Recursion operators and the hierarchies of MKdV equations related to D (1) 4 , D (2) 4 and D (3) 4 Kac-Moody algebras
V. S. Gerdjikov, A.A. Stefanov, I. D. Iliev, G. P. Boyadjiev, A. O. Smirnov, V. B. Matveev, M. V. Pavlov
aa r X i v : . [ n li n . S I] J un Recursion operators and the hierarchies of MKdV equationsrelated to D (1)4 , D (2)4 and D (3)4 Kac-Moody algebras
V. S. Gerdjikov , , , A.A. Stefanov , , I. D. Iliev , G. P. Boyadjiev ,A. O. Smirnov, V. B. Matveev, , M. V. Pavlov Institute of Mathematics and Informatics, Bulgarian Academy of Sciences,Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria National Research Nuclear University MEPHI,31 Kashirskoe Shosse, 115409 Moscow, Russian Federation Institute for Advanced Physical Studies, New Bulgarian University,21 Montevideo Street, Sofia 1618, Bulgaria Faculty of Mathematics and Informatics, Sofia University ”St. Kliment Ohridski” Sankt-Petersburg State University of Aerospace InstrumentationSt-Petersburg, B.Morskaya, 67A, St-Petersburg, 1900000, Russia Sankt-Petersburg department of Steklov Mathematical Instituteof Russian Academy of Sciences,St-Petersburg, Russia Institut de Math´ematiques de Bourgogne (IMB),Universit´e de Bourgogne - France Comt´e, Dijon, France P.N. Lebedev Physical Institute of Russian Academy of SciencesLeninskij Prospekt, 53, Moscow, 119991, Russia.
Abstract
We constructed the three nonequivalent gradings in the algebra D ≃ so (8). The first one isthe standard one obtained with the Coxeter automorphism C = S α S α S α S α using its dihedralrealization. In the second one we use C = C R where R is the mirror automorphism. The thirdone is C = S α S α T where T is the external automorphism of order 3. For each of these gradingswe constructed the basis in the corresponding linear subspaces g ( k ) , the orbits of the Coxeter au-tomorphisms and the related Lax pairs generating the corresponding mKdV hierarchies. We foundcompact expressions for each of the hierarchies in terms of the recursion operators. At the end wewrote explicitly the first nontrivial mKdV equations and their Hamiltonians. For D (1)4 these are infact two mKdV systems, due to the fact that in this case the exponent 3 has multiplicity 2. Eachof these mKdV systems consist of 4 equations of third order with respect to ∂ x . For D (2)4 this is asystem of three equations of third order with respect to ∂ x . Finally, for D (3)4 this is a system of twoequations of fifth order with respect to ∂ x . ontents D -type 3 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 The Coxeter automorphisms of D ( s )4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 D ( a )4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 D (1)4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 The MKdV for D (2)4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 The MKdV for D (3)4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 The works of Gardner, Green, Kruskal and Miura [12] and Lax [34] can be considered the foundation ofmodern soliton science. Initially many researchers believed that the methods presented in those papersapply only to the Korteweg-de-Vries (KdV) equation.The second soliton equation discovered by Zakharov and Shabat in 1971 [51] was the nonlinearSchr¨odinger equation (NLS); soon after that the third integrable equation, the mKdV appeared [47].It was followed by an explosion of interest in soliton equations. Mathematicians were excited by thefact that both KdV and NLS provided the first examples of infinite dimensional completely integrableHamiltonian equations [48, 45, 4]. The physicists appreciated the new stable nonlinear waves that hadpurely elastic interaction and appeared in various physical processes: hydrodynamics, plasma physics,nonlinear optics etc.It is worth mentioning some of the milestones in this development.The seminal paper by Ablowitz, Kaup, Newell and Segur [1] demonstrated new techniques of workingout with the Lax pairs and formulated new important idea. The proved that the inverse scattering method(ISM) can be understood as a generalized Fourier transform (GFT), which allows one to linearize thesoliton equations. They introduced the notion of the recursion operator Λ, that generated the hierarchyof soliton equations and the GFT were the spectral expansions of Λ. The proof of this idea was completedby establishing the completeness relations for the ‘squared solutions‘ of the Lax operator L [31, 17, 32, 29].The next examples of soliton equations such as the N -wave equations [49], the principal chiral field[50] and the massive Thirring model [33] quickly led to the necessity to extend the ISM to new classes ofLax operators depending polynomially on the spectral parameter λ , see also [2, 11, 18, 19, 20]. Indeed,the AKNS system provided the simplest nontrivial Lax operator Lψ ≡ i ∂ψ∂x + ( Q ( x, t ) − λJ ) ψ ( x, t, λ ) = 0 (1)which was linear in the spectral parameter λ , with J = σ and whose potential Q ( x, t ) took valuesin the algebra sl (2). We note also that the ISP for this Lax operator has been developed earlier byGelfand, Levitan and Marchenko [13]. At the same time the N -wave equations required Lax operators1ike (1) but with potentials taking values in the algebra sl ( n ) and J being real constant diagonal matrix J = diag ( a , a , . . . , a n ). The principle chiral fields and the massive Thirring models demonstrated theneed to study the spectral properties of operators, that had more complicated dependence of λ : rationalfor the chiral field and polynomial in λ and λ − .The direct and inverse scattering problems (ISP) for the n × n operators linear in λ were solvedintroducing the notion of a fundamental analytic solution of L [43, 44]. As a result it became clear thatthe ISP is equivalent to a Riemann-Hilbert problem (RHP). The next important step here was proposedby Zakharov and Shabat, who developed the dressing method [52, 53, 40] for constructing the solitonsolutions of the relevant soliton equations. Rather quickly it was demonstrated that the AKNS idea ofinterpreting the ISM as a GFT can be generalized for Lax operators related not only to sl ( n ) [21], butalso to any simple Lie algebra g [14]. Lax operators polynomial in λ were used to integrate the so-calledderivative NLS equation [32] and its gauge equivalent GI equation [18, 19]. And again it was possible todemonstrate that the ISM is a GFT [20, 18, 19].Another important class of generalizations applied to the soliton equations was established by Kulishand Fordy [10]. The discovered the fact that using symmetric spaces one can construct multicomponentgeneralizations of the corresponding NLS or GI equations, see [16]. Again one can naturally extend themain tools of the soliton equations like the RHP and the dressing Zakharov-Shabat method for obtainingthe soliton solutions. The notion of ‘squared solutions‘ and the ideas that ISP is a GFT are also naturallygeneralized, see [16] and references therein.Another important question that came up was: given a NLEE can we check if it is integrable ornot? A way to answer it was to check whether the equation possesses an infinite set of integrals ofmotion, or symmetries. Following this ideas Shabat, Zhiber, Mikhailov [54, 39, 38] developed a methodfor classification of all integrable NLEE of given form, see [36, 41] and the references therein. Some ofthese equations, like the one now known as Tsitseica eq. [46]: u xt = e u − e − u , (2)became for some time a challenge. It was known to have physical applications [6], it was known to havean infinite number of integrals of motion but its Lax representation for some time was unknown. Thereason for that was not only that the relevant Lax operator was related to the sl (3) algebra, but alsoin the fact that it had very special symmetry. Solving this problem A. V. Mikhailov introduced the socalled reduction group [35] and discovered the family of 2-dimensional Toda field theories (see also [37]),of which Tsitseica equation was a member. These trend was later extended to treat generalizations ofmKdV equations in [36, 35, 41].Solving the ISP for Lax operator of the form (1) with Z h reduction group leads to the necessity toconsider J with complex-valued eigenvalues. The construction of the fundamental analytic solutions forthis class of Lax operators was achieved by Beals and Coifman [3] for systems related to sl ( n ) algebras.Later their results were generalized to any simple Lie algebras [24]. The completeness of the ‘squaredsolutions‘ and the ideas of GFT [18, 19, 20, 21, 27] were combined with the Mikhailov reduction groupin [25, 26].Another important aspect, namely that there is a connection between soliton equations and Kac-Moody algebras was discovered by Drifneld and Sokolov [7, 8]. At that time the Lax pairs for the KdVand mKdV equations were often formulated using scalar differential operators of third order. Drinfeldand Sokolov demonstrated that scalar differential operators of order n can be conveniently rewritten asfirst order n × n operator with conveniently applied Z n reduction. They extended this result by showingthe deep connection between the soliton equations and the Kac-Moody algebras. The latter can beconstructed starting from simple Lie algebra g graded by its Coxeter automorphism C , for details seeSection 2 below.This present paper is an extension of our previous results reported in [22, 23]. Its main purpose isto present the modified Korteweg-de-Vries (mKdV) equations related to Kac-Moody algebras of type2 ( k )4 , k = 1 , ,
3. We assume that the reader is familiar with the theory of simple Lie algebras [28] andwith the basic ideas for constructing Kac-Moody algebras [5]. In Section 2 we outline the constructionof the three nonequivalent gradings of the algebra D ≃ so (8) which give rise to the three Kac-Moodyalgebras of height 1, 2 and 3. In Section 3 we formulate the Lax pairs related to each of the threegradings. Extending the AKNS ideas we solve the recurrent relations for each of the hierarchies of mKdVequations. To this end we have to introduce several types of elementary recursion operators Λ a . We alsointroduce a master recursion operator Λ which is an ordered product of the elementary ones. Then wefind that with each exponent of the Kac-Moody algebra one can relate a hierarchy of mKdV equationsgenerated by the master recursion operator Λ. An exception is the case D (1)4 for which the exponent 3is double-valued. As a result with this exponent one can relate two nonequivalent hierarchies of mKdVequations. In Section 4 we provide explicitly the simplest mKdV equations. For the case D (1)4 these aretwo systems of 4 equations of third order with respect to ∂ x . For the case D (2)4 this is a system of threeequations of third order with respect to ∂ x . Finally for D (3)4 this is a system two equations of fifth orderwith respect to ∂ x . We also briefly analyze their relations with the results of [36]. In the last Section webriefly discuss the results and outline their possible extensions. D -type We assume that the reader is familiar with the basic facts about the simple Lie algebras [28, 30].
Let g be a finite-dimensional Lie algebra over C . Then g [ λ, λ − ] = ( m X i = n v i λ i : v i ∈ g , n, m ∈ Z ) ,f [ λ ] = ( m X i =0 f i λ i : f i ∈ g , m ∈ Z ) . (3)There is a natural Lie algebraic structure on g [ λ, λ − ]. Let ϕ be an automorphism of g of order s . Then L ( g , ϕ ) = (cid:26) f ∈ g [ λ, λ − ] : ϕ ( f ( λ )) = f (cid:20) λ exp (cid:18) πis (cid:19)(cid:21)(cid:27) . (4) L ( g , ϕ ) is a Lie subalgebra of g [ λ, λ − ]. If g is simple then L ( g , ϕ ) is called a Kac-Moody algebra. Itis obvious that Kac-Moody algebras are graded algebras. Note that commonly the central extension of L ( g , ϕ ) is called a Kac-Moody algebra. The definition given above is the one used in [7, 8].The above definition of Kac-Moody algebras can be stated in simpler words - the elements of a Kac-Moody algebras are formal series in λ with coefficients in some properly graded finite-dimensional simpleLie algebra.As is shown in [30], two Kac-Moody algebras L ( g , ϕ ) , L ( g , ϕ ) are isomorphic if g is isomorphic to g and the automorphisms of the Dynkin diagram determined by ϕ and ϕ are conjugate. Since thereare simple Lie algebras with non-trivial outer automorphisms, for those simple Lie algebras there will bemore than one Kac-Moody algebra. Again, every automorphism ϕ of g can be uniquely represented inthe form ϕ = f ◦ ϕ τ . The order of ϕ τ is called the height of L ( g , ϕ ). Commonly, Kac-Moody algebras ofheight greater than one are called twisted Kac-Moody algebras.In analogy with finite dimensional simple Lie algebras, C is a Coxeter automorphism of L ( g , C ) if g is Abelian and C is of minimal order. The number r = dim( g ) is called the rank of L ( g , C ). The order3lgebra Coxeter automorphism Coxeter number Exponents Rank D (1)4 C = S α S α S α S α , , , D (2)4 C = S α S α S α R , , , D (3)4 C = S α S α T
12 1 , , ,
11 2Table 1:
A realization of the Coxeter automorphisms and Coxeter numbers for D (1)4 , D (2)4 and D (3)4 .Here S α i denotes reflection with respect to the simple root α i , R is the second order outer automorphismthat exchanges α and α , and T is the third order outer automorphism (a triality transformation) thatsends α → α → α . h of C is called the Coxeter number of L ( g , C ). We will be using Coxeter automorphisms to constructthe Kac-Moody algebras needed in this work.Let p is a permutation of the simple roots that preserves the Dynkin diagram of g . Every such p induces an outer automorphism P of g . Assume that a Coxeter automorphism C of a Kac-Moodyalgebra L ( g , C ) is of the form C = ˜ C ◦ P, (5)where ˜ C is an inner automorphism induced by a Weyl group element group element ˜ c . Every such C is induced by a linear mapping c = ˜ c ◦ p acting in the root space g . By analogy with simple Lie algebras,the eigenvalues of c are called exponents of the Kac-Moody algebra L ( g , C ).A basis of L ( g , C ) can be constructed as follows: Each element X of L ( g , C ) is of the form X = m X k = n X ( k ) λ k , n, m ∈ Z , (6)where X ( k ) ∈ g ( k mod h ) . Each of the subspaces g ( k ) has a basis given by: E ( k ) α = h − X s =0 ω − sk C s ( E α ) , H ( k ) j = h − X s =0 ω − sk C s ( H j ) . (7)Note that H ( k ) j is non-vanishing only if k is an exponent. This means that the number of elements in g ( k ) is r + 1 if k is an exponent and r otherwise, where r is the rank of L ( g , C ). The roots α are chosen asfollows: c splits the root system of g into r non-intersecting orbits. From each orbit we select only oneroot α .This work is mainly concerned with Kac-Moody algebras of type D . Note that, since there are twotypes of outer automorphisms of D , there are three Kac-Moody algebras - D (1)4 , D (2)4 and D (3)4 (here theupper index denotes the height of the algebra). The Coxeter numbers and the exponents of those algebrasare given in Table 1 (the values are taken from [8]). The table also contains the Coxeter automorphismswe use. The construction of each of those automorphisms is given in the corresponding section. Theexplicit form of the basis for each Kac-Moody algebra of type D is given below. D We will review the most important properties of the simple Lie algebra D ≡ so (8). Let e i be a standardbasis in the root space of D (from now on, if we don’t specify otherwise, we will always assume that this Not every Coxeter automorphism can be realized in this way. For example, Coxeter automorphisms of the form C ( X ) = cF ( X ) c − , where c is a diagonal matrix and F is some properly chosen outer automorphism, can never beconstructed from a Weyl group element. α α α a) α α α α b) α α α α c) Figure 1: a) Dynkin diagram of the simple Lie algebra D ; a) Dynkin diagram of the simple Lie algebra D with the mirror automorphism R ; a) Dynkin diagram of the simple Lie algebra D with the third orderouter automorphism T . is the basis of the root space). For the root system of D we have ∆ = ∆ + ∪ ∆ − , where∆ + = { e i ± e j : i, j = 1 ... , i < j, } , ∆ − = {− ( e i ± e j ) : i, j = 1 ... , i < j. } (8)The simple roots are given by α = e − e , α = e − e , α = e − e , α = e + e . (9)With each vertex of the Dynkin diagram of D we can associate a simple root (Fig 1). D is usually represented by a 8 × X ∈ D satisfies SX + ( SX ) T = 0 , (10)where the matrix S is given by S = − − − − . (11)This way the Cartan subalgebra is given by diagonal matrices. The Cartan-Weyl generators of D aregiven by H i = e ii − e − i, − i , i = 1 , . . . , , E α j = e j,j +1 + e − j, − j , j = 1 , , , E α = e , + e , ,E e i − e j = e i,j − ( − i + j e − j, − i , E e i + e j = e i, − j − ( − i + j e j, − i , , E − α j = ( E α j ) T , (12)where 1 ≤ i < j ≤
4. By e ij we denote a matrix that has a one at the i − th row and j − th column andis zero everywhere else. The exponents of D are 1 , , , .3 The Coxeter automorphisms of D ( s )4 In what follows by S α i we will denote the Weyl reflection with respect to the simple root α i . It is wellknown that each element of the Weyl group naturally induces an inner automorphism. S α i ( E β ) = s i E β s − i = n α i ,β E S αi ( β ) , S α i ( H β ) = H S αi ( β ) (13)where n α i ,β = ± s i are easy to find.The Coxeter automorphism for D (1)4 coincides with the Coxeter automorphism for D and is given by C = S α S α S α S α . (14)In root space it has the form C = − − . (15) C splits the root system of D into 4 orbits, each containing 6 elements: O : e − e → − ( e − e ) → − ( e + e ) → − ( e − e ) → e − e → e + e , O : e − e → e + e → e + e → − ( e − e ) → − ( e + e ) → − ( e + e ) , O : e − e → − ( e − e ) → − ( e + e ) → − ( e − e ) → e − e → e + e , O : e + e → − ( e + e ) → − ( e − e ) → − ( e + e ) → e + e → e − e . When building the basis (7) we will average only the Weyl generators corresponding to the simple roots α = e − e , α = e − e , α = e − e , α = e + e . In the algebra C is realized as an innerautomorphism , i.e. a similarity transformation C ( X ) = c Xc − , X ∈ D (1)4 , c = − − − − . (16)It is easy to check that C = g ( k ) by taking weightedaverage of E α i over the action of the Coxeter automorphism: E ( k ) i = X s =0 ω − sk C s ( E α i ) , H ( k ) i = X s =0 ω − sk C s ( H e i ) . (17)One can check that both E ( k ) i and H ( k ) i must belong to g ( k ) ; indeed, it is easy to see that C ( E ( k ) i ) = ω k E ( k ) i and C ( H ( k ) i ) = ω k H ( k ) i . Another important remark is that H ( k ) i is not vanishing if and only if k is anexponent of D (1)4 . In addition for k = 1 and k = 5 it is enough to consider only H ( k )1 ; considering H ( k ) i for i = 2 , , H ( k )1 . The exception here is only for the case k = 3;6hen we have two linearly independent Cartan elements: H ( k )1 and H ( k )4 . Skipping the details we list theresults: E ( k )1 = E e − e − ω − p E − e + e − ω − p E − e − e − ω − p E − e + e + ω − p E e − e + ω − p E e + e , E ( k )2 = E e − e − ω − p E e + e + ω − p E e + e − ω − p E − e + e + ω − p E − e − e − ω − p E − e − e , E ( k )3 = E e − e − ω − p E − e + e + ω − p E − e − e + ω − p E − e + e − ω − p E e − e + ω − p E e + e , E ( k )4 = E e + e − ω − p E − e − e + ω − p E − e + e + ω − p E − e − e − ω − p E e + e + ω − p E e − e , H ( s )1 = 2( H e + ω s H e + ω − s H e ) , H (3)4 = 6 H e , s = 1 , , . (18)These results allow us also to calculate the commutation relations between the basis in (18). In particular[ H ( s )1 , E ( k ) i ] = α i ( H ( s )1 ) E ( k + s ) i , [ H (3)4 , E ( k ) i ] = α i ( H (3)4 ) E ( k +3) i . (19)where α ( H ( s )1 ) = 2(1 − ω s ) , α ( H ( s )1 ) = 2( ω s − ω − s ) , α ( H ( s )1 ) = 2 ω − s , α ( H ( s )1 ) = 2 ω − s ,α ( H (3)4 ) = 0 , α ( H (3)4 ) = 0 , α ( H (3)4 ) = − , α ( H (3)4 ) = 6 . (20)The Coxeter automorphism for D (2)4 is given by C = S α S α S α R, (21)where R is the outer automorphism that exchanges α and α . In the root space of G we have C = −
10 1 0 0 . (22)From (22) it is easy to check that C = ω , ω , ω , ω , where ω = exp(2 πi/ D (2)4 in Table 1. C splits the root system of D into 3 orbits, each containing 8 elements: O : e − e → e − e → e + e → e − e → − ( e − e ) → − ( e − e ) →− ( e + e ) → − ( e − e ) , O : e − e → − ( e − e ) → − ( e + e ) → − ( e + e ) → − ( e − e ) → e − e → e + e → e + e , O : e − e → e + e → e + e → e + e → − ( e − e ) → − ( e + e ) →− ( e + e ) → − ( e + e ) . The roots chosen in (7) are α = e − e , α = e − e , α = e − e .In the algebra R is realized as a similarity transformation with a matrix r , given by r = . (23)7he Coxeter automorphism of D (2)4 is realized as C ( X ) = c Xc − , X ∈ D (2)4 , c = − −
10 0 0 0 0 1 0 0 . (24)The analog of the Cartan-Weyl basis in the subspaces g ( k ) is given by E ( k ) i = X s =0 ω − sk C s ( E α i ) , H ( k ) i = X s =0 ω − sk C s ( H e i ) . (25)where we take only the roots α , α and α ; each of them specifies a different orbit of C . Obviously that C ( E ( k ) i ) = ω k E ( k ) i and C ( H ( k ) i ) = ω k H ( k ) i . Besides H ( k ) i is not vanishing if and only if k is an exponentof D (1)4 . It is enough to consider only H ( k )1 ; the Cartan elements H ( k ) i for i = 2 , , H ( k )1 . Skipping the details we list the results: E ( k )1 = E e − e + ω − p E e − e + ω − p E e + e − ω − p E e − e − ω − p E − e + e − ω − p E − e + e − ω − p E − e − e + ω − p E − e + e , E ( k )2 = E e − e + ω − p E − e + e + ω − p E − e − e + ω − p E − e − e − ω − p E − e + e − ω − p E e − e − ω − p E e + e − ω − p E e + e , E ( k )3 = E e − e + ω − p E e + e + ω − p E e + e + ω − p E e + e − ω − p E − e + e − ω − p E − e − e − ω − p E − e − e − ω − p E − e − e , H ( p )1 = 2( H e + ω − p H e − ω − p H e + ω − p H e ) . (26)These results allow us also to calculate the commutation relations between the basis in (26). In particular[ H ( s )1 , E ( k ) i ] = α i ( H ( s )1 ) E ( k + s ) i , (27)where α ( H ( s )1 ) = 2(1 − ω − s ) , α ( H ( s )1 ) = 2( ω − s − ω s ) ,α ( H ( s )1 ) = 2( ω s − ω − s ) , α ( H ( s )1 ) = 2( ω s + ω − s ) , (28)The Coxeter element for D (3)4 is given as C = S α S α T, (29)where T (known in some literature as triality transformation) is the third order outer automorphism forwhich T : α α α (30)and is stationary on α . In root space T can be realized as a matrix T = 12 − − − − − − . (31)8nfortunately, it seems that there is no similarity transformation that realizes T in the algebra. Byknowing the action on the Weyl generators E α i one can construct the action over the whole algebra usingthe fact that T (cid:0)(cid:2) E α i , E α j (cid:3)(cid:1) = (cid:2) T ( E α i ) , T ( E α j ) (cid:3) . (32)This leads to T : E α → E α → E α ,T : E α + α → − E α + α → − E α + α ,T : E α + α + α → − E α + α + α → E α + α + α , (33)with stationary elements E α , E α + α + α + α and E α +2 α + α + α The action on the negative roots isobtained by transposing the above. Thus we obtain the following realization of C in the root space of D . C = 12 − − − −
11 1 1 1 − − . (34)From (34) it is easy to check that C = ω , ω , ω , ω , where ω = exp(2 πi/ D (3)4 in Table 1.Therefore C splits the roots of D into 2 orbits, each containing 12 elements: O : e − e → e − e → e − e → e + e → e − e → e + e →− ( e − e ) → − ( e − e ) → − ( e − e ) → − ( e + e ) → − ( e − e ) → − ( e + e ) O : e + e → − ( e − e ) → − ( e − e ) → − ( e + e ) → − ( e + e ) → − ( e + e ) →− ( e + e ) → e − e → e − e → e + e → e + e → e + e . In (7) we choose the roots α = e − e and α = e + e .The Coxeter automorphism of D (3)4 is then realized as C ( X ) = S S T ( X ) S S . (35)The Cartan-Weyl basis in the subspaces g ( k ) takes the form E ( k ) i = X s =0 ω − sk C s ( E α i ) , H ( k ) i = X s =0 ω − sk C s ( H e i ) . (36)where we take only the roots α and α ; each of them specifies a different orbit of C . Obviously that C ( E ( k ) i ) = ω k E ( k ) i and C ( H ( k ) i ) = ω k H ( k ) i . Besides H ( k ) i is not vanishing if and only if k is an exponentof D (3)4 . It is enough to consider only H ( k )1 ; the Cartan elements H ( k ) i for i = 2 , , H ( k )1 . Skipping the details we list the results: E ( k )2 = E e − e + ω − p E e − e + ω − p E e − e + ω − p E e + e + ω − p E e − e + ω − p E e + e − ω − p E − e + e − ω − p E − e + e − ω − p E − e + e − ω − p E − e − e − ω − p E − e + e − ω − p E − e − e , E ( k )4 = E e + e + ω − p E − e + e − ω − p E − e + e − ω − p E − e − e − ω − p E − e − e + ω − p E − e − e − ω − p E − e − e − ω − p E e − e + ω − p E e − e + ω − p E e + e − ω − p E e + e − ω − p E e + e , H (1)1 = √ √ − i ( √ i − − i , H (5)1 = √ √ − − i − ( √ − i i , H (7)1 = H (5) , ∗ , H (11)1 = H (1) , ∗ . (37)9hese results allow us also to calculate the commutation relations between the basis in (37). In particular[ H ( s )1 , E ( k ) i ] = α i ( H ( k )1 ) E ( k + s ) i , (38)where α ( H ( s )1 ) = √ i √
3) for , s = 1 , √ − i √
3) for s = 5 , √ − − i √
3) for s = 7 , √ − i √
3) for s = 11 , , α ( H ( s )1 ) = √ − − i (2 + √ s = 1 , √ i (2 − √ s = 5 , √ − i (2 − √ s = 7 , √ − i (2 + √ s = 11 . (39) Remark 1.
In order to simplify the usage of indices, we have used the same letters and types of indicesto denote the Cartan-Weyl basis in g ( k ) for each of the Kac-Moody algebras D ( s )4 . Of course it will beclear from the context which of them we have in mind. Let us consider a generic Lax pair, Lψ ≡ ( i∂ x + U ( x, t, λ )) ψ ( x, t, λ ) = 0 , M ψ ≡ ( i∂ t + V ( x, t, λ )) ψ ( x, t, λ ) = 0 ,U ( x, t, λ ) = Q ( x, t ) − λJ, V ( x, t, λ ) = n − X k =0 λ k V k ( x, t ) − λ n K, (40)whose potentials U ( x, t, λ ) and V ( x, t, λ ) are elements of the Kac-Moody algebra D ( s )4 ; in fact most ofthe results in this subsection will be valid for any Kac-Moody algebra. This means that Q ( x, t ) ∈ g (0) , V k ( x, t ) ∈ g ( k ) , K ∈ g ( n ) ∩ h , J ∈ g (1) ∩ h . (41)Such choice for the potentials of the Lax pair means that they involve Z h as their reduction group [35]: C ( U ( x, t, λ )) = U ( x, t, ωλ ) , C ( V ( x, t, λ )) = V ( x, t, ωλ ) . (42)with ω = e πih is the Coxeter number. We also assume that J and K are constant elements of the Cartansubalgebra.We request that the operators L and M commute identically with respect to λ . In particular, since[ J, K ] = 0 this means that the leading power n in V ( x, t, λ ) must be of the form n = n h + n , where n must be an exponent of D ( s )4 . To simplify the notation we will often omit writing the explicit dependenceon x and t . This implies the following recursion relations λ n +1 : (cid:2) J, K (cid:3) = 0 ,λ n : (cid:2) J, V n − (cid:3) + (cid:2) Q, K (cid:3) = 0 ,λ n − : i∂ x V n − + (cid:2) Q, V n − (cid:3) = (cid:2) J, V n − (cid:3) ,λ s : i∂ x V ( s ) + (cid:2) Q, V s (cid:3) = (cid:2) J, V s − (cid:3) ,λ : − i∂ t Q + i∂ x V + (cid:2) Q ( x, t ) , V (cid:3) = 0 . (43)Now we view eq. (43) as a set of recurrence relations and aim to resolve them and express all V s ( x, t )in terms of Q ( x, t ). Doing this we have to take into account that the operator ad J X ≡ [ H, X ] has a10ernel. Therefore we need to split each V s into a sum of diagonal and off-diagonal parts. Rememberingthe results of the previous section we set s = s h + s and consider two cases: V s ( x, t ) = ( V f s ( x, t ) if s is not an exponent V f s ( x, t ) + w s ( x, t ) H ( s )1 if s is an exponent , V f s ( x, t ) = r X p =1 V s,p ( x, t ) E ( s ) p . (44) Remark 2.
Note that using proper gauge transformation we can always transform away the diagonalpart of V n − ; so using the commutation relations (19), (27) or (38) From the second of the equations(43) we obtain: V n − ( x, t ) ≡ V f n − ( x, t ) = r X p =1 α p ( K ) α p ( J ) q p ( x, t ) E ( n − p . (45)Now let us assume that s is an exponent and split the third equation in (43) into diagonal andoff-diagonal parts. Evaluating the Killing form of this equation with H h − s we obtain: w s ( x, t ) = ic s ∂ − x D [ Q, V f s ] , H h − s E + const , c s = D H s , H h − s E . (46)In what follows for simplicity we will set all these integration constants to 0. A diligent reader can easilywork out the more general cases when some of these constants do not vanish. The off-diagonal part ofthe third equation in (43) gives: i∂ x V f s + [ Q, V f s ] f + [ Q, w s H s ] = [ J, V s − ] . (47)i.e. V f s − = ad − J (cid:0) i∂ x V f s + [ Q, V f s ] f + [ Q, w s H s ] (cid:1) = Λ s V f s . (48)Thus we obtained the integro-differential operator Λ s which acts on any Z ≡ Z f ∈ g ( s ) by:Λ s Z = ad − J (cid:18) i∂ x Z + [ Q, Z ] f + ic s [ Q, H s ] ∂ − x D [ Q, Z ] , H h − s E(cid:19) (49)If s is not an exponent we have only to work out the off-diagonal part of the third equation in (43) withthe result: V f s − = ad − J (cid:0) i∂ x V f s + [ Q, V f s ] f (cid:1) = Λ V f s , Λ Z = ad − J (cid:0) i∂ x Z + [ Q, Z ] f (cid:1) . (50)Now Λ is a differential operator.In the case of D (1)4 the exponent 3 has multiplicity 2. Therefore the recursion operator Λ must bereplaced by ˜Λ which has the form:˜Λ Z = ad − J (cid:18) i∂ x Z + [ Q, Z ] f + ic [ Q, H ] ∂ − x (cid:10) [ Q, Z ] , H (cid:11) + ic ′ [ Q, H ] ∂ − x (cid:10) [ Q, Z ] , H (cid:11)(cid:19) (51)where c = (cid:10) H , H (cid:11) , c ′ = (cid:10) H , H (cid:11) . (52)Here we also used the fact that (cid:10) H , H (cid:11) = 0. 11 .2 The hierarchies of MKdV related to D ( a )4 The last of the equations in (43) provides the corresponding set of MKdV equations that can be solvedapplying the ISM to the corresponding Lax operator. In fact this last equations simplifies into ∂ t Q ( x, t ) = ∂ x V ( x, t ) . (53)because the subalgebra g (0) is commutative. Let us now describe the class of the sets of MKdV equationsusing the recursion operators Λ and Λ s . With each of the Lax operators L we can relate 4 series ofNLEE whose dispersion laws are monomial in λ .Let us first start with D (1)4 . Skipping the details we write them compactly as follows: n = 6 n + 1 ∂ t Q = ∂ x ( Λ n Q ( x, t )) , f ( λ ) = λ n +1 H (1)1 ,n = 6 n + 3 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ ad − J [ a H + b H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +3 ( a H + v H ) ,n = 6 n + 5 ∂ t Q = ∂ x (cid:16) Λ n Λ Λ ˜Λ Λ ad − J [ H , Q ( x, t )] (cid:17) , f ( λ ) = λ n +5 H . (54)where Λ = Λ Λ ˜Λ Λ Λ Λ . The fact that the exponent 3 is double valued leads to the fact that witheach dispersion law proportional to λ n +3 we have a one-parameter family of NLEE. Indeed, we canrescale the time t → τ = t/a which will make the parameter a = 1; however the other parameter b → b/a can not be taken away.Similarly we can treat the hierarchies related to D (2)4 . The results are: n = 8 n + 1 ∂ t Q = ∂ x ( Λ n Q ( x, t )) , f ( λ ) = λ n +1 H (1)1 ,n = 8 n + 3 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +3 H ,n = 8 n + 5 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +5 H ,n = 8 n + 7 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ Λ Λ Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +7 H , (55)where Λ = Λ Λ Λ Λ Λ Λ Λ Λ .Finally for D (3)4 we get: n = 12 n + 1 ∂ t Q = ∂ x ( Λ n Q ( x, t )) , f ( λ ) = λ n +1 H (1)1 ,n = 12 n + 5 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +5 H ,n = 12 n + 7 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +7 H ,n = 12 n + 11 ∂ t Q = ∂ x (cid:0) Λ n Λ Λ Λ Λ Λ Λ ad − J [ H , Q ( x, t )] (cid:1) , f ( λ ) = λ n +11 H , (56)where Λ = Λ Λ Λ Λ Λ Λ Λ Λ .We end this Section by the simple remark, which follows directly from the structure of the recursionoperators and from the grading conditions of the algebras. Indeed, since J ∈ g (1) and Q ∈ g (0) , thenad J : g ( p ) → g ( p +1) , ad − J : g ( p ) → g ( p − , ad Q : g ( p ) → g ( p ) , Λ k : g ( p ) → g ( p − , Λ k : g ( p ) → g ( p − , Λ : g ( p ) → g ( p ) . (57)Thus it is easy to check that both sides of the NLEE (54), (55) and (56) obviously take values in g (0) . The first non-trivial member of the hierarchy is a set of mKdV equations which is obtained from (54),(55) and (56) setting n = 3, n = 3 and n = 5 respectively.12ere we will briefly describe the Hamiltonian formulation of the equations from (53). Every equationin (53) has infinitely many integrals of motion Every integral of motion can be viewed as a Hamiltonianwith a properly chosen Poisson structure. We will use the integral of motion given by [25, 26] I = Z ∞−∞ i∂ − x Dh Q, Λ V (0) i , H (1)1 E dx, (58)where ∂ − x f ( x ) = R f ( x ) dx and we have set any constants of integration to be zero. The Hamiltonian H is proportional to (58). Hamilton’s equations are ∂ t q i = { q i , H } (59)with a Poisson bracket given by { F, G } = Z R ω ij ( x, y ) δFδq i δGδq j dxdy, (60)where we sum over repeating indexes. The Poisson structure tensor is ω ij ( x, y ) = 12 δ ij ( ∂ x δ ( x − y ) − ∂ y δ ( x − y )) . (61)In this case (59) reduces to ∂ t q i = ∂ x δHδq i . (62) D (1)4 The algebra D (1)4 has 3 as a double exponent. This means that the element K in (40) involves twoarbitrary parameters: K = 12 a H (3)1 + 16 b H (3)4 . (63)We can also say, that there are two nonequivalent mKdV equations related to D (1)4 . Each equation isdetermined by its own dispersion law: the first one by f = λ H (3)1 , the other one – by f = λ H (3)4 .Below we will write down the two systems of equations separately; the generic mKdV equation will haveas dispersion law λ K which is a linear combination of f and f . ∂ t q = ∂ x (cid:16) ∂ x q − √ q ∂ x q + 3 q ∂ x q + 3 q ∂ x q ) + 3( − q + q + q ) q (cid:17) ,∂ t q = ∂ x (cid:16) √ q ∂ x q − q ∂ x q − q ∂ x q ) + 3( − q + q + q ) q (cid:17) ,∂ t q = ∂ x (cid:16) − ∂ x q + √ q ∂ x q + 3 q ∂ x q ) + 3 (cid:0) q + q − q (cid:1) q (cid:17) ,∂ t q = ∂ x (cid:16) − ∂ x q + √ q ∂ x q + q ∂ x q ) + 3 (cid:0) q + q − q (cid:1) q (cid:17) . (64)The second set of mKdV eqs. are given by: ∂ t q = ∂ x (cid:16) √ q ∂ x q − q ∂ x q ) + 3( q − q ) q (cid:17) ,∂ t q = ∂ x (cid:16) −√ q ∂ x q − q ∂ x q ) + 3( q − q ) q (cid:17) ,∂ t q = ∂ x (cid:16) − ∂ x q + √ q ∂ x q + q ∂ x q + q ∂ x q ) + 3 (cid:0) q − q (cid:1) q (cid:17) ,∂ t q = ∂ x (cid:16) ∂ x q − √ q ∂ x q + q ∂ x q + 2 q ∂ x q ) + 3 (cid:0) q − q (cid:1) q (cid:17) . (65)13he Hamiltonian densities of these equations are given by: H a = − ( ∂ x q ) + 12 ( ∂ x q ) + 12 ( ∂ x q ) + 3 √ q q ∂ x q − √
32 (2 q − q − q ) ∂ x q + 32 ( q + q )( q + q ) − q q − q q . (66)and H b = 12 ( ∂ x q ) −
12 ( ∂ x q ) + √ (cid:0) ( q − q ) ∂ x q − q q ∂ x q + 2 q q ∂ x q (cid:1) + 32 ( q − q )( q − q ) . (67)Notice that the equation for q in (64), as well as the equations for q and q in (65) do not containthird order derivatives with respect to x . This fact directly reflects the structure of the Hamiltonians.Indeed, in the kinetic past of H a there is no term proportional to ( ∂ x q ) . Likewise the kinetic part of H a there are no term proportional to ( ∂ x q ) and ( ∂ x q ) . Another important property of these Hamiltoniansis that their kinetic parts are neither positive nor negative definite. D (2)4 Since the rank of D (2)4 is 3 then the mKdV is a set of three equations for three functions. Since the 3 isan exponent, then the simplest mKdV will contain third order derivatives with respect to x . The detailsof the calculations and the explicit form of these equations can be found in [23]. ∂ t q = 2 ∂ x (cid:16) (4 + 3 √ ∂ x q − √ q ∂ x q − √ q ∂ x q − √ q ∂ x q + 2 q (3 q − q + q − q q ) − q q (cid:17) ,∂ t q = 2 ∂ x (cid:16) ∂ x q + 3(2 + √ q ∂ x q + 3(2 − √ q ∂ x q + 3 √ q ∂ x q − √ q ∂ x q − q (3 q + q + 3 q + 12 q q ) (cid:17) ,∂ t q = 2 ∂ x (cid:16) (4 − √ ∂ x q − − √ q ∂ x q + 6 √ q ∂ x q + 3 √ q ∂ x q + 2 q ( q − q + 3 q − q q ) − q q (cid:17) . (68)The Hamiltonian density of this system is: H = − (4 + 3 √ ∂ x q ) − ( ∂ x q ) − (4 − √ ∂ x q ) − √ q + (2 − √ q ) ∂ x q − √ q ( q ∂ x q − q ∂ x q ) + 4 q q ( q + q − q q ) − q (6 q + q + 6 q + 24 q q ) . (69)Like in the previous cases, the kinetic part of H (2) is not positive definite. However now it containsall three fields q i , so each of the equations will contain terms with third order derivatives with respect to x . D (3)4 The rank of D (3)4 is 2 then the mKdV is a set of two equations for two functions. Now the set of exponentsis 1, 5, 7 and 11. Therefore the simplest mKdV equations will be a set of two equations of fifth order14ith respect to x . Skipping the details we write down the equations with dispersion law f = λ H (5)1 : ∂ t q = ∂∂x (cid:16) (3 √ ∂ x q + 10( q − q ) ∂ x q + 5 (cid:16) (5 √ ∂ x q + ( √ ∂ x q (cid:17) ∂ x q − (cid:16) ( √ q + q ) + q q (cid:17) ∂ x q + 5 (cid:16) ( √ ∂ x q − ( √ ∂ x q (cid:17) ∂ x q − (cid:16) ( √ q − (3 − √ q (cid:17) ∂ x q − (cid:16) √ q + q (cid:17) ( ∂ x q ) − (cid:16) q + 2( √ q (cid:17) ∂ x q ∂ x q + 10 (cid:16) √ − q + (5 − √ q (cid:17) ( ∂ x q ) + 20( q − q ) (cid:16) (2 √ q + 4 q q − (2 √ − q (cid:17) ∂ x q + 10 √ (cid:0) q − q q + 2 q q + 8 q q + q q (cid:1) − √ q (cid:17) . (70) ∂ t q = ∂∂x (cid:16) (3 √ − ∂ x q − q − q ) ∂ x q − (cid:16) (5 − √ ∂ x q + ( √ − ∂ x q (cid:17) ∂ x q − (cid:16) (3 + √ q − (3 − √ q ) (cid:17) ∂ x q + 5 (cid:16) − ( √ − ∂ x q + (9 − √ ∂ x q (cid:17) ∂ x q + 20 (cid:16) − ( √ − q + q ) + q q (cid:17) ∂ x q − (cid:16) (2 √ q − √ q (cid:17) ( ∂ x q ) − (cid:16) √ − q − q (cid:17) ∂ x q ∂ x q + 10 (cid:16) q − √ − q (cid:17) ( ∂ x q ) − q − q ) (cid:16) (2 √ q + 4 q q − (2 √ − q (cid:17) ∂ x q + 10 √ q ( q + q ) − √ q ( q − q q + 5 q ) (cid:17) . (71) H = 12 h (3 √ ∂ x q ) + (3 √ − ∂ x q ) + 5( q − q q ) ∂ x q + 5( q − q q ) ∂ x q i + 52 (cid:20) (2 + √ ∂ x q ( ∂ x q ) − (3 + 53 √ ∂ x q ) − (3 − √ ∂ x q ) + (2 − √ ∂ x q ) ∂ x q (cid:21) + 10 h ( √ q + q ) + q q i ( ∂ x q ) + 10 h ( √ − q + q ) − q q i ( ∂ x q ) + 4( q − q ) h (1 + 2 √ q + 4 q q + (1 − √ q i ( q ∂ x q − q ∂ x q )+ 10 h ( √ q + ( √ − q i ∂ x q ∂ x q + √ (cid:2) q + q ) − q + q ) q q + 80 q q (cid:3) (72)We note that the equations (70) and (71) can be written down in the form: ∂ t q = ∂ x ( P ( q , q ) + Q ( q , q )) ∂ t q = ∂ x ( P ( q , q ) − Q ( q , q )) (73)15here P ( u, v ) = 3 √ ∂ x u + 10( u − v ) ∂ x v + 45 ∂ x u∂ x u + 5 ∂ x v∂ x u + 5 ∂ x u∂ x v − ∂ x v∂ x v − √ u + v ) ∂ x u − √ u + v ) ∂ x v − √ u ( ∂ x u ) + 20 √ u ( ∂ x v ) − √ v ( ∂ x v ) − √ v∂ x u∂ x v + 20( u + 3 u v − uv − v ) ∂ x v + 2 √ u − u v + 10 u v + 40 u v + 5 uv − v ) ,Q ( u, v ) = 5 ∂ x u + 25 √ ∂ x u∂ x u + 5 √ ∂ x v∂ x u + 5 √ ∂ x u∂ x v − √ ∂ x v∂ x v − u + uv + v ) ∂ x u − u − v ) ∂ x v − u ( ∂ x u ) − v ( ∂ x u ) − u ( ∂ x v ) + 50 v ( ∂ x v ) − u∂ x u∂ x v − v∂ x u∂ x v + 40 √ u − u v − uv + v ) ∂ x v. (74) We constructed the three nonequivalent Coxeter gradings in the algebra D ≃ so (8). The first of themis the standard one obtained with the Coxeter automorphism C = S α S α S α S α using its dihedralrealization. In the second one we use C = C R where R is the mirror automorphism. The thirdone is C = S α S α T where T is the external automorphism of order 3. For each of these gradings weconstructed the basis in the corresponding linear subspaces g ( k ) , the orbits of the Coxeter automorphismsand the related Lax pairs generating the corresponding mKdV hierarchies. We found compact expressionsfor each of the hierarchies in terms of the recursion operators. At the end we wrote explicitly the firstnontrivial mKdV equations and their Hamiltonians. For D (1)4 these are in fact two mKdV systems, dueto the fact that in this case the exponent 3 has multiplicity 2. Each of these mKdV systems consist of 4equations of third order with respect to ∂ x . For D (2)4 this is a system of three equations of third orderwith respect to ∂ x . Finally, for D (3)4 this is a system of two equations of fifth order with respect to ∂ x .The fact that these mKdV equations have the structure outlined above is a consequence of theseminal papers by Mikhailov [35] and Drinfeld and Sokolov [8]. However the explicit formulation of theLax operators as well as the explicit form of the equations themselves and their Hamiltonians, especiallythe ones for D (2)4 and D (3)4 are not so well known and deserve additional studies. Indeed, the fact thatthe mKdV equations (70) and (71) can be cast into the form (73), (74) means, that the Lax pair for D (3)4 case has a symmetry that interchanges the q ↔ q combined with a Weyl reflection S e − e S e + e whichinterchanges the orbits O ↔ O . Similar more complicated symmetries exist also for D (1)4 and D (2)4 ;they will be studied in next publications.We note, that the problem of constructing systems of higher mKdV equations for two functions hasbeen attacked by using the symmetry formalism developed by Shabat and his collaborators, see [39] andthe references therein. In the [36, 41] a system of two mKdV equations of order 5 with exponents 1, 5,7 and 11 has been reported. It does not coincide with the system (70) and (71) found above. On theother hand the only Kac-Moody algebra that has rank 2 and exponents 1, 5, 7 and 11 is D (3)4 , so thetwo systems must be equivalent. In other words one should be looking for a (gauge) transformation thatrelates the two equations.Another important aspect in the studies of these equations is related to the spectral theory of thecorresponding Lax operators. This will require further elaboration of the results in [25, 26] specifyingthem to the relevant choices of Q ( x, t ) and J in the Lax operators. One can expect deeper understandingof the expansions over the ‘squared‘ solutions of L . As a result one could see that even for Lax operatorspossessing deep reductions the inverse scattering problem can be related to a Riemann-Hilbert problem,and can be interpreted as a generalized Fourier transform [25, 26]. In particular one can expect to derivethe symplectic form of the ‘squared solutions’ [17, 29] and as a result to derive explicit expressions forthe action-angle variables for the mKdV hierarchy in terms of the scattering data. All these results16ill be naturally compatible with the existence of the hierarchy of Hamiltonian structures of the mKdVequations [8, 9, 15, 16, 26, 27] and the hierarchy of Lagrangian structures [42]. Acknowledgements
We are grateful to Ms S. Sushko for careful reading of the manuscript. One of us (VSG) is grateful toprofessor A. V. Mikhailov and professor V. S. Novikov for useful discussions and comments. This workhas been supported by the Bulgarian Science Foundation (grant NTS-Russia 02/101 from 23.10.2017)and by the RFBR (grant 18-51-18007). Two of us (VSG and AAS) are grateful to the organizing commit-tee of the IX-th International Conference (SCT-19) “Solitons, collapses and turbulence” Achievements,Developments and Perspectives in honor of Vladimir Zakharov’s 80th birthday, held in Yaroslavl, RussiaAugust 5-9, 2019 for their support and hospitality.
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