u D 1 L u L u D 1 R u L v D 1 R v R v D 1 L v D R v D 1 R u L v D 1 L u. C6

18 J. Math. Phys., Vol. 40, No. 12, December 1999 On construction of recursion operators from Non-Abelian modified KdV equation The standard non-abelian modified KdV equation is given by u t 1 4u xxx 3 4u x u 2 3 4u 2 u x. C7 The Lax representation of this equation is given L D u 0 0 u. C8 The recursion operator R can be found from 60 and 61. In our case the automorphism is the same as in Example 6, and formulas 60 and 61 give us where 0 u n 1 v n u n v n 0 R n,l, C9 Using C9 we find a n,b n,c n,d n from the following: R n 0 a n b n c n d n. C10 b n a n u n, a n,x a n u ua n c n d n 0, b n,x b n u ub n d n c n 0, d n,x c n,x c n d n,u, The resulting recursion operator is given by u n 1 d n,x d n,u. R 1 4 D ad u D 1 ad u D L u R u D 1 L u R u. C11 1 P. J. Oliver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 2nd ed. Springer- Verlag, New York, 1993, Vol P. J. Olver, J. Math. Phys. 18, M. Gürses and A. Karasu, J. Math. Phys. 36, M. Gürses and A. Karasu, Phys. Lett. A 214, M. Gürses and A. Karasu, J. Math. Phys. 39, M. Gürses and A. Karasu, Phys. Lett. A 251, S. I. Svinolupov, Theor. Math. Phys. 87, J. Krasil shchik, Contemp. Math. 219, A. S. Fokas and R. L. Anderson, J. Math. Phys. 23, A. S. Fokas, Stud. Appl. Math. 77, A. P. Fordy and J. Gibbons, J. Math. Phys. 22, P. M. Santini and A. S. Fokas, Commun. Math. Phys. 115, A. S. Fokas and P. M. Santini, Commun. Math. Phys. 116, W. Symes, J. Math. Phys. 20, M. Adler, Invent. Math. 50, I. Ya. Dorfman and A. S. Fokas, J. Math. Phys. 33, A. S. Fokas and I. M. Gel fand, in Important Developments in Soliton Theory, Springer Series in Nonlinear Dynamics, edited by A. S. Fokas and V. E. Zakharov Springer-Verlag, Berlin, 1993, pp M. Antonowicz and A. P. Fordy, in Nonlinear Evolution Equations and Dynamical Systems (NEEDS 87), edited by J. Leon World Scientific, Singapore, 1988, pp M. Antonowicz and A. P. Fordy, in Soliton Theory: A Survey of Results, edited by A. P. Fordy Manchester University Press, Manchester, England, See also the related references therein. 20 I. M. Gel fand and L. A. Dikii, Funct. Anal. Appl. 10, F. Magri, J. Math. Phys. 19, V. V. Sokolov, Sov. Math. Dokl. 30,

19 6490 J. Math. Phys., Vol. 40, No. 12, December 1999 Gurses, Karasu, and Sokolov 23 Here we note that in 1980 that Shabat and Sokolov independently found the recursion operator for the Sawada Kotera equation. This result was published in Ref. 24. In Ref. 22, Sokolov found the recursion operator for the Krichever Novikov equation. 24 N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics Reidel, Boston, V. G. Drinfeld and V. V. Sokolov, J. Sov. Math. 30, V. G. Drinfeld and V. V. Sokolov, Proc. Sobolev Sem. Novosibirsk 2, in Russian. 27 I. Z. Golubchik and V. V. Sokolov, Theor. Math. Phys. 112, K. Sawada and T. Kotera, Prog. Theor. Phys. 51, J. Springael, X. B. Hu, and I. Loris, J. Phys. Soc. Jpn. 65, J. Satsuma and R. Hirota, J. Phys. Soc. Jpn. 51, G. Wilson, Phys. Lett. A 89,