Recursion operators of some equations of hydrodynamic type

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1 Recursion operators of some equations of hydrodynamic type M. Gürses and K. Zheltukhin Citation: J. Math. Phys. 4, 309 (00); doi: 0.063/ View online: View Table of Contents: Published by the American Institute of Physics. Additional information on J. Math. Phys. Journal Homepage: Journal Information: Top downloads: Information for Authors: Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

2 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 4, NUMBER 3 MARCH 00 Recursion operators of some equations of hydrodynamic type M. Gürses a) and K. Zheltukhin Department of Mathematics, Faculty of Sciences, Bilkent University, Ankara Turkey Received 7 August 000; accepted for publication 8 November 000 We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give several examples for N and N3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators. We also discuss a reduction of N systems to N systems of some new equations of hydrodynamic type. 00 American Institute of Physics. DOI: 0.063/ I. INTRODUCTION Most of the integrable nonlinear partial differential equations admit Lax representations, L t A,L, where L is a pseudo-differential operator of order m and A is a pseudo-differential operator. Recently we established a new method for such integrable equations to construct their recursion operators. This method uses the hierarchy of equations, L tn A n,l, and the Gel fand Dikkii construction of the A n -operators. Defining an operator R n in the form A n LA nm R n, 3 one then obtains relations among the hierarchies, L tn LL tnm R n ;L. 4 This equation allows to find L tn in terms of L tnm. It is important to note that one does not need to know the exact form of A n. For further details of the method see Ref.. In Ref. we introduced a direct method to determine a recursion operator of a system of evolution equations when its Lax representation is known. It has no direct reference to the Hamiltonian operators. Hence one may be able to determine the recursion operators when any one of the Hamiltonian operators are degenerate. In the same paper we gave several applications of the method. In all these examples we have considered the Lax representation is given either in a pseudo-differential operator or in matrix form taking values in some lower dimensional Lie algebras. We call such Lax representations as standard Lax representation. On the other hand there are some systems of evolution equations, such as the equations of hydrodynamic type, which are obtained by nonstandard Lax represenations used in the present paper. We first show that the method introduced in Ref. is also applicable here in the case of systems of equations of hydrodynamic types and we give several examples for illustration. These equations and their Hamila Electronic mail: gurses@fen.bilkent.edu.tr /00/4(3)/309/7/$ American Institute of Physics Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

3 30 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin tonian formulation sometimes called the dispersion-less KdV system were studied by Dubrovin and Novikov. 3 See Ref. 4 for more details on this subject see also Ref. 5. It is known that these equations admit a nonstandard Lax representation, L t A,L k, 5 where A,L are differentiable functions of t,x, p on a Poisson manifold M with local coordinates (x,p) and, k is the Poisson bracket. On M we take this Poisson bracket, k p k,, where, is the canonical Poisson bracket and k is an integer. For more information on Poisson manifolds see Refs. 6 and 7. Equations of hydrodynamic type with the above Lax representations were studied in Refs. 8. Having such a Lax representation, we can consider a whole hierarchy of equations, L t n A n,l k. 6 We can also represent function A n in the form given in 3 and apply our method for the construction of a recursion operator for the equation 6. There are some other works 4 which also give recursion operators of some equations of hydrodynamic type. The form of these operators are different than the recursion operators presented in this work. Our method produces recursion operators for hydrodynamic type of equations in the form RAB D where A and B are functions of dynamical variables and their derivatives. All higher symmetries obtained by the repeated application of this recursion operator to translational symmetries also belong to the hydrodynamic type of equations. The recursion operators obtained in Refs. 4 are of the form RC DAB D E, where A,B,C, and E are functions of dynamical variables and their derivatives. In the next section we discuss the Lax representation with Poisson brackets for polynomial Lax functions. In Sec. III we give the method of construction of the recursion operators following Ref.. In Sec. IV we give several examples for k0 and k. In Sec. V we consider the Poisson bracket for general k and let LpSPp, 7 and find the Lax equations and the corresponding recursion operator for N. In Sec. VI we consider the Lax function v Lp u p, 8 and take k0. We obtain the equations corresponding to the polytropic gas dynamics and its recursion operators. 6,0 It is interesting to note that the systems of equations and their recursion operators obtained in Secs. V and VI are transformable into each other. In Sec. VII we give a method reduction from an N system to an N system and from an N system to an N system by letting one of the symmetrical variables defined in the text either to zero or equating to another variable. The systems obtained by the reduction are equivalent to the systems obtained by the Lax function in symmetrical variables having zeros with multiplicities greater than one. Reduced systems are shown to be also integrable, i.e., they admit recursion operators. II. LAX FORMULATION WITH POISSON BRACKET We start with the definition of the standard Poisson bracket. Let f (x,p) and g(x,p) be differentiable functions of their arguments. Then the standard Poisson bracket is defined by see Refs. 6 and 9 for more details Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

4 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 3 f,g f p g x f g x p. 9 We give a slight modification of this bracket as 9 f,g k p k f,g, 0 where k is an integer. It is easy to prove that, k also defines a Poisson bracket for all kz. Although this bracket is equivalent to,, under p k (d/dp) d/dq where q is the new variable, we shall keep using it. The main reason is technical. There is a nice duality between the systems obtained by polynomial Lax representation, Lp N, with Poisson bracket, k and by Lax represention Lp p N with Poisson bracket,. For illustration we have examples, equations governing the polytropic gas dynamics, given in Propositions 6 and 7. For each kz we can consider hierarchies of equations of hydrodynamic type, defined in terms of the Lax function, N Lp N i p i S i x,t, by the Lax equation L t n L n/n k ;L k, where n jl(n) and j,,...,(n),ln. So we have a hierarchy for each k and j,...,(n). Also, we require nk to ensure that (L n/(n) ) k is not zero. With the choice of Poisson brackets, k, we must take a certain part of the series expansion of L n/(n) to get the consistent equation. This part is (L n/(n) ) k. The Lax function can also be written in terms of symmetric variables u,...,u N, N L pu p j, j 3 that is u,...,u N are roots of the polynomial p N S N p N...S p. In new variables the equation is invariant under transposition of variables. III. RECURSION OPERATORS For each hierarchy of the equations, depending on the pair (N,k), we can find a recursion operator. Lemma : For any n, L n LL n(n) R n ;L k, 4 where function R n has a form N R n i0 p ik A i S...S N,S / t nn...s N / t nn. 5 Proof: Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

5 3 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin So, L n/n k LL n/n k LL n/n k k. L n/n k LL n/n k LL n/n k k LL n/n k k. 6 If we put R n LL n/n k k LL n/n k k, then L n/n k LL n/n k R n. Hence, L n L n/n k ;L k LL n/n k R n ;L k LL n(n) R n ;L k, 7 and 4 is satisfied. Evaluating powers of L(L n/(n) ) k k and L(L n/(n) ) k k we get that R n has form 5. Lemma : A recursion operator for the hierarchy () is given by equalities, for mn,n3,...,, m S m t n j S j S m j m t nn j m jka j S m j,x j m ja j,x S m j, 8 where to simplify the above formula we have defined that S N and S N,x 0, (S N / t n )0. Coefficients A N,A N3,...,A 0 can be found from the recursion relations, for mn,...,, N NA m,x jm S j S Nm j t nn N jm jka j S Nm j,x N jm Nm ja j,x S Nm j. 9 Proof: Let us write the equality 4, using 5 for R n, N N N p N p i S i i i i p i S i t n N j N p j S j,x p k j0 p i S Nm j t nn k N p j0 jkp jk A j N p jk A j,xnp N j jp j S j. To have the equality, the coefficients of p N3,...,p N and p must be zero; it gives recursion relations to find A N,...,A 0. The coefficients of p N,...,p give the expressions for S N / t n,..., S / t n. Although the recursion operator R, given by 8, is a pseudo-differential operator, but it gives a hierarchy of local symmetries starting from the equation itself. Indeed, equalities 8, 9 Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

6 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 33 give expressions S N / t n,..., S / t n in terms of S N,..., S and S N / t n(n),..., S / t n(n). Hence, the recursion operator R is constructed in such a way that L n/n k ;L k RL n/n k ;L k. 0 IV. SOME INTEGRABLE SYSTEMS We shall consider first some examples for k0, k and the general case in the next section. A. Multicomponent hierarchy containing also the shallow water wave equations, kä0 This hierarchy corresponds to the case k0. Let us give the first equation of hierarchy and a recursion operator for N,3. Proposition : In the case N one has the Lax function, LpSP p, and the Lax equation for n, given by (47), when k0, S t SS x P x, P t SP x PS x, and the recursion operator, given by 48, R SS xd x PP x D x These equations are known as the shallow water wave equations or as the equations of polytropic gas dynamics for See Sec. VI. The first two symmetries of the system are given by S t S 3 6SP x,. S P t 3S P3P x, 3 S t S 4 S P6P x, P t 4S 3 PSP x. 4 These are all commuting symmetries. Remark : In symmetric variables the system () is written as u t uvu x uv x, v t vu x uvv x, 5 and the recursion operator () takes the form R uvu xd x vv x D x uu x D x uvv x D x. 6 Proposition : In the case N3 one has the Lax function Lp pspp Q, Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

7 34 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin and the Lax equation with n3 is 3 S t P 8 S S x SP x Q x, 3 P t QS x 8 S PP x SQ x, 7 3 Q t 4 SQS x QP x 8 S PQ x. The recursion operator, corresponding to this equation, is R S 4 PP xd x S x 4 D x S S S x D x 3 3Q Q x P xs D x P x 4 D x S P P x D x S SQ 4 SQ x S xq D x Q x 4 D x S 3Q Q x D x P. 8 Proof: Using 9 we find the function R n and using 8 we find the recursion operator 8. Remark : In symmetric variables the equation (7) is written as 3 u t 8 u uvuwvw 8 vw u x 4 u 4 uv 3 4uwv x 4u 4uw 4uvw 3 x, 3v t 4v 4uv 3 4vwu x 4v 4vw 3 4uvw x 8v uvuwvw 8uw v x, 9 3w t 4w 4uw 4wvu 3 x 4w 4wv 3 4uwv x 8w uvuwvw 8vu w x, and the recursion operator takes the form (A) given in the Appendix. B. Toda hierarchy kä Toda hierarchy corresponds to the case k. 9 Let us give the first equation of hierarchy and a recursion operator for N and N3. Proposition 3: In the case N and n one has the Lax function and the Lax equation for n, given by (4), LpSP p, S t P x, P t PS x, 30 and the recursion operator, given by (4), R S P xd x P P SS x PD. 3 x P The first two symmetries of the equation 30 are given by Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

8 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 35 S t SP x, P t PPS x, S t 3S P3P x, P t P6PSS 3 x Remark 3: In symmetric variables the equation (30) is written as u t uv x, v t vu x, 34 and the recursion operator (3) takes the form R uvuv xd x u uuv x D x v vvu x D x u uvvu x D. 35 x v Proposition 4: In the case N3 and n one has the Lax function and the Lax equation with n is Lp ps Pp Q, S t P x SS x, P t Q x, 36 Q t QS x. The recursion operator, corresponding to this equation, is P 4S R P x 4SS x D x S 3Q x D x Q 3 Q Q x D x P SSQ x D x Q. 37 4SQ 4S 3 x QD x Q PP x QD x Q Proof: Using equalities 9 we find the function R n and using 8 we find the recursion operator 37. Remark 4: In symmetric variables the equation (36) is written as u t uu x v x w x, v t vu x v x w x, 38 w t wu x v x w x, and the recursion operator takes the form (A) given in the Appendix. V. LAX EQUATION FOR GENERAL k We shall only consider the case where N. We have the Lax function LpSPp, 39 Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

9 36 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin and the Lax equation L t n L n k ;L k. 40 We consider two cases k and k0. A. The first case kð Proposition 5: In the case N and k one has the Lax equation S t kp k P x, P t kp k S x, 4 and the recursion operator for this equation is R SkS xd x PkP x D x kp k P x D x P k SkS x P k D x P k. 4 Proof: The smallest power of p in L n is n. To have powers less than k we must put nk. If there are no such powers then Poisson brackets are (L n );L k 0. Let us calculate the Lax equation, L t L k k ;L k L k k ;L k. We have (L k ) k (pspp ) k k P k p k, thus L t P k p k ;pspp k. And we get the equation 4. Using 8, 9 we find the recursion operator 4. First two symmetries are given as follows: S t kp k S x, k S P t kp P k. x 43 S t kk Pk Pk S, k x k S3 P t kkp PS k. 6 x 44 Remark 5: In symmetric variables the equation (4) is written as u t ku k v k v x, v t ku k v k u x, 45 and the recursion operator (4) takes the form Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

10 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 37 uvku x D x R uku x D x ku k v k v x D x u k v k ku k v k v x D x u k v k vkv x D x uvkv x D x k. 46 ku k v k u x D x u k v k ku k v k u x D x u k v B. The second case kï0 Proposition 6: In the case N and k0 one has the Lax equation S t kkss x kp x, P t kksp x ks x P, 47 and the recursion operator for this equation is R SkS xd x PkP x D x kp k P x D x P k SkS x P k D x P k. 48 Proof: The largest power of p in L n is p n. To have powers larger than k we must put nk. Then we have thus Then the Lax equation becomes L k k pspp k k p k ; L t p k ;pspp k. S t S x, P t P x. This is a trivial equation; let us calculate the second symmetry. We have (L k ) k (p SPp ) k k p k (k)sp k, thus L t p k ksp k ;pspp k. We get the equation 47. Using 8, 9 we find the recursion operator 48. First two symmetries are given as follows: S t kk3p S 6kS 3 x, P t kk3ss x P ks P x PP x, S t k3k4k P S P 6 S4, k x P t k3k4k S S x P 6 ks3 P x SPP x k P S x. Remark 6: In symmetric variables the equation (47) is written as u t kkuvu x kuv x, Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

11 38 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin v t kvu x kkuvv x, 5 and the recursion operator (48) takes the form uvku x D x R uku x D x ku k v k v x D x u k v k ku k v k v x D x u k v k vkv x D x uvkv x D x k. 5 ku k v k u x D x u k v k ku k v k u x D x u k v In this section, to obtain the recursion operators we have considered two different cases k 0 and k to simplify some technical problems in the method. At the end we obtained recursion operators having the same forms 4 and 48. Hence any one of these represent the recursion operator for kz. It seems, comparing the results, that the systems of equations in one case are symmetries of the other case. For instance, the system 47 is a symmetry of system 4. Hence we may consider only one case with recursion operator 4 for all integer values of k. VI. LAX FUNCTION FOR POLYTROPIC GAS DYNAMICS In this section we consider another Lax function, introduced in Ref. 0, v Lp u p, 53 and the Lax equation L t gives the equations of the polytropic gas dynamics. Proposition 7: The Lax equation corresponding to (54) is L/,L 0, 54 u t uu x v v x 0, v t uv x Proof: Expanding the function 53 around the point p, we have p u v / p p pu...; all other terms have negative powers of p. Therefore L / p pu, and the Lax equation 54 corresponds to 55. Proposition 8: The recursion operator for the equation (55) is u x u R D x v v x D x v v x u u xd x D x. 56 Proof: Using the equation Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

12 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 39 L L L R t n t n,l, n in the same way as for the polynomial Lax function one can find the recursion operator 56. It is interesting to note that the equation 47 and equations of polytropic gas dynamics 55 are related by the following change of variables: u S kk, P v/k k, 57 where (k)/(k). We note that under this change of variables recursion operator 48 is mapped to the recursion operator 56. VII. REDUCTION In this section we consider reductions of the equation, written in symmetric variables, by setting u 0, or u u N,..., or u u,u N. These reductions correspond to the Lax equations with different Lax functions. For reduction u 0 we have a polynomial Lax function with simple roots L(pu N ) (pu ) and for reduction u N u we have a polynomial Lax function with a root of multiplicity two L (/p)(pu N ) (pu N )...(pu ), etc. We note that instead of working on the Lax functions with higher multiplicities like the last example one can take a polynomial Lax function without any multiplicities and perform the reductions we propose in this section. A. Reduction u Ä0 Let us write the equation as u N,...,u 0, 58 where is a differential operator. Then u N,...,u u 0 un,...,u, 0, 59 where is another differential operator. Indeed, following Ref. 8 for the Lax function L (/p) N j (pu j ) we have and L L N t L j L N x L j p L p j u j,t pu j, u j,x pu j, N pu j. Thus u j,t Res pu j M,L k, where M(L n/(n) ) k. The Lax equation can be written as Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

13 30 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin N j N u j,t p k M pu j p j u j,x p k M pu j x N p j pu j. Note that p k M x and p k M p are polynomials. So, if we put u 0 and calculate the residue of the right hand side at p0 we get 59. A new equation, 60 un,...,u 0, 6 is also integrable and a recursion operator of this equation can be obtained as a reduction of the recursion operator of the equation 58. Let R be the recursion operator of 58 given by Lemma, then Indeed, we found the recursion operator using formula 4. This formula can be written as N u N j,tn u LL j pu n(n) p k j,x R j n,p p k R j pu j n,x N p j pu j 63 and in the same way as for the reduction of 58 we have 6; note, that p k R n,x and p k R n,p are also polynomials. Lemma 3: The operator R is a recursion operator of the equation (6). Proof: Equation 6 is an evolution equation, so, to prove that R is a recursion operator we must prove that for any solution (u N,...,u )of6 the following equality holds see Ref. 6: D R R D, where D is a Frechet derivative of. If (u N,...,u ) is a solution of 6 then (u N,...,u,u 0) is a solution of 58 and for the solution (u N,...,u,u 0) we have 6 D RR D. 64 Next and Hence by 64 we have D R R D. Calculating the Frechet derivative, we take derivatives with respect to one variable, considering other variables as constants. Thus, to calculate D we can put u 0 and differentiate with respect to other variables or we can first differentiate and then put u 0. It means that D D and D R R D. Let us consider the reduction of systems, given by Remark and Remark 4 and their recursion operators. Proposition 9: Putting w0 in (38) and (A) we obtain a new system, Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

14 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 3 u t uu x v x, v t vu x v x, 65 u uv 4 R uv u 4 uv u 4 u xv x D x u 4 u xv x D x v 4 uv v 4 u xv x D x uv v 4 uv v 4 u xv x D, x 66 and its recursion operator, respectively. Proposition 0: Putting w0 in (9) and (A) we obtain a new system, 3u t 8u uv 8v u x 4u 4uvv x, 3v t 4v 4uvu x 8v uv 8u v x, 67 and its recursion operator, u 4 R 3uv 4 u xv uv x D x u 4 uv u xv uv x D x u x 4 D x u u x 4 D x v u x 4 D x u u x 4 D x v v 4 uv uv x u xv D x v 3uv 4 4 uv x u xv D x v x 4 D x u v x 4 D x v v x 4 D x u v x 4 D x v, 68 respectively. It is worth mentioning that by reduction we obtain a new equation. For example, consider the case k0. The equation 5, corresponding to N, and reduction of the equation 9, corresponding to N3, are not related by a linear transformation of variables. Indeed, in the equation 5 coefficients of u x,v x are linear in u,v but in the equation 67 coefficients of u x,v x contain quadratic terms. Hence they cannot be related by a linear transformation. B. Reduction u N Äu It follows from 60 that the Lax equation can be written as N u i,t j h i j u N,...,u u j,x, 69 where i, j,...,n and h i j h (u i,u N,...,û i,...,u ) when i j and h i i h (u i,u N,...,û i,...,u ), the overcaret denotes the absence of the corresponding variable. It also follows from 60 that the functions h (x N,...,x ) and h (x N,...,x ) are symmetric under permutations of variables x N,...,x. Reduction u N u gives us a new integrable equation, Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

15 3 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin u N,t h N N u N,u N,...,u,u N h N u N,u N,...,u,u N u N,x N j h j N u N,u N,...,u,u N u j,x, N u i,t h i N u N,u N,...,u,u N u N,x j h i j u N,u N,...,u,u N u j,x, 70 where i(n),...,. The Frechet derivative of 69, under condition u N u, has the form a a a (N) a N a a a (N) a D un u ] ] ] ], 7 a (N) a (N) a (N)(N) a (N) a N a a (N) a where a ij, i, j,...,n are differential operatos. So, the Frechet derivative of 70 can be writen as a a(n) a a a (N) D aan 7 ] ] ] a (N) a (N) a (N)(N). Now let us write the recurcion operator of 69, given by Lemma. From 63 it follows that, under condition u N u, it has the form b b b (N) b N b b b (N) b R un u ] ] ] ], 73 b (N) b (N) b (N)(N) b (N) b N b b (N) b where b ij, i, jn,...,aredifferential operators. Now we can write a recursion operator for Eq. 70, b b(n) b b b (N) R bbn 74 ] ] ] b (N) b (N) b (N)(N). The form of 74 can be deduced by applaing operator R un u to a symmetry (u N / t n,u N / t n,..., u / t n, u n / t n ). Lemma 4: The operator R in (74) is a recursion operator of the equation (70). Proof: Equation 70 is an evolution equation, so, to prove that R is a recursion operator we must prove that for any solution (u N,...,u )of70 the following equality holds see Ref. 6: D R R D. Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

16 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 33 If (u N,...,u ) is a solution of 70 then (u N,...,u,u u N ) is a solution of 69 and for the solution (u N,...,u,u u N ) we have D RR D. 75 One can show that from commutation of 7 and 73 follows the commutation of 7 and 74 that is equality 75. Let us consider reduction of systems, given by Remark and Remark 4 and their recursion operators. Proposition : Putting wu in (38) and (A) we obtain a new system, u t uv x, v t vu x v x, 76 and its recursion operator R uvu uv 3 4uuv 3 u uv x D x 4uu x v x D x uuv x D x u uuv x D x v. 77 vuv3uv uvu 4vuv vu x v x D x 4vu x v x D x vuv x D x u vu x D x v Proposition : Putting wu in (9) and (A) we obtain a new system, 3u t u uv 8v u x u 4uvv x, 3v t v uvu x 8v uvu v x, 78 and its recursion operator, 7 uvu Ru uv x D x u x D x v 4uv v uv x D x u 4uv u uvd x u x D x u 4u x D x v. 79 4v 3 uvu uv x D x v x D x v v x D x u 4v x D x v We may go on introducing new reductions. For instance a reduction of the type u u u N, (N3), reduces an N-system to an (N)-system. One may obtain this (N)-system also from the polynomial Lax function having the form Lp (pu ) 3 (pu 3 ) (pu N ) a zero of L with multiplicity three. In this way one obtains an infinite number of different classes of N, N3 systems. VIII. CONCLUSION We have constructed the recursion operators of some equations of hydrodynamic type. The form of the these operators fall into the class of pseudo-differential operators AB D where A and B are functions of dynamical variables and their derivatives. The generalized symmetries of these equations are local and all belong to the same class i.e., they are also equations of hydro- Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

17 34 J. Math. Phys., Vol. 4, No. 3, March 00 M. Gürses and K. Zheltukhin dynamic type. We have introduced a method of reduction which leads also to integrable classes. Depending upon the type of reductions we may obtain infinitely many different classes of Nk systems. These properties, the bi-hamiltonian structure of the equations we obtained and equations with rational Lax functions, will be communicated elsewhere. ACKNOWLEDGMENTS We thank Burak Gürel and Atalay Karasu for several discussions. We also thank the referee for his several suggestions on this work. This work is partially supported by the Scientific and Technical Research Council of Turkey and by the Turkish Academy of Sciences. APPENDIX: RECURSION OPERATORS FOR NÄ3 SYSTEMS 9 AND 38 Recursion operators of the systems 9 and 38 are, respectively, given by u uvuwwv u 4 uvw 3uw u 4 uvw 3uv u x vwd x u x vwd x u x vwd x R u x 4 D x u u x 4 D x v u x 4 D x u u x 4 D x v u x 4 D x u u x 4 D x v u x 4 D x w u x 4 D x w u x 4 D x w v 3vw uvw v uvvwuw v 3uv uvw 4 v x uwd x v x uwd x v x uwd x v u xw x D x v u xw x D x v u xw x D x v x 4 D x u v x 4 D x v v x 4 D x u v x 4 D x v v x 4 D x u v x 4 D x v w 4 v x 4 D x w v x 4 D x w v x 4 D x w uvw 3vw w 4 uvw 3uw w uwvwuv w x uvd x w x uvd x w x uvd x, u v xw x D x u v xw x D x u v xw x D x w u xv x D x w u xv x D x w u xv x D x w x 4 D x u w x 4 D x v w x 4 D x u w x 4 D x v w x 4 D x u w x 4 D x v w x 4 D x w w x 4 D x w w x 4 D x w A Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at:

18 J. Math. Phys., Vol. 4, No. 3, March 00 Recursion operators of equations hydrodynamic type 35 uvuwvw u 4 uvw u 4 uvw R u 4 u xv x w x D x u 4 u xv x w x D x u 4 u xv x w x D x uwv x vw x D x u uwv x vw x D x v uwv x vw x D x w v 4 uvw uvuwvw v 4 uvw 3vw v 4 uvw 3uv. v 4 u xv x w x D x v 4 u xv x w x D x v 4 u xv x w x D x vwu x uw x D x u vwu x uw x D x v vwu x uw x D x w w 4 uvw w 4 uvw uvuwvw u 4 uvw 3uw 3uv 3uw w 4 u xv x w x D x 3vw w 4 u xv x w x D x w 4 uvw w 4 u xv x w x D x wuv x vu x D x u wuv x vu x D x v wuv x vu x D x w A M. Gürses, A. Karasu, and V. V. Sokolov, On construction of recursion operator from Lax representation, J. Math. Phys. 40, I. M. Gel fand and L. A. Dikkii, Asymptotic behavior of the re-solvent of Sturm Liouville equations and the algebra of the Korteweg-de Vries equations, Funct. Anal. Appl. 0, B. A. Dubrovin and S. P. Novikov, Hamiltonian formalism of one-dimensional systems of hydrodynamic type, Sov. Math. Dokl. 7, E. V. Ferepantov, Hydrodynamic-type systems, in CRC Handbook of Lie Group Analysis of Differential Equations, edited by N. H. Ibragimov CRC, New York, 994, Vol., pp Preliminary version of this work was first reported in M. Gürses and K. Zhelthukin, On construction of recursion operators for some equations of hydrodynamic type, in International Conference on Complex Analysis, Differential Equations and Related Topics, 9 May 3 June 000, Ufa, Russia. 6 P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Text in Mathematics, nd ed. Springer- Verlag, New York, 993, Vol A. Weinstein, Poisson geometry, Diff. Geom. Applic. 9, D. B. Fairlie and I. A. B. Strachan, The algebraic and Hamiltonian structure of the dispersion-less Benney and Toda hierarchies, Inverse Probl., I. A. B. Strachan, Degenerate Frobenius manifolds and the bi-hamiltonian structure of rational Lax equations, J. Math. Phys. 40, J. C. Brunelli and A. Das, A Lax description for polytropic gas dynamics, Phys. Lett. A 35, J. C. Brunelli, M. Gürses, and K. Zhelthukin, On the integrability of some Monge Ampere equations, Rev. Math. Phys. in press. M. B. Sheftel, Generalized hydrodynamic-type systems, in CRC Handbook of Lie Group Analysis of Differential Equations, edited by N. H. Ibragimov CRC, New York, 996, Vol. 3, pp V. M. Teshukov, Hyperbolic systems admitting a nontrivial Lie Backlund group, LIIAN, 06, A. P. Fordy and B. Gürel, A new construction of recursion operators for systems of hydrodynamic type, Theor. Math. Fiz., Downloaded 08 May 03 to This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: