Recursion operators of some equations of hydrodynamic type

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M. Gürses, and K. Zheltukhin

Citation: Journal of Mathematical Physics 42, 1309 (2001); doi: 10.1063/1.1346597 View online: http://dx.doi.org/10.1063/1.1346597

View Table of Contents: http://aip.scitation.org/toc/jmp/42/3 Published by the American Institute of Physics

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Recursion operators of some equations

of hydrodynamic type

M. Gu¨rsesa) and K. Zheltukhin

Department of Mathematics, Faculty of Sciences, Bilkent University, 06533 Ankara—Turkey

共Received 7 August 2000; accepted for publication 28 November 2000兲

We give a general method for constructing recursion operators for some equations of hydrodynamic type, admitting a nonstandard Lax representation. We give sev-eral examples for N⫽2 and N⫽3 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⫹1 systems to N systems of some new equations of hydrodynamic type. © 2001 American Institute of Physics.

关DOI: 10.1063/1.1346597兴

I. INTRODUCTION

Most of the integrable nonlinear partial differential equations admit Lax representations,

Lt⫽关A,L兴, 共1兲

where L is a pseudo-differential operator of order m and A is a pseudo-differential operator. Recently1we established a new method for such integrable equations to construct their recursion operators. This method uses the hierarchy of equations,

L_t

n⫽关An,L兴, 共2兲

and the Gel’fand–Dikkii2construction of the An-operators. Defining an operator Rn in the form

An⫽LAn⫺m⫹Rn, 共3兲

one then obtains relations among the hierarchies,

Lt_n⫽LLt_n_⫺m⫹关Rn;L兴. 共4兲

This equation allows to find L_t

n in terms of Ltn⫺m. It is important to note that one does not need to know the exact form of An. For further details of the method see Ref. 1.

In Ref. 1 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 Hamil-tonian 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 共taking values in some lower dimensional Lie algebras兲. 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. 1 is also applicable here in the case of systems of equations of hydro-dynamic types and we give several examples for illustration. These equations and their Hamil-a兲_{Electronic mail: gurses@fen.bilkent.edu.tr}

1309

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tonian formulation共sometimes called the dispersion-less KdV system兲 were studied by Dubrovin and Novikov.3See Ref. 4 for more details on this subject共see also Ref. 5兲. It is known that these equations admit a nonstandard Lax representation,

⳵L

⳵t ⫽兵A,L其k, 共5兲

where A,L are differentiable functions of t,x, p on a Poisson manifold M with local coordinates (x, p) and兵,其_kis the Poisson bracket. On M we take this Poisson bracket兵,其_k⫽pk兵,其, where兵,其 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. 8–11. Having such a Lax representation, we can consider a whole hierarchy of equations,

⳵L

⳵t_n⫽兵An,L其k. 共6兲

We can also represent function A_n in the form given in 共3兲 and apply our method1 for the construction of a recursion operator for the equation共6兲. There are some other works12–14which also give recursion operators of some equations of hydrodynamic type. The form of these opera-tors are different than the recursion operaopera-tors presented in this work. Our method1 produces recursion operators for hydrodynamic type of equations in the formR⫽A⫹B D⫺1 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. 12–14 are of the form

R⫽C D⫹A⫹B D⫺1_{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. 1. In Sec. IV we give several examples for k⫽0 and k⫽1. In Sec. V we consider the Poisson bracket for general k and let

L⫽p⫹S⫹Pp⫺1, 共7兲

and find the Lax equations and the corresponding recursion operator for N⫽2. In Sec. VI we consider the Lax function

L⫽p␥⫺1⫹u⫹ v ␥⫺1

共␥⫺1兲2p⫺␥⫹1, 共8兲

and take k⫽0. We obtain the equations corresponding to the polytropic gas dynamics and its recursion operators.6,10 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⫹1 system to an N system and from an N⫹1 system to an N⫺1 system by letting one of the symmetrical variables共defined in the text兲 either to zero or equating to another variable. The systems obtained by the reduction are equivalent to the systems obtained by the Lax function共in symmetrical variables兲 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共see Refs. 6 and 9 for more details兲

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兵f ,g其⫽⳵f ⳵p ⳵g ⳵x⫺ ⳵f ⳵x ⳵g ⳵p. 共9兲

We give a slight modification of this bracket as9

兵f ,g其k⫽pk兵f ,g其, 共10兲

where k is an integer. It is easy to prove that 兵,其k also defines a Poisson bracket for all k苸Z.

Although this bracket is equivalent to兵,其, under pk(d/d p)⫽ 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, L⫽pN⫹¯ , with Poisson bracket 兵,其k and by Lax

represention L⫽p␥关pN⫹¯兴 with Poisson bracket兵,其. For illustration we have examples, equa-tions governing the polytropic gas dynamics, given in Proposiequa-tions 6 and 7.

For each k苸Z we can consider hierarchies of equations of hydrodynamic type, defined in terms of the Lax function,

L⫽pN⫺1⫹

兺

i_⫽⫺1 N⫺2

piSi共x,t兲, 共11兲

by the Lax equation

⳵L

⳵tn⫽兵共L n/共N⫺1兲_兲

⭓⫺k⫹1;L其k, 共12兲

where n⫽ j⫹l(N⫺1) and j⫽1,2, . . . ,(N⫺1),l苸N. So we have a hierarchy for each k and j ⫽1, . . . ,(N⫺1). Also, we require n⭓⫺k⫹1 to ensure that (Ln/(N⫺1)₎

⭓⫺k⫹1 is not zero. With the choice of Poisson brackets兵,其_k, we must take a certain part of the series expansion of Ln/(N⫺1) to get the consistent equation共12兲. This part is (Ln/(N⫺1))_⭓⫺k⫹1.

The Lax function共11兲 can also be written in terms of symmetric variables u1, . . . ,uN,

L⫽1 p

兿

j⫽1

N

共p⫺uj兲, 共13兲

that is u1, . . . ,uN are roots of the polynomial

pN⫺1⫹SN⫺2pN⫺2⫹ . . . ⫹S⫺1p⫺1.

In new variables the equation共12兲 is invariant under transposition of variables.

III. RECURSION OPERATORS

For each hierarchy of the equations共12兲, depending on the pair (N,k), we can find a recursion operator.

Lemma 1: For any n,

L_n⫽LL_n_⫺(N⫺1)⫹兵R_n;L其_k, 共14兲 where function Rn has a form

Rn⫽

兺

i⫽0 N⫺2

pi⫺kAi共S⫺1. . . SN⫺2,⳵S⫺1/⳵tn⫺共N⫺1兲. . .⳵SN⫺2/⳵tn⫺共N⫺1兲兲. 共15兲

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兺

j⫽⫺1 N_⫺2 j pj⫺1S_j

冊

. To have the equality, the coefficients of p2N⫺3, . . . , pN⫺1and p⫺2must be zero; it gives recursion relations to find AN⫺2, . . . ,A0. The coefficients of pN⫺2, . . . , p⫺1 give the expressions for

⳵SN⫺2/⳵tn, . . . ,⳵S⫺1/⳵tn. 䊐

Although the recursion operator R, given by 共18兲, is a pseudo-differential operator, but it gives a hierarchy of local symmetries starting from the equation itself. Indeed, equalities共18兲, 共19兲

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give expressions⳵SN_⫺2/⳵tn, . . . ,⳵S_⫺1/⳵tnin terms of SN_⫺2, . . . , S_⫺1 and⳵SN_⫺2/⳵tn_{⫺(N⫺1), . . . ,}

⳵S_⫺1/⳵tn_⫺(N⫺1). Hence, the recursion operatorR is constructed in such a way that

兵共Ln/共N⫺1兲 ⫹1_兲

⭓⫺k⫹1;L其k⫽R共兵共Ln/共N⫺1兲兲⭓⫺k⫹1;L其k兲. 共20兲 IV. SOME INTEGRABLE SYSTEMS

We shall consider first some examples for k⫽0, k⫽1 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 k⫽0. Let us give the first equation of hierarchy and a recursion operator for N⫽2,3.

Proposition 1: In the case N⫽2 one has the Lax function, L⫽p⫹S⫹P p⫺1, and the Lax equation for n⫽2, given by (47), when k⫽0,

1

2St⫽SSx⫹Px,

共21兲 1

2Pt⫽SPx⫹PSx,

and the recursion operator, given by共48兲,

R⫽

冉

S⫹SxDx

⫺1 ₂

2 P⫹P_xD_x⫺1 S

冊

. 共22兲

These equations are known as the shallow water wave equations or as the equations of polytropic gas dynamics for␥⫽2 共See Sec. VI兲.

The first two symmetries of the system共21兲 are given by S_t 1⫽共S 3_⫹6SP兲 x, Pt₁⫽共3S2P⫹3P2兲x, 共23兲 St₂⫽共S4⫹12S2P⫹6P2兲x, Pt₂⫽共4S3P⫹12SP2兲x. 共24兲

These are all commuting symmetries.

Remark 1: In symmetric variables the system (21) is written as 1

2ut⫽共u⫹v兲ux⫹uvx,

共25兲 1

2vt⫽vux⫹共u⫹v兲vx,

and the recursion operator (22) takes the form

R⫽

冉

u⫹v⫹uxDx

⫺1 _2u_⫹u

xDx⫺1

2v⫹vxDx⫺1 u⫹v⫹vxDx⫺1

冊

. 共26兲

Proposition 2: In the case N⫽3 one has the Lax function L⫽p2⫹pS⫹P⫹p⫺1Q,

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and the Lax equation with n⫽3 is 1 3St⫽共12P⫺ 1 8S 2_兲S x⫹12S Px⫹Qx, 1 3Pt⫽ 1 2QSx⫹共 1 8S 2_⫹1 2P兲Px⫹SQx, 共27兲 1 3Qt⫽14SQSx⫹12Q Px⫹共18S 2_⫹1 2P兲Qx.

The recursion operator, corresponding to this equation, is

. 共28兲

Proof: Using 共19兲 we find the function Rn and using 共18兲 we find the recursion operator

共28兲. 䊐

Remark 2: In symmetric variables the equation (27) is written as 1 3ut⫽共⫺ 1 8u 2_⫹1 2共uv⫹uw⫹vw兲⫹ 1 8共v⫹w兲 2_兲u x⫹共 1 4u 2_⫹1 4uv⫹ 3 4uw兲vx⫹共 1 4u 2_⫹1 4uw⫹ 3 4uv兲wx, 1 3vt⫽共14v 2_⫹1 4uv⫹ 3 4vw兲ux⫹共14v 2_⫹1 4vw⫹ 3 4uv兲wx⫹共⫺18v 2_⫹1 2共uv⫹uw⫹vw兲⫹ 1 8共u⫹w兲 2_兲v x, 共29兲 1 3wt⫽共 1 4w 2_⫹1 4uw⫹ 3 4wv兲ux⫹共 1 4w 2_⫹1 4wv⫹ 3 4uw兲vx⫹共⫺ 1 8w 2_⫹1 2共uv⫹uw⫹vw兲⫹ 1 8共v⫹u兲 2_兲w x,

and the recursion operator takes the form (A1) given in the Appendix.

B. Toda hierarchy„kÄ1…

Toda hierarchy corresponds to the case k⫽1.9Let us give the first equation of hierarchy and a recursion operator for N⫽2 and N⫽3.

Proposition 3: In the case N⫽2 and n⫽1 one has the Lax function L⫽p⫹S⫹P p⫺1,

and the Lax equation for n⫽1, given by (41), St⫽Px,

Pt⫽PSx, 共30兲

and the recursion operator, given by (42),

R⫽

冉

S 2⫹PxDx

⫺1_•P⫺1 2 P S⫹SxPDx⫺1•P⫺1

冊

. 共31兲

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St₁⫽共2SP兲x, 共32兲 Pt₁⫽P共2P⫹S2兲x, St₂⫽共3S2P⫹3P2兲x, 共33兲 P_t 2⫽P共6PS⫹S 3_兲 x.

Remark 3: In symmetric variables the equation (30) is written as ut⫽uvx,

共34兲

vt⫽vux,

R⫽

冉

u⫹v⫹uvxDx

⫺1_•u⫺1 _2u_⫹uv

xDx⫺1•v⫺1

2v⫹vuxDx⫺1•u⫺1 u⫹v⫹vuxDx⫺1•v⫺1

冊

. 共35兲

Proposition 4: In the case N⫽3 and n⫽1 one has the Lax function L⫽p2⫹pS1⫹P⫹p⫺1Q,

and the Lax equation with n⫽1 is

S_t⫽P_x⫺1 2SSx,

Pt⫽Qx, 共36兲

Q_t⫽1 2QSx.

The recursion operator, corresponding to this equation, is

R⫽

冉

P⫺14S 2_⫹共1 2Px⫺ 1 4SSx兲Dx⫺1 1 2S 3⫹2QxDx⫺1•Q⫺1 3 2Q⫹ 1 2QxDx⫺1 P 2S⫹共SQ兲xDx⫺1•Q⫺1 1 4SQ⫹ 1 4SxQDx⫺1 3 2Q P⫹PxQDx⫺1•Q⫺1

冊

. 共37兲

Proof: Using equalities 共19兲 we find the function R_n and using 共18兲 we find the recursion

operator共37兲. 䊐

Remark 4: In symmetric variables the equation (36) is written as ut⫽ 1 2u共⫺ux⫹vx⫹wx兲, vt⫽12v共⫹ux⫺vx⫹wx兲, 共38兲 wt⫽ 1 2w共⫹ux⫹vx⫺wx兲,

and the recursion operator takes the form (A2) given in the Appendix.

V. LAX EQUATION FOR GENERALk

We shall only consider the case where N⫽2. We have the Lax function

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and the Lax equation

⳵L

⳵tn

⫽兵共Ln_兲

⭓⫺k⫹1;L其k. 共40兲

We consider two cases k⭓1 and k⭐0.

A. The first casekÐ1

Proposition 5: In the case N⫽2 and k⭓1 one has the Lax equation St⫽kPk⫺1Px,

共41兲 P_t⫽kPkS_x,

and the recursion operator for this equation is

R⫽

冉

S⫹共1⫺k兲SxDx ⫺1 ₂_⫹kPk⫺1_P xDx⫺1•P⫺k 2 P⫹共1⫺k兲P_xD_x⫺1 S⫹kS_xPk_D x ⫺1_•P⫺k

冊

. 共42兲

Proof: The smallest power of p in Ln is⫺n. To have powers less than ⫺k⫹1 we must put n⫽k. If there are no such powers then Poisson brackets are兵(Ln);L其_k⫽0.

Let us calculate the Lax equation,

Lt⫽兵共Lk兲⭓⫺k⫹1;L其k⫽⫺兵共Lk兲⭐⫺k;L其k.

We have (Lk)_⭐⫺k⫽关(p⫹S⫹Pp⫺1)k兴_⭐⫺k⫽Pkp⫺k, thus Lt⫽⫺兵Pkp⫺k; p⫹S⫹Pp⫺1其k.

And we get the equation共41兲. Using 共18兲, 共19兲 we find the recursion operator 共42兲. 䊐 First two symmetries are given as follows:

x .

Remark 5: In symmetric variables the equation (41) is written as u_t⫽kuk_vk⫺1_v

x,

共45兲

vt⫽kuk⫺1vkux,

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R⫽

冉

u⫹v⫹共1⫺k兲u_xD_x⫺1⫹ 2u⫹共1⫺k兲u_xD_x⫺1⫹ kukvk⫺1vxDx⫺1•u⫺kv⫺k⫹1 ku k vk⫺1vxDx⫺1•u⫺k⫹1v⫺k 2v⫹共1⫺k兲vxDx⫺1⫹ u⫹v⫹共1⫺k兲vxDx⫺1⫹ kuk⫺1vk_u xDx⫺1•u⫺kv⫺k⫹1 ku k⫺1_vk_u xDx⫺1•u⫺k⫹1v⫺k

冊

. 共46兲

B. The second casekÏ0

Proposition 6: In the case N⫽2 and k⭐0 one has the Lax equation St⫽共⫺k⫹2兲共⫺k⫹1兲SSx⫹共⫺k⫹2兲Px,

共47兲 Pt⫽共⫺k⫹2兲共⫺k⫹1兲SPx⫹共⫺k⫹2兲SxP,

and the recursion operator for this equation is

R⫽

冉

S⫹共1⫺k兲SxDx

⫺1 ₂_⫹kPk⫺1_P

xDx⫺1•P⫺k

2 P⫹共1⫺k兲PxDx⫺1 S⫹kSxPkDx⫺1•P⫺k

冊

. 共48兲

Proof: The largest power of p in Ln is pn. To have powers larger than⫺k⫹1 we must put n⫽⫺k⫹1. Then we have

共L⫺k⫹1_兲

⭓⫺k⫹1⫽关共p⫹S⫹Pp⫺1兲⫺k⫹1兴⭓⫺k⫹1⫽p⫺k⫹1; thus

Lt⫽兵p⫺k⫹1; p⫹S⫹Pp⫺1其k.

Then the Lax equation becomes

St⫽Sx,

Pt⫽Px.

This is a trivial equation; let us calculate the second symmetry. We have (L⫺k⫹2)_⭓⫺k⫹1⫽关(p ⫹S⫹Pp⫺1₎⫺k⫹1_兴

⭓⫺k⫹1⫽p⫺k⫹2⫹(⫺k⫹2)Sp⫺k⫹1, thus

Lt⫽兵p⫺k⫹2⫹共⫺k⫹2兲Sp⫺k⫹1; p⫹S⫹Pp⫺1其k.

We get the equation共47兲. Using 共18兲, 共19兲 we find the recursion operator 共48兲. 䊐 First two symmetries are given as follows:

S_t 1⫽共k⫺2兲共k⫺3兲共P S⫹ 1 6共1⫺k兲S 3_兲 x, 共49兲 Pt₁⫽共k⫺2兲共k⫺3兲共SSxP⫹ 1 2共1⫺k兲S 2_P x⫹PPx兲, St₂⫽共2⫺k兲共3⫺k兲共4⫺k兲

冉

1 2S 2_P_⫹1 6S 4_⫹ 1 2共2⫺k兲P 2

冊

x , 共50兲 P_t 2⫽共2⫺k兲共3⫺k兲共4⫺k兲

冉

1 2S 2_S xP⫹ 1 6共1⫺k兲S 3_P x⫹SPPx⫹ 1 共2⫺k兲P2Sx

冊

.

Remark 6: In symmetric variables the equation (47) is written as ut⫽共⫺k⫹2兲共1⫺k兲共u⫹v兲ux⫹共⫺k⫹2兲uvx,

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vt⫽共⫺k⫹2兲vux⫹共⫺k⫹2兲共1⫺k兲共u⫹v兲vx, 共51兲

R⫽

冉

u⫹v⫹共1⫺k兲uxDx⫺1⫹ 2u⫹共1⫺k兲uxDx⫺1⫹ kukvk⫺1vxDx⫺1•u⫺kv⫺k⫹1 ku k vk⫺1vxDx⫺1•u⫺k⫹1v⫺k 2v⫹共1⫺k兲vxDx⫺1⫹ u⫹v⫹共1⫺k兲vxDx⫺1⫹ kuk⫺1vkuxDx⫺1•u⫺kv⫺k⫹1 ku k_⫺1 vkuxDx⫺1•u⫺k⫹1v⫺k

冊

. 共52兲

In this section, to obtain the recursion operators we have considered two different cases k ⭐0 and k⭓1 to simplify some technical problems in the method. At the end we obtained recur-sion operators having the same forms共42兲 and 共48兲. Hence any one of these represent the recur-sion operator for k苸Z. It seems, comparing the results, that the systems of equations in one case are symmetries of the other case. For instance, the system 共47兲 is a symmetry of system 共41兲. Hence we may consider only one case with recursion operator共42兲 for all integer values of k.

VI. LAX FUNCTION FOR POLYTROPIC GAS DYNAMICS

In this section we consider another Lax function, introduced in Ref. 10,

L⫽p␥⫺1⫹u⫹ v ␥⫺1

共␥⫺1兲2p⫺␥⫹1, 共53兲

and the Lax equation

⳵L

⳵t⫽

␥⫺1

␥ 兵共L␥/共␥⫺1兲兲⭓1,L其0, 共54兲

gives the equations of the polytropic gas dynamics.

Proposition 7: The Lax equation corresponding to (54) is ut⫹uux⫹v␥⫺2vx⫽0,

vt⫹共uv兲x⫽0. 共55兲

Proof: Expanding the function共53兲 around the point p⫽⬁, we have

冉

p␥⫺1⫹u⫹ v ␥⫺1 共␥⫺1兲2p⫺␥⫹1

冊

␥/共␥⫺1兲 ⫽p␥_⫹ ␥ ␥⫺1pu⫹ . . . ; all other terms have negative powers of p. Therefore

共L␥/共␥⫺1兲兲⭓1⫽p␥⫹_␥_⫺1␥ pu,

and the Lax equation共54兲 corresponds to 共55兲. 䊐 Proposition 8: The recursion operator for the equation (55) is

R⫽

冉

u⫹ ux ␥⫺1Dx⫺1 2v␥⫺2 ␥⫺1 ⫹ 共v␥⫺2_兲 x ␥⫺1 Dx⫺1 2v ␥⫺1 ⫹ vx ␥⫺1Dx⫺1 u⫹ ␥⫺2 ␥⫺1uxDx⫺1

冊

. 共56兲

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⳵L

⳵tn⫹1

⫽L_⳵⳵L tn

⫹兵Rn,L其,

in the same way as for the polynomial Lax function one can find the recursion operator 共56兲. 䊐 It is interesting to note that the equation共47兲 and equations of polytropic gas dynamics 共55兲 are related by the following change of variables:

S⫽ u 共⫺k⫹2兲共⫺k⫹1兲, 共57兲 P⫽ v 1/共⫺k⫹1兲 共⫺k⫹2兲2,

where ␥⫽ (⫺k⫹2)/(⫺k⫹1). We note that under this change of variables recursion operator 共48兲 is mapped to the recursion operator 共56兲.

VII. REDUCTION

In this section we consider reductions of the equation共12兲, written in symmetric variables, by setting u1⫽0, or u1⫽uN,..., or u1⫽u2⫽¯ ,⫽uN. These reductions correspond to the Lax

equations with different Lax functions. For reduction u1⫽0 we have a polynomial Lax function with simple roots L⫽(p⫺uN)¯(p⫺u2) and for reduction uN⫽u1 we have a polynomial Lax

function with a root of multiplicity two L⫽ (1/p) (p⫺uN)2( p⫺uN⫺1) . . . ( p⫺u2), 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. Reductionu1Ä0

Let us write the equation共12兲 as

⌬共uN, . . . ,u1兲⫽0, 共58兲 where⌬ is a differential operator. Then

⌬共uN, . . . ,u1兲兩u₁⫽0⫽

冉

⌬˜共uN, . . . ,u2兲,

0

冊

, 共59兲

right hand side at p⫽0 we get 共59兲. A new equation,

⌬˜共uN, . . . ,u2兲⫽0, 共61兲 is also integrable and a recursion operator of this equation can be obtained as a reduction of the recursion operator of the equation共58兲. Let R be the recursion operator of 共58兲 given by Lemma 2, then

共62兲

also polynomials.

Lemma 3: The operator R˜ is a recursion operator of the equation (61).

Proof: Equation共61兲 is an evolution equation, so, to prove that R˜ is a recursion operator we

must prove that for any solution (u_N, . . . ,u₂) of共61兲 the following equality holds 共see Ref. 6兲:

D_⌬˜R˜⫽R˜D_⌬˜, where D_⌬˜ is a Frechet derivative of⌬˜.

If (u_N, . . . ,u₂) is a solution of共61兲 then (u_N, . . . ,u₂,u₁⫽0) is a solution of 共58兲 and for the solution (u_N, . . . ,u₂,u₁⫽0) we have

D_⌬R⫽R D_⌬. 共64兲

Next and

Hence by 共64兲 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

u1⫽0 and differentiate with respect to other variables or we can first differentiate and then put

u1⫽0. It means that D˜⫽D⌬˜ and

D_⌬˜R˜⫽R˜D_⌬˜.

䊐

Let us consider the reduction of systems, given by Remark 2 and Remark 4 and their recursion operators.

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ut⫽

1

2u共⫺ux⫹vx兲,

共65兲

vt⫽12v共⫹ux⫺vx兲,

and its recursion operator,

R⫽

冉

⫺uv⫹u 4共u⫹v兲 ⫺ u 4共u⫹v兲 ⫹u₄共ux⫺vx兲Dx⫺1 ⫹ u 4共ux⫺vx兲Dx⫺1

⫺v₄共u⫹v兲 ⫺uv⫹v₄共u⫹v兲 ⫹v₄共⫺ux⫹vx兲Dx⫺1 ⫹

v

4共⫺ux⫹vx兲Dx⫺1

冊

, 共66兲

respectively. 䊐

Proposition 10: Putting w⫽0 in (29) and (A1) we obtain a new system, 1 3ut⫽共⫺18u 2_⫹1 2uv⫹ 1 8v 2_兲u x⫹共14u 2_⫹1 4uv兲vx, 共67兲 1 3vt⫽共 1 4v 2_⫹1 4uv兲ux⫹共⫺ 1 8v 2_⫹1 2uv⫹ 1 8u 2_兲v x,

, 共68兲 respectively. 䊐

It is worth mentioning that by reduction we obtain a new equation. For example, consider the case k⫽0. The equation 共25兲, corresponding to N⫽2, and reduction of the equation 共29兲, corre-sponding to N⫽3, are not related by a linear transformation of variables. Indeed, in the equation 共25兲 coefficients of ux,vx are linear in u,v but in the equation共67兲 coefficients of ux,vx contain

quadratic terms. Hence they cannot be related by a linear transformation.

B. ReductionuNÄu1

It follows from共60兲 that the Lax equation 共12兲 can be written as

ui,t⫽

兺

j⫽1 N

h_ij共uN, . . . ,u1兲uj,x, 共69兲

where i, j⫽1, . . . ,N and h_ij⫽h1(ui,uN, . . . ,uˆi, . . . ,u1) when i⫽ j and hi i

⫽h2(ui,uN, . . . ,uˆi, . . . ,u1), the overcaret denotes the absence of the corresponding variable. It also follows from共60兲 that the functions h1(xN, . . . ,x1) and h2(xN, . . . ,x1) are symmetric under permutations of variables xN⫺1, . . . ,x1.

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u_N,t⫽„h_NN共u_N,u_N_⫺1, . . . ,u₂,u_N兲⫹h_N1共u_N,u_N_⫺1, . . . ,u₂,u_N兲…u_N,x ⫹

兺

j⫽2 N⫺1 h_Nj共uN,uN_⫺1, . . . ,u2,uN兲uj,x, 共70兲 ui,t⫽2hi N_共u N,uN⫺1, . . . ,u2,uN兲uN,x⫹

兺

j⫽2 N⫺1 h_ij共uN,uN⫺1, . . . ,u2,uN兲uj,x, where i⫽(N⫺1), . . . ,2.

The Frechet derivative of共69兲, under condition u_N⫽u1, has the form

D_⌬兩u_N⫽u₁⫽

冉

a11 a12 ¯ a1(N⫺1) a1N a₂₁ a₂₂ ¯ a_2(N_⫺1) a₂₁ ] ] ¯ ] ] a_(N_⫺1)1 a_(N_⫺1)2 ¯ a(N_⫺1)(N⫺1) a_(N_⫺1)1 a1N a12 ¯ a1(N⫺1) a11

冊

, 共71兲

where ai j, i, j⫽1, . . . ,N are differential operatos. So, the Frechet derivative of 共70兲 can be writen

as D_⌬¯⫽

冉

a11⫹a1N a₁₂ ¯ a_1(N_⫺1) 2a21 a22 ¯ a2(N⫺1) ] ] ¯ ] 2a(N⫺1)1 a(N⫺1)2 ¯ a(N⫺1)(N⫺1)

冊

. 共72兲

Now let us write the recurcion operator of 共69兲, given by Lemma 2. From 共63兲 it follows that, under condition uN⫽u1, it has the form

R兩u_N⫽u1⫽

冉

b₁₁ b₁₂ ¯ b_1(N_⫺1) b_1N b21 b22 ¯ b2(N⫺1) b21 ] ] ¯ ] ] b(N⫺1)1 b(N⫺1)2 ¯ b(N⫺1)(N⫺1) b(N⫺1)1 b1N b12 ¯ b1(N⫺1) b11

冊

, 共73兲

where bi j, i, j⫽N, . . . ,1 are differential operators.

Now we can write a recursion operator for Eq.共70兲,

R¯_⫽

冉

b11⫹b1N b12 ¯ b1(N⫺1) 2b21 b22 ¯ b2(N⫺1) ] ] ¯ ] 2b(N⫺1)1 b(N⫺1)2 ¯ b(N⫺1)(N⫺1)

冊

. 共74兲

The form of 共74兲 can be deduced by applaing operator R兩u_N⫽u₁ to a symmetry

(⳵u_N/⳵t_n,⳵u_N_⫺1/⳵t_{n, . . . ,}⳵u₂/⳵t_n,⳵u_n/⳵t_n).

Lemma 4: The operator R¯ in (74) is a recursion operator of the equation (70).

Proof: Equation共70兲 is an evolution equation, so, to prove that R¯ is a recursion operator we must prove that for any solution (uN, . . . ,u2) of共70兲 the following equality holds 共see Ref. 6兲:

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If (uN, . . . ,u2) is a solution of共70兲 then (uN, . . . ,u2,u1⫽uN) is a solution of共69兲 and for the

solution (uN, . . . ,u2,u1⫽uN) we have

D_⌬R⫽R D_⌬. 共75兲

One can show that from commutation of共71兲 and 共73兲 follows the commutation of 共72兲 and 共74兲

that is equality共75兲. 䊐

Let us consider reduction of systems, given by Remark 2 and Remark 4 and their recursion operators.

Proposition 11: Putting w⫽u in (38) and (A2) we obtain a new system, ut⫽ 1 2uvx, 共76兲 vt⫽ 1 2v共2ux⫺vx兲,

and its recursion operator

R⫽

冉

⫺共2uv⫹u2_兲⫺3 2uv ⫺ 1 4u共2u⫹v兲⫺ 3 2u 2 ⫹1 2uvxDx⫺1 ⫹ 1 4u共2ux⫺vx兲Dx⫺1

⫺2u共uv兲xDx⫺1•u⫺1 ⫺u共uv兲xDx⫺1•v⫺1

⫺1 2v共2u⫹v兲⫺3uv ⫺共2uv⫹u 2_兲⫹1 4v共2u⫹v兲 ⫹1 2v共⫺2ux⫹vx兲Dx⫺1 ⫹ 1 4v共2⫺ux⫹vx兲Dx⫺1

⫺2v共uv兲xDx⫺1•u⫺1 ⫺v共u

2_兲

xDx⫺1•v⫺1

冊

. 共77兲

䊐 Proposition 12: Putting w⫽u in (29) and (A1) we obtain a new system,

1 3ut⫽共u2⫹2uv⫹ 1 8v 2_兲u x⫹共u2⫹ 1 4uv兲vx, 共78兲 1 3vt⫽共12v 2_⫹2uv兲u x⫹共⫺18v 2_⫹uv⫹u2_兲v x,

R⫽

冉

u2⫹72uv⫹共u 2_⫹uv兲 xDx⫺1 2u 2_⫹1 4uv⫹ 1 2共u 2_⫹uv兲D x ⫺1 ⫹1 2uxDx⫺1•v ⫹ 1 2uxDx⫺1•u⫺ 1 4uxDx⫺1•v 4uv⫹1 2v 2_⫹2共uv兲 xDx⫺1 ⫺ 1 4v 2_⫹3 2uv⫹u 2_⫹共uv兲 xDx⫺1 ⫹1 2vxDx⫺1•v ⫹ 1 2vxDx⫺1•u⫺ 1 4vxDx⫺1•v

冊

. 共79兲䊐 We may go on introducing new reductions. For instance a reduction of the type u1⫽u2⫽uN,

(N⬎3), reduces an N-system to an (N⫺2)-system. One may obtain this (N⫺2)-system also from the polynomial Lax function having the form L⫽p⫺1( p⫺u1)3( p⫺u3)¯(p⫺uN_⫺1)共a zero

of L with multiplicity three兲. In this way one obtains an infinite number of different classes of N⫽2, N⫽3 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 A⫹B D⫺1 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共i.e., they are also equations of

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hydro-dynamic type兲. 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 N⫽k systems. These properties, the bi-Hamiltonian structure of the equations we obtained and equa-tions with rational Lax funcequa-tions, will be communicated elsewhere.

ACKNOWLEDGMENTS

We thank Burak Gu¨rel 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 FORNÄ3 SYSTEMS„29…AND„38…

Recursion operators of the systems共29兲 and 共38兲 are, respectively, given by

R⫽

¨

⫺u 2 4 ⫹ 3 4共uv⫹uw兲⫹wv u 4共u⫹v⫹w兲⫹ 3uw 2 u 4共u⫹v⫹w兲⫹ 3uv 2 ⫹ux 2 共v⫹w兲Dx ⫺1 _⫹ux 2 共v⫹w兲Dx ⫺1 _⫹ux 2 共v⫹w兲Dx ⫺1 ⫹u 2共vx⫹wx兲Dx ⫺1 _⫹u 2共vx⫹wx兲Dx ⫺1 _⫹u 2共vx⫹wx兲Dx ⫺1 ⫺ux 4 Dx ⫺1_•u⫹ux 4 Dx ⫺1_•v _⫹ux 4 Dx ⫺1_•u⫺ux 4 Dx ⫺1_•v _⫹ux 4 Dx ⫺1_•u⫹ux 4 Dx ⫺1_•v ⫹u₄xDx⫺1•w ⫹ ux 4 Dx ⫺1_•w _⫺ux 4 Dx ⫺1_•w v 4共u⫹v⫹w兲⫹ 3vw 2 ⫺ v2 4 ⫹ 3 4共uv⫹vw兲⫹uw v 4共u⫹v⫹w兲⫹ 3uv 2 ⫹vx 2 共u⫹w兲Dx⫺1 ⫹ vx 2 共u⫹w兲Dx⫺1 ⫹ vx 2 共u⫹w兲Dx⫺1 ⫹v₂共ux⫹wx兲Dx⫺1 ⫹ v 2共ux⫹wx兲Dx⫺1 ⫹ v 2共ux⫹wx兲Dx⫺1 ⫺vx 4 Dx⫺1•u⫹ vx 4 Dx⫺1•v ⫹ vx 4 Dx⫺1•u⫺ vx 4 Dx⫺1•v ⫹ vx 4 Dx⫺1•u⫹ vx 4 Dx⫺1•v ⫹vx 4 Dx⫺1•w ⫹ vx 4 Dx⫺1•w ⫺ vx 4 Dx⫺1•w w 4共u⫹v⫹w兲⫹ 3vw 2 w 4共u⫹v⫹w兲⫹ 3uw 2 ⫺ w2 4 ⫹ 3 4共uw⫹vw兲⫹uv ⫹wx 2 共u⫹v兲Dx ⫺1 _⫹wx 2 共u⫹v兲Dx ⫺1 _⫹wx 2 共u⫹v兲Dx ⫺1 ⫹w 2共ux⫹vx兲Dx ⫺1 _⫹w 2共ux⫹vx兲Dx ⫺1 _⫹w 2共ux⫹vx兲Dx ⫺1 ⫺wx 4 Dx ⫺1_•u⫹wx 4 Dx ⫺1_{•v ⫹}wx 4 Dx ⫺1_•u⫺wx 4 Dx ⫺1_{•v ⫹}wx 4 Dx ⫺1_•u⫹wx 4 Dx ⫺1_•v ⫹w₄xDx⫺1•w ⫹ wx 4 Dx ⫺1_•w _⫺wx 4 Dx ⫺1_•w

©

, 共A1兲

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R⫽

¨

⫺共uv⫹uw⫹vw兲 ⫺u 4共u⫹v⫹w兲 ⫺ u 4共u⫹v⫹w兲 ⫹u₄共u⫹v⫹w兲 ⫺3uw₂ ⫺3u₂v ⫹u₄共ux⫺vx⫺wx兲Dx⫺1 ⫹ u 4共ux⫺vx⫺wx兲Dx⫺1 ⫹ u 4共ux⫺vx⫺wx兲Dx⫺1

⫺u共wvx⫹vwx兲Dx⫺1•u⫺1 ⫺u共wvx⫹vwx兲Dx⫺1•v⫺1 ⫺u共wvx⫹vwx兲Dx⫺1•w⫺1

⫺v 4共u⫹v⫹w兲 ⫺共uv⫹uw⫹vw兲 ⫺ v 4共u⫹v⫹w兲 ⫺3vw₂ ⫹v₄共u⫹v⫹w兲 ⫺3u₂v ⫹v₄共⫺ux⫹vx⫺wx兲Dx⫺1 ⫹ v 4共⫺ux⫹vx⫺wx兲Dx⫺1 ⫹ v 4共⫺ux⫹vx⫺wx兲Dx⫺1

⫺v共wux⫹uwx兲Dx⫺1•u⫺1 ⫺v共wux⫹uwx兲Dx⫺1•v⫺1 ⫺v共wux⫹uwx兲Dx⫺1•w⫺1

⫺w₄共u⫹v⫹w兲 ⫺w₄共u⫹v⫹w兲 ⫺共uv⫹uw⫹vw兲

⫺3uw₂ ⫺3vw₂ ⫹w₄共u⫹v⫹w兲 ⫹w₄共⫺ux⫺vx⫹wx兲Dx⫺1 ⫹ w 4共⫺ux⫺vx⫹wx兲Dx⫺1 ⫹ w 4共⫺ux⫺vx⫹wx兲Dx⫺1

⫺w共uvx⫹vux兲Dx⫺1•u⫺1 ⫺w共uvx⫹vux兲Dx⫺1•v⫺1 ⫺w共uvx⫹vux兲Dx⫺1•w⫺1

©

.

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