On construction of recursion operators from Lax representation

Metin Gürses, Atalay Karasu, and Vladimir V. Sokolov

Citation: J. Math. Phys. 40, 6473 (1999); doi: 10.1063/1.533102

View online: http://dx.doi.org/10.1063/1.533102

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On construction of recursion operators from Lax

representation

Metin Gu¨rsesa)

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

Vladimir V. Sokolov

Landau Institute, Moscow, 117940 Russia

共Received 28 June 1999; accepted for publication 16 August 1999兲

In this work we develop a general procedure for constructing the recursion opera-tors for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations. © 1999 American Institute of Physics. 关S0022-2488共99兲03212-0兴

I. INTRODUCTION

It is well known that most of the integrable nonlinear partial differential equations,

ut⫽F共t,x,u,ux,...,unx兲, 共1兲

admit a Lax representation,

Lt⫽关A,L兴, 共2兲

so that the inverse scattering method is applicable. The generalized symmetries1of共1兲 have also Lax representations with the same L operator,

L_t

n⫽关An,L兴, n⭓1. 共3兲

The recursion operatorR, satisfying the equation 共see Ref. 2兲

Rt⫹关DF,R兴⫽0, 共4兲

where DFis the Freche´t derivative of the function F, generates symmetries of共1兲 starting from the

simplest ones. In general,R is a nonlocal operator 共a pseudodifferential operator兲.

The construction of the recursion operator of a given integrable system共1兲 is not an easy task. Several works are devoted to this subject. Among these works, most of the authors use共4兲 for the construction of the recursion operator.3–8There are several difficulties in this direct approach. The main problems are the choices of the order ofR and the structure of the nonlocal terms. This is an approach having no relation with the Lax representation共2兲.

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 operator9–13and are based on finding an eigenvalue equation for the squared eigenfunctions of the Lax operator. The operator correspond-ing to this eigenvalue equation turns out to be the adjoint of the recursion operator.

a兲_{Electronic mail: gurses@fen.bilkent.edu.tr}

6473

There is an alternative use of the Lax representation to construct recursion operators. This approach is based on the explicit construction of the Anoperators共3兲. It was first used by Symes,14

Adler15 共see also Dorfman–Fokas,16 Fokas–Gel’fand17兲 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–Dikii20 construction of the An operators. On the other hand,

Antonowicz–Fordy determines these operators from integrability condition共3兲 and by using an ansatz for An. Their basic aim is to determine the Hamiltonian operators ␪1 and ␪221 of the

equations under consideration. The recursion operator is simply given by R⫽␪2␪₁⫺1. Their ap-proach is based on some explicit formulas for coefficients of the Anoperator. 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 operatorR, we shall show in this work that it is possible to succeed this without any concrete information of the coefficients of An

operators. We use only an ansatz A˜⫽PA⫹R that relates An operators for different n. HereP 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 共2兲 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

Lt⫽关A,L兴, 共5兲

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共3兲. It is well known that one can define operators A_n by the following formula:20

An⫽共Ln/m兲_⫹, 共6兲

where Ln/m is a pseudodifferential series of the form Ln/m⫽兺_⫺⬁n viDi and (Ln/m)⫹⫽兺i⫽0 n

viDi.

Hereviare some concrete functions depending on the coefficients of L and D is the total

deriva-tive 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 inte-grable 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 共DS兲 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.

A. Gel’fand–Dikii systems

In this section we shall consider the case where L is a differential operator,

L⫽Dm⫹u_m⫺2Dm⫺2⫹¯⫹u₀, 共7兲

where ui, i⫽0,1,...,m⫺2 are functions of x, t. In the framework of Ref. 25, this corresponds to the

Kac–Moody algebras of the type A_m(1)_⫺1.

To show that共3兲 is equivalent to a system of (m⫺1) evolution equations with respect to ui

one can use the following standard reasoning. Set

Ln/m⫽共Ln/m兲_⫹⫹共Ln/m兲_⫺, 共8兲

where (Ln/m)_⫹ is the differential part of the series Ln/m and (Ln/m)_⫺ is a series of order⭐⫺1. Since关L,Ln/m兴⫽0 we have

关共Ln/m_兲

⫹,L兴⫽关L,共Ln/m兲⫺兴. 共9兲

The left-hand side of 共9兲 is a differential operator, but the right side is a series of order⭐n⫺2. Thus, both sides of共3兲 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 Di, where 0,...,m⫺2 in 共3兲.

Since L(n⫹m)/m⫽LLn/m, then we have

Am⫹n⫽共LLn/m兲⫹⫽L共Ln/m兲⫹⫹„L共Ln/m兲⫺…⫹, 共10兲

which leads directly to

Lt_n⫹m⫽关An⫹m,L兴⫽LLt_n⫹关„L共Ln/m兲⫺…⫹,L兴. 共11兲

The above equation共11兲 has been given also by Symes14 共see also Adler’s paper15兲. In his work Symes expressed the coefficients of the both parts of共11兲, 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_t

n⫹min terms of Lt_n. 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_t

n⫹m⫽LLt_n⫹关Rn,L兴. 共12兲

Obviously, it follows from the following.

Proposition 1: For any n,

A_n⫹m⫽LA_n⫹R_n, 共13兲

where Rn is a differential operator of order⭐m⫺1.

Proof: The relation共13兲 coincides with 共10兲 if we put

Rn⫽„L共Ln/m兲⫺…⫹. 共14兲

Since (Ln/m₎

⫺is a series of order⭐⫺1, then ord(Rn)⭐m⫺1.

Remark 1: It follows from the formula

An⫹m⫽共Ln/mL兲⫹⫽共Ln/m兲⫹L⫹共共Ln/m兲⫺L兲⫹, 共15兲

An⫹m⫽AnL⫹R¯n, 共16兲

and

Lt_n⫹m⫽Lt_nL⫹关L,R¯n兴, 共17兲

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共12兲. It is easy to see that in this comparison of the coefficients of Di, i⫽2m⫺2,...,m⫺1 we determine Rn in terms of the coefficients of operators L and Lt_n. It is important that the resulting

formulas turn out to be linear in the coefficients of Lt_n. The remaining coefficients of Di, i⫽m ⫺2,...,0 in 共12兲 give us the relation

冉

冊

冉

冊

whereR is a recursion operator. Instead of 共12兲 one can use 共17兲. The corresponding recursion operators coincide.

Example 1. KdV equation: The KdV equation, ut⫽

1

4共u3x⫹6uux兲, 共19兲

has a Lax representation with

L⫽D2_{⫹u, A⫽共L}3/2_兲

⫹. 共20兲

Since in this case L_t

n⫹2⫽ut_n⫹2⬅un⫹2 and Lt_n⫽ut_n⬅un, the main relation共12兲 takes the form

un⫹2⫽共D2⫹u兲•un⫹关Rn,L兴, 共21兲

with Rn⫽anD⫹bn.

Now if we equate successively to zero the coefficients of D2, D, and D0 in the above equation, we obtain an⫽ 1 2D⫺1共un兲, bn⫽ 3 4un, and u_n⫹2⫽共1 4D 2_⫹u⫹1 2uxD⫺1兲un,

that gives the standard recursion operator for the KdV equation,

R⫽1 4D

2_⫹u⫹1

2uxD⫺1. 共22兲

In the same way one can find a recursion operator for the Boussinesq equation共see Appendix A兲.

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⫽⌺aiDi. Its adjoint L* is given by L*⫽⌺(⫺D)i•ai. 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 Anare defined by共6兲, where n takes odd

integer values. This condition provides that (An)*⫽⫺An that is necessary for 共3兲 to be

compat-ible.

If L*⫽L, the formula An⫹m⫽(LLn/m)⫹⫽(L(n⫹m)/m)⫹ gives a correct An operator since n ⫹m is an odd integer. Thus, in this case Proposition 1 remains valid and the recursion operator can be found from共12兲 or 共17兲.

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 An operator. In this 共skew adjoint兲 case we must take

An⫹2m⫽共L共n⫹2m兲/m兲⫹⫽共L2Ln/m兲⫹,

to find the recursion operator. Following the proof of Proposition 1 we obtain Proposition 2.

Proposition 2: If L*⫽⫺L then

An⫹2m⫽L2An⫹Rn, 共23兲

where ord(Rn)⬍2 ord(L). It follows from 共23兲 that

Lt_n⫹2m⫽L2Lt_n⫹关Rn,L兴. 共24兲

Remark 2: Instead of共23兲 we can use the ansatz

An_⫹2m⫽LAnL⫹R˜n, 共25兲

or

An⫹2m⫽AnL2⫹R˜˜n. 共26兲

The recursion operators obtained by the utility of共23兲, 共25兲, and 共26兲 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⫺1is a differential operator. In this general reductions, as well, possible An operators are given by 共6兲, 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. 共12兲, 共24兲 accordingly to obtain the recursion operators.

Example 2. Kupershmidt equation: This equation,

ut⫽u5x⫹10uu3x⫹25uxu2x⫹20u2ux, 共27兲

has the Lax pair

L⫽D3⫹2uD⫹ux, A⫽共L5/3兲⫹. 共28兲

In this case L*⫽⫺L; therefore we use Eq. 共24兲 with

R

˜_n⫽anD5_⫹b

nD4⫹cnD3⫹dnD2⫹enD⫹ fn. 共29兲

By equating the coefficients of powers of D in共24兲, we obtain

an⫽ 2 3D⫺1共un兲, bn⫽ 11 3un, cn⫽ 1

9„20uD⫺1共un兲⫹73un,x…,

dn⫽

1

en⫽

1

27„70u2xD⫺1共un兲⫺2D⫺1共u2xun兲⫹40u2D⫺1共un兲⫺8D⫺1共u2un兲 ⫹134un,3x⫹212uun,x⫹184uxun…,

fn,x⫽

1

27„20u4xD⫺1共un兲⫹74u3xun⫹126u2xun,x⫹40uu2xD⫺1共un兲⫹40ux

2

D⫺1共un兲 ⫹136uxun,2x⫹27uuxun⫹28un,5x⫹64uun,3x⫹16u2un,x…,

and the recursion operator for the Kupershmidt equation:

R⫽D6_⫹12uD4_⫹36u

xD3⫹共49u2x⫹36u2兲D2⫹5共7u3x⫹24uux兲D⫹13u4x⫹82uu2x⫹69ux

2

⫹32u3_⫹2u

xD⫺1共u2x⫹4u2兲⫹2共u5x⫹10uu3x⫹25uxu2x⫹20u2ux兲D⫺1. 共30兲

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共13兲 and 共23兲 the Rn operator also becomes a

pseudodif-ferential operator.

It follows from these formulas that the structure of the nonlocal terms in Rn is, in general,

similar to the nonlocal terms in L since An⫹m and An are differential operators.

For skew-symmetric case, An may be defined by either共23兲 or 共25兲, or 共26兲. In the

pseudo-differential case they are not equivalent, in the sense that the nonlocal part of Rndepends 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, 共31兲

then the formulas共13兲 and 共25兲 are much suitable then 共16兲, 共23兲, and 共26兲.

Lemma: Let L†⫽⑀L, where⑀⫽⫾1. Then

Rn⫽Dm⫺1⫹¯⫹a0, for ⑀⫽1, 共32兲

where Rn is defined by共13兲, and

R ˜ n⫽D 2m_{⫺1⫹¯⫹a} ⫺1D⫺1, for ⑀⫽⫺1, 共33兲 where R˜n is defined by共25兲.

Proof: If L⫽MD⫺1 then L†⫽⑀L implies M*⫽⫺⑀M . It is easy to show that (L1/m)† ⫽⫺L1/m_{. Hence (L}n/m₎†_⫽⫺Ln/m _{for an odd integer n. Define now a series K}

n by

Ln/m⫽DKn.

It is easy to prove that K_n*⫽K_n. Since K_n⫽(K_n)_⫹⫹(K_n)_⫺and (K_n)*⫽K_n, we have 共Kn兲_⫹*⫽共Kn兲⫹, 共Kn兲_⫺*⫽共Kn兲⫺.

From the last formula it follows that ord(K_n)_⫺⭐⫺2, which leads to an important result,

A_n⫽共Ln/m_兲

⫹⫽D共Kn兲⫹.

This implies that

LAn⫽M共Kn兲⫹ 共34兲

is a differential operator. Now using共34兲 in 共13兲 and 共25兲 for the cases⑀⫽1 and⑀⫽⫺1, respec-tively, we find the ansatz for An given by共32兲 and 共33兲.

Example 3 (⑀⫽⫺1): It is known that the KdV equation has, besides the standard Lax

repre-sentation, the following Lax pair:

L⫽共D2⫹u兲D⫺1, A⫽共L3兲_⫹. 共35兲

The L operator satisfies the reduction L†⫽⫺L. According to the formula 共33兲 we have

R ˜

n⫽anD⫹bn⫹cnD⫺1.

It follows from共25兲 that

an⫽D⫺1共un兲, bn⫽un, cn⫽⫺un,x⫺uD⫺1共un兲.

The remaining equation in共25兲 gives the recursion operator

R⫽D2_⫹4u⫹2u

xD⫺1. 共36兲

Example 4 (⑀⫽1). DSIII system: The DSIII system25,26is given by

ut⫽⫺u3x⫹6uux⫹6vx,

共37兲

vt⫽2v3x⫺6uvx.

The nonlocal Lax representation for this system is

L⫽共D5⫺2uD3⫺2D3u⫺2Dw⫺2wD兲D⫺1,

共38兲

A⫽共L3/4兲_⫹,

where w⫽v⫺u2x. Since L†⫽L we can use 共32兲, which gives us

Rn⫽anD3⫹bnD2⫹cnD⫹dn. 共39兲

By equating the coefficients of the powers of D in共25兲, we obtain

an⫽D⫺1共un兲, bn⫽4un,

cn⫽

1

2„⫺6uD⫺1共un兲⫹11un,x⫹2D⫺1共uun兲⫹2D⫺1共vn兲…,

d_n,x⫽⫺1

2„6u2xD⫺1共un兲⫹10uxun⫺5un,3x⫹4uun,x⫺6vn,x….

The recursion operator of the DSIII is found as

R⫽

冉

冊

xD⫺8u2x⫹16u2⫹16v⫹共⫺2u3x⫹12uux⫹12vx兲D⫺1⫹4uxD⫺1u,

R1 0_⫽⫺10D2_⫹8u⫹4u xD⫺1, 共41兲 R0 1_⫽10v xD⫹12v2x⫹共4v3x⫺12uvx兲D⫺1⫹4vxD⫺1u, R1 1_⫽⫺4D4_⫹16uD2_⫹8u xD⫹16v⫹4vxD⫺1.

III. MATRIXL 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

L⫽Dx⫹␭a⫹q共x,t兲. 共42兲

A. General case

Let us consider the Lax operator共42兲, 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共ada兲丣Im共ada兲. 共43兲

First, let us recall the procedure25of constructing the A operators for the Lax operator 共42兲.

Proposition 3: There exist unique series,

u⫽u_⫺1␭⫺1⫹u_⫺2␭⫺2⫹¯ , ui苸Im共ada兲, 共44兲

h⫽h₀⫹h_⫺1␭⫺1⫹h_⫺2␭⫺2⫹¯ , h_i苸Ker共ad_a兲, 共45兲

such that

eadu共L兲⫽L⫹关u,L兴⫹1

2†u,关u,L兴‡⫹¯⫽Dx⫹a␭⫹h. 共46兲

Let b be a constant element of g such that 关b,Ker(ad_a)兴⫽兵0其. It follows from 共45兲 that 关b␭n_,D

x⫹a␭⫹h兴⫽0. Hence 关⌽b,n,L兴⫽0, where

⌽b,n⫽e⫺adu共b␭n兲. 共47兲

Then the corresponding A operator of the form

Ab,n⫽b␭n⫹an⫺1␭n⫺1⫹¯⫹a0, 共48兲

is defined by the formula

Ab,n⫽共⌽b,n兲⫹, 共49兲 where 共⌺_⫺⬁n _␣ i␭i兲⫹⫽⌺0 n_␣ i␭i. 共50兲 According to共47兲, ⌽b,n⫹1⫽␭⌽b,n. 共51兲 Hence Ab,n_⫹1⫽共␭⌽b,n兲_⫹⫽␭共⌽b,n兲_⫹⫹„␭共⌽b,n兲_⫺…_⫹. 共52兲

The last formula shows that

Ab,n⫹1⫽␭Ab,n⫹Rn, Rn苸g, 共53兲

where Rn does not depend on␭. Substituting 共53兲 into the Lax equation Lt_n⫹1⫽关Ab,n_⫹1,L兴, we

get

Using the ansatz 共54兲, one can easily find the corresponding recursion operator.

Example 5: The system

ut⫽⫺ 1 2uxx⫹u2v, 共55兲 vt⫽ 1 2vxx⫺v2u,

is equivalent to the nonlinear Schro¨dinger equation, has a Lax operator

L⫽D⫹

冉

1 0

0 ⫺1

冊

冉

0 u

v 0

冊

The Lie algebra g in this example coincides with sl(2). Using共54兲 with Rn⫽

冉

冊

and the recursion operator of the system共55兲 is given by

R⫽

冉

冊

B. Reductions in matrix case

In the general case considered in the previous section the An operators belong to the Lie

algebra,

a_⫹⫽兵⌺i␬⫽0ai␭i, ai苸g, ␬苸Z⫹其, 共58兲

that is a subalgebra of the Lie algebra, a⫽兵⌺_⫺⬁␬ _a

i␭i, ␣i苸g, ␬苸Z其. 共59兲

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 gi be an eigenspace corresponding to eigenvalue ⑀i. Then the following reduction aj 苸gi, where i⫽ j(mod␬) in共58兲 and 共59兲 is compatible with Eqs. 共3兲. Note that according to this

definition a苸g₁, and the potential q(x,t) in共42兲 belongs to g₀ 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, 共60兲

where

Rn⫽r␬⫺1␭␬⫺1⫹¯⫹r0, ri苸gi. 共61兲

Further generalizations are associated with modifications of sign ‘‘⫹’’ in 共50兲, which corre-sponds to the simplest decomposition of algebraa into the direct sum of two subalgebras,

a⫽a⫹丣a⫺, 共62兲

wherea_⫹ is given by共58兲 and

a⫺⫽兵⌺⫺⬁⫺1ai␭i, ai苸g其. 共63兲

The sign ‘‘⫹’’ in 共50兲 is the projection of onto a_⫹parallel toa_⫺. If we have a different decom-position 共62兲, then the construction from Proposition 3 is also valid, but we have the following condition:

R_n苸a_⫹艚␭a_⫺, 共64兲

instead of Rn苸g. If we also have the␴ reduction, we must use the most general ansatz 共60兲,

where

Rn苸a⫹艚␭␬a⫺. 共65兲

Example 6: Let us consider the following equation: ut⫽ 1 4uxxx⫺ 3 8uxxu⫹ 3 8uuxx⫺ 3 8uuxu, 共66兲

where u is a square matrix of arbitrary size, or more generally, u belongs to an arbitrary associa-tive algebra K. This equation has a Lax representation with

L⫽D⫹

冉

0 1

1 0

冊

冉

u 0

0 0

冊

Here 1 is the unity ofK. The reduction 共67兲 can be described as follows 共see Ref. 27兲. The Lie algebra g is the algebra of all 2⫻2 matrices with entries belonging to K. The automorphism␴is defined by

␴共X兲⫽TXT⫺1_{, 共68兲}

where

T⫽

冉

0 ⫺1

冊

Obviously ␴2_{⫽Id and eigenvalues of␴are 1 and⫺1. The corresponding eigenspaces are} g0⫽

再

* 0

0 _*

冎

再

0 _*

* 0

冎

and therefore the coefficients aiin共59兲 have the following structure:

a2 j⫽

冉

* 0

0 _*

冊

冉

0 _*

* 0

冊

The subalgebraa_⫹is given by共58兲, where the coefficients have the structure 共70兲 and, addition-ally,

a0⫽

冉

* 0

0 0

冊

a_⫺⫽⌺_⫺⬁0 _a

i␭i, 共71兲

where a0 is of the form

a₀⫽

冉

0 ␣

冊

The A operator for共66兲 is given by formula A⫽(⌽a,3)⫹关see 共49兲兴, where

a⫽

冉

0 1

1 0

冊

According to共65兲, Rn is of the form

R_n⫽

冉

冊

冉

冊

冉

冊

un⫺an,x⫹关an,u兴⫹bn⫺cn⫽0 cn⫺bn⫺an,x⫽0,

dn⫺bn,x⫺ubn⫽0, dn⫹cn,x⫺cnu⫽0,

un_⫹2⫽⫺dn,x⫹关dx,u兴.

Finding an, bn, cn, and dn from this system, we obtain the following recursion operator:

R⫽⫺共D⫹adu兲共⫺D⫹Ru兲共2D⫹adu兲⫺1共D⫹Lu兲D共2D⫹adu兲⫺1, 共74兲

where Ru and Lu are the operators of right and left multiplications by u, respectively.

Note that in the commutative case共66兲 coincides with the modified KdV equation. It is easy to verify that 共74兲 becomes the standard recursion operator of a modified KdV equation. All factors in共74兲 have to be regarded as operators acting on a 共noncommutative兲 polynomial depend-ing on u,ux,uxx,... .

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 共TUBITAK兲. M. G. is a member of the Turkish Academy of Sciences 共TUBA兲. V. S. is supported by Russian Foundation of Basic Researches 共RFBR兲 Grant No. 99-01-00294 and INTAS.

APPENDIX A: EXAMPLE TO SEC. II A The Boussinesq equation,

utt⫽⫺13„u4x⫹2共u2兲2x…, 共A1兲

can be expressed in the form of a pair of first-order evolution equations,

ut⫽vx,

vt⫽⫺

1

3共u3x⫹8uux兲. 共A2兲

This system has a Lax pair,

L⫽D3⫹2uD⫹ux⫹v, A⫽共L2/3兲⫹. 共A3兲

To construct the recursion operator for this system, we use Eq.共12兲 with the differential operator,

Rn⫽anD2⫹bnD⫹cn.

By equating the coefficients of the powers of D in共12兲, we find

an⫽ 2 3D⫺1共un兲, bn⫽ 1 3„5un⫹D⫺1共vn兲…, cn⫽ 1

9„6vn⫹8uD⫺1共un兲⫹10un,x…,

and after that we obtain the recursion operator of the form共40兲 for 共A2兲 with

R0 0_⫽3v⫹2v xD⫺1, R1 0_⫽D2_⫹2u⫹u xD⫺1, 共A4兲 R0 1_⫽⫺„1 3D 4_⫹10 3uD 2_⫹5u xD⫹3u2x⫹163u 2_⫹共2 3u3x⫹163uux兲D⫺1…, R1 1_⫽3v⫹v xD⫺1.

APPENDIX B: EXAMPLES TO SEC. II B

1. Sawada–Kotera equation

The Lax pair for the Sawada–Kotera equation,28

ut⫽u5x⫹5uu3x⫹5uxu2x⫹5u2ux, 共B1兲

is given by

L⫽D3⫹uD, A⫽共L5/3兲_⫹. 共B2兲

In this example, L†⫽⫺L, where L†⫽D⫺1L*D and L is skew-symmetric, then we use共24兲. The

operator R˜n has the same form as 共29兲, with the coefficients given by

a_n⫽1

3D⫺1共un兲, bn⫽

5

3un, cn⫽

1

9„5uD⫺1共un兲⫹29un,x…,

dn⫽

1

en⫽

1

27„10u2xD⫺1共un兲⫺2D⫺1共u2xun兲⫺D⫺1共u2un兲⫹5u2D⫺1共un兲 ⫹28un,3x⫹32uun,x⫹32uxun…,

f_n⫽0.

The recursion operator is given as

R⫽D6_⫹6uD4_⫹9u

xD3⫹共9u2⫹11u2x兲D2⫹共10u3x⫹21uux兲D⫹5u4x⫹16uu2x⫹6ux

2_⫹4u3

⫹共u5x⫹5uu3x⫹5uxu2x⫹5u2ux兲D⫺1⫹uxD⫺1共u2⫹2u2x兲. 共B3兲 2. DSI system

The DSI system,25,26

ut⫽3vvx,

共B4兲

vt⫽2v3x⫹2uvx⫹vux,

has a Lax representation with

L⫽关D3⫹共u⫹v兲D⫹12共u⫹v兲x兴关D3⫹共u⫺v兲D⫹

1

2共u⫺v兲x兴,

共B5兲

A⫽共L1/2兲_⫹.

Here Rnis a differential operator of order 5, and since L is symmetric we again use Eq.共12兲. The

expressions for the coefficients of the operator Rnare very long and complicated. Hence we do not

display them here. We find that the recursion operatorR of this system is of the form 共40兲, where

R0 0_⫽⫺4D6_⫺24uD4_⫺27u xD 3_{⫹2共⫺49u} 2x⫺18u 2_⫹42v2_兲D2_{⫹10共⫺7u} 3x⫺12uux⫹30vvx兲D ⫺26u4x⫺82uu2x⫺69ux

2_⫹222vv

x⫹141vx

2_⫺16u3_⫹48v2_u

⫹2共⫺2u5x⫺10uu3x⫺25uxu2x⫺10u2ux⫹15v2ux⫹30vv3x⫹45vxv2x⫹30uvvx兲D⫺1 ⫹2uxD⫺1共3v2⫺2u2⫺u2x兲,

R1

0_⫽168vD4_⫹204vD3_⫹6共21v

2x⫹32uv兲D2⫹6共40vux⫹7v3x⫹22uvx兲D

⫹6共13vu2x⫹10uxvx⫹v4x⫹5uv2x⫹4vu2⫹12v3兲⫹108vvxD⫺1v⫹2uxD⫺1共6uv⫹9v2x兲,

共B6兲

R0

1_⫽56vD4_⫹268v

xD3⫹2共243v2x⫹32uv兲D2⫹2共36vux⫹219v3x⫹106uvx兲D ⫹2共27vu2x⫹92uxvx⫹99vax⫹99uv2x⫹4vu

2_⫹12v3_{兲⫹2共10vu}

3x⫹35u2xvx⫹45uxv2x

⫹10uvux⫹18v5x⫹30uv3x⫹10u2vx⫹15v2vx兲D⫺1⫹2vxD⫺1共3v2⫺2u2⫺u2x兲,

R1

1_⫽108D6_⫹216uD4_⫹432u

xD3⫹6共81u2x⫹18u2⫹22v2兲D2⫹6共45u3x⫹36uux⫹70vvx兲D ⫹3共18u4x⫹18uu2x⫹9ux

2_⫹98vv 2x⫹67vx 2_⫹32uv2_{兲⫹36共2v} 3x⫹2vxu⫹vux兲D⫺1v ⫹2vxD⫺1共6uv⫹9v2x兲. 3. DSII system

ut⫽3vx,

共B7兲

vt⫽⫺2共v3x⫹uvx⫹vux兲,

has a Lax representation with

L⫽共D5⫹uD3⫹D3u⫹共v⫹1 2u 2_{兲D⫹D共v⫹}1 2u 2_兲兲D, A⫽共L1/2兲_⫹. 共B8兲

Since L is symmetric we again use Eq.共12兲. In this case the operator Rn is given as follows:

Rn⫽anD5⫹bnD4⫹cnD3⫹dnD2⫹enD, 共B9兲 where an⫽ 1 3D⫺1共un兲, bn⫽ 5 3un, c_n⫽1

9关5uD⫺1共un兲⫹3D⫺1共vn兲⫹29un,x兴,

dn⫽

1

9关5uxD⫺1共un兲⫹26un,2x⫹14uun⫹12vn兴,

en⫽

1

27关5共2u2x⫹u2⫹3v兲D⫺1共un兲⫺3D⫺1共vun⫹uvn兲⫹9uD⫺1共vn兲 ⫺2D⫺1_共u

2xun⫹

1 2u

2_u

n兲⫹54uxun⫹28un,3x⫹32共uun,x⫺unux兲⫹42vn,x兴.

The recursion operator共40兲 for the system can be found as29

R0

0_⫽⫺D6_⫺6uD4_⫺9u

xD3⫺共11u2x⫹9u2⫹42v兲D2⫹共⫺10u3x⫺21uux⫺30vx兲D ⫺5u4x⫺16uu2x⫺6ux

2_⫺60v

2x⫺4u3⫺24vu⫹共⫺u5x⫺5uu3x⫺5uxu2x

⫺5u2_u

x⫺15vux⫺15v3x⫺15uvx兲D⫺1⫺uxD⫺1共2u2x⫹u2⫹3v兲,

R1

0_⫽⫺42D4_⫺48uD2_⫺87u

xD⫺6共7u2x⫹u2⫺6v兲⫹27vxD⫺1⫺3uxD⫺1u,

共B10兲

R0

1_⫽28vD4_⫹106v

xD3⫹共165v2x⫹32uv兲D2⫹共54vux⫹132v3x⫹74vxu兲D⫹30vu2x⫹79uxvx ⫹54v4x⫹57uv2x⫹4u2v⫺24v2⫹共10vu3x⫹25vxu2x⫹30uxv2x⫹10uvux⫹9v5x⫹15uv3x

⫹5u2 vx⫺15vvx兲D⫺1⫺vxD⫺1共3v⫹u 2_⫹2u 2x兲, R1 1_⫽27D6_⫹54uD4_⫹135u

xD3⫹3共54u2x⫹9u2⫺22v兲D2⫹3共36u3x⫹27uux⫺28vx兲D ⫹3共9u4x⫹9uu2x⫹9ux

2_⫺21v

2x⫺16vu兲⫺18共v3x⫹uxv⫹vxu兲D⫺1⫺3vxD⫺1u.

4. DSIV system

The DSIV system,25,26which is also known as the Hirota–Satsuma system,30,31

ut⫽

1

2u3x⫹3uux⫺6vvx,

共B11兲

vt⫽⫺v3x⫺3uvx,

has Lax representation with

Since the operator L is symmetric we use Eq.共12兲. In this case the operator Rnhas the same form

as 共39兲, with coefficients given by

an⫽ 1 2D⫺1共un兲, bn⫽ 7 4un⫺ 1 2vn, cn⫽ 1

8关6uD⫺1共un兲⫹2D⫺1共uun兲⫺4D⫺1共vvn兲⫹17un,x⫺12vn,x兴,

d_n,x⫽1

16关6u2xD⫺1共un兲⫺12v2xD⫺1共un兲⫹30uxun⫺8uxvn⫹24uun,x ⫹15un,3x⫺12vxvn⫺8uvn,x⫺20vvn,x⫺28vn,3x兴.

The recursion operator共40兲 for the given system is

R0 0_⫽1

4D

4_⫹2uD2_⫹3u

xD⫹2u2x⫹4共u2⫺v2兲⫹共3uux⫺6vvx⫹

1 2u3x兲D⫺1⫹uxD⫺1u, R1 0_⫽⫺5vD2_⫺4v xD⫺v2x⫺4uv⫺2uxD⫺1v, 共B13兲 R0 1_⫽⫺5 2vxD⫺3v2x⫺共v3x⫹3uvx兲D⫺1⫹vxD⫺1u, R1 1_⫽⫺D4_⫺4uD2_⫺2u xD⫺4v2⫺2vxD⫺1v. 5.Nⴝ3 Hirota–Satsuma system This system is given by29

ut⫽ 1 4u3x⫹3uux⫹3共⫺v2⫹w兲x, vt⫽⫺ 1 2v3x⫺3uvx, 共B14兲 wt⫽⫺12w3x⫺3uwx.

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⫽共D2⫹2u⫺2v兲共D2⫹2u⫹2v兲⫹4w, A⫽共L3/4兲_⫹. 共B15兲 In this case the operator L is symmetric and hence Rn has the same form as 共39兲, with the

coefficients an⫽D⫺1共un兲, bn⫽ 7 2un⫹vn, cn⫽ 1

4关12uD⫺1共un兲⫹4D⫺1共uun⫹wn⫺2vvn兲⫹17un,x⫹12vn,x兴,

dn,x⫽18关12u2xD⫺1共un兲⫹24v2xD⫺1共un兲⫹60uxun⫹16uxvn⫹15un,3x⫹48uun,x⫹24vxun ⫺40vxvn⫹20vn,3x⫹16vvn,x⫹20wn,x兴.

The recursion operator is given by

R⫽

冉

冊

R0 0_⫽1 4D 4_⫹4uD2_⫹6u xD⫹4共u2x⫹4u2⫺4v2⫹4w兲 ⫹4共1 4u3x⫹3uux⫺6vvx⫹3wx兲D⫺1⫹4uxD⫺1u, R1 0 ⫽⫺2共5vD2_⫹4v xD⫹v2x⫹8uv⫹4uxD⫺1v兲, R2 0_⫽5D2_⫹8u⫹4u xD⫺1, R0 1_⫽⫺5u xD⫺6v2x⫺2共v3x⫹6vxu兲D⫺1⫹4vxD⫺1u, R1 1_⫽⫺D4_⫺8uD2_⫺4u xD⫹8共8w⫺2v2兲⫺8vxD⫺1v⫺8D⫺1wx, R2 1 ⫽4共vxD⫺1⫹2D⫺1vx兲, R0 2_⫽⫺5w xD⫺6w2x⫺2共v3x⫹6wxu兲D⫺1⫹4wxD⫺1u. R1 2_{⫽⫺16vD⫺1} wx⫺8wxD⫺1v, R2 2_⫽⫺D4_⫺8uD2_⫺4u xD⫹16共w⫺v2兲⫹4wxD⫺1⫹16vD⫺1vx. 共B17兲

APPENDIX C: EXAMPLES TO SEC. III

1. Non-Abelian Schro¨ dinger equation This is the system given by

ut⫽⫺

1

2uxx⫹uvu,

共C1兲

vt⫽12vxx⫹vuv,

where u andv belong toK 共see Example 6 for the notations兲. The Lax operator of 共C1兲 is given

by L⫽D⫹

冉

冊

冉

冊

The corresponding formula共54兲 reduces to

冉

冊

冉

冊

冉

冊

The formula共C3兲 gives us both an,bn,cn and the recursion operator R. They are given by

an⫽ 1 2D⫺1共unv⫹uvn兲, bn⫽ 1 2un, cn⫽⫺ 1 2un, 共C5兲 R⫽1 2

冉

冊

2. Non-Abelian modified KdV equation

The standard non-Abelian modified KdV equation is given by

ut⫽ 1 4uxxx⫺ 3 4uxu2⫺ 3 4u 2_u x. 共C7兲

The Lax representation of this equation is given

L⫽D⫹

冉

0 1

1 0

冊

冉

u 0

0 ⫺u

冊

The recursion operatorR can be found from 共60兲 and 共61兲. In our case the automorphism␴is the same as in Example 6, and formulas共60兲 and 共61兲 give us

冉

冊

冉

冊

冉

冊

冉

冊

Using共C9兲 we find a_n,b_n,c_n,d_n from the following:

bn⫺an⫽un, ⫺an,x⫺anu⫺uan⫹cn⫺dn⫽0, ⫺bn,x⫹bnu⫹ubn⫹dn⫺cn⫽0, dn,x⫹cn,x⫽关cn⫺dn,u兴,

un⫹1⫽dn,x⫹关dn,u兴.

The resulting recursion operator is given by

R⫽1

4共D⫺adu•D⫺1•adu兲„D⫺共Lu⫹Ru兲D⫺1共Lu⫹Ru兲…. 共C11兲

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22

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