MKDV EQUATIONS RELATED TO THE D (2)

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1 MKDV EQUATIONS RELATED TO THE D ) ALGEBRA V. S. GERDJIKOV 1, D. M. MLADENOV, A. A. STEFANOV,3,, S. K. VARBEV,5 1 Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 7 Tzarigradsko chaussee, Sofia 178, Bulgaria Theoretical Physics Department, Faculty of Physics, Sofia University St. Kliment Ohridski, 5 James Bourchier Blvd, 116 Sofia, Bulgaria 3 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria Faculty of Mathematics, Sofia University St. Kliment Ohridski, 5 James Bourchier Blvd, 116 Sofia, Bulgaria 5 Institute of Solid State Physics, Bulgarian Academy of Sciences, 7 Tzarigradsko chaussee, Sofia 178, Bulgaria Received February 9, 015 We present a one-parameter families of mkdv-type equations related to g D 1) and g D ). They are a set of partial differential equations, integrable via the inverse scattering method. They admit a Hamiltonian formulation and the corresponding Hamiltonians are also given. The Riemann-Hilbert problems for the two Lax operators are formulated on a set of h rays l ν. We show that to each ray l ν one can relate a subalgebra of g which is direct sum of sl) subalgebras. Key words: Soliton equations, Integrable models, Kac-Moody algebras. PACS: 0.30.Ik, Lm, Tg, 5.35.Sb. 1. INTRODUCTION The discovery of the inverse scattering method by Gardner, Green, Kruskal and Miura [1] and the subsequent development by Lax [] led to the birth of modern soliton science. What followed was an explosion of interest in soliton equations and non-linear systems in general a good review on the history of solitons can be found in [3, ]), and the development of the techniques needed to solve them. It is worth mentioning some of the milestones in this development. One such example is the paper by Ablowitz, Kaup, Newell and Segur [5], where the integrability of the Zaharov- Shabat system is proved. Another important aspect, namely that there is a connection between soliton equations and Kac-Moody algebras was discovered by Drifneld and Sokolov [6, 7]. Quite independently the notion of the reduction group was introduced by Mikhailov [8] and used to construct new integrable systems, starting with the family of -dimensional Toda field theories. These two papers provide a natural ground for analyzing the connections between the Lax operators, the Kac-Moody algebras and the properties of the related nonlinear evolution equations NLEE). RJP Rom. 61Nos. Journ. Phys., 1-), Vol , Nos. 1-, 016) P , c) 016 Bucharest, - v.1.3a*

2 mkdv equations related to the D ) algebra 111 This paper is a part of a larger series of articles, where the connection between soliton equations and the Kac-Moody algebras based on the classical Lie algebra D is examined. Here we present only what is commonly called modified Korteveg - de Vries mkdv) equations related to the affine Kac-Moody algebras D 1) and. They are a set of non-linear partial differential equations, integrable via the inverse scattering method. They possess soliton solutions, which can be constructed, for example by using the dressing method for details on the dressing method, see [8, 9]). The equations can be derived by a number of different techniques, for example by using Drinfeld-Sokolov reduction [6] or with the help of recursion operators [10 1]. They can also be computed by directly solving the corresponding recursion relations. The paper is organized as follows. In Section we give some basic preliminaries, needed for the derivation of the equations and formulate the results from our previous paper [16] concerning the mkdv equations related to the D 1) algebra. In D Section 3 we derive the equations related to D ) and in Section se briefly address the spectral properties for both Lax operators Section contains some concluding remarks.. PRELIMINARIES We assume that the reader is familiar with the basic facts about finite and infinite dimensional Lie algebras. An introductory treatment can be found in [6, 13]. We will only highlight the information needed for the present paper. Let g be a finite-dimensional simple Lie algebra. By ad X we denote the linear operator defined by ad X Y ) = [ X, Y ], X,Y g. 1) This operator has a kernel and can only be inverted on its image. We denote that inverse by ad 1 X. If X is diagonalizable then ad 1 X can be expressed as a polynomial of ad X. Let, be the Killing-Cartan form on g which is defined by X,Y = tradx ad Y ). ) Let ϕ be an automorphism of g of finite order. Every such ϕ introduces a grading in g by g = s 1 g k), 3) k=0 such that ) πi ϕx) = ω k X, ω = exp, s X g k), ) RJP 61Nos. 1-), ) c) v.1.3a*

3 11 V. S. Gerdjikov et al. 3 Table 1 A realization of the Coxeter automorphisms and Coxeter numbers for D 1),D) and D 3). Here Sαi denotes reflection with respect to the simple root α i, R is the second order outer automorphism that exchanges α 3 and α, and T is the third order outer automorphism a triality transformation) that sends α 1 α 3 α. Algebra Coxeter automorphism Coxeter number Exponents D 1) C 1 = S α S α1 S α3 S α 6 1,3,5,3 D ) C = S α S α1 S α3 R 8 1,3,5,7 D 3) C 3 = S α S α1 T 1 1,5,7,11 where s is the order of ϕ. Obviously the grading condition holds [ g k),g l)] g k+l), 5) where k + l is taken modulo s. As is well known, Kac-Moody algebras are graded algebras for a definitions and details see [6, 1, 15]). We will use the definition of a Kac-Moody algebra given in [6]. In our case the gradings are done using the Coxeter automorphisms given in Table 1. A basis compatible with the grading is given by averaging the standard Cartan- Weyl basis of D for more details see [16]) over the action of the corresponding Coxeter automorphism E k) i = Ẽ k) i = 5 s=0 7 s=0 ω sk 1 C s 1E αi ), H k) j = ω sk C s E αi ), H k) j = 5 s=0 7 s=0 ω sk 1 C s 1H j ), ford 1), ω sk C s H j ), ford ), where ω 1 = expπi/6), ω = expπi/8). H i are the basis elements of the Cartan subalgebra of D and E αi is the Weyl generator corresponding to the simple root α i. Note that H k) j is non-vanishing only if k is an exponent. This means that the dimension of each subspace g k) is equal to r + m s where m s is the multiplicity of the exponent s and r is the rank of the corresponding algebra. Remark 1 It is important to note, that basis in 6) is correct provided we have chosen the Coxeter automorphism in its dihedral form. For the D 1) it is C = w 1 w, w 1 = S α, w = S α1 S α3 S α. 7) where α 1 = e 1 e, α = e e 3, α 3 = e 3 e and α = e 3 + are the simple roots D 1). With this choice one can check that through each simple root passes just one orbit of C [15]. RJP 61Nos. 1-), ) c) v.1.3a* )

4 mkdv equations related to the D ) algebra 113 The explicit form of 6) for D 1) and D ) can be found in [16] and [17] correspondingly..1. LAX PAIR AND RECURSION RELATIONS We will consider a Lax pair of the following form: L = i x + Qx,t) λj, M = i t + V 0) x,t) + λv 1) x,t) + λ V ) x,t) λ 3 K, i.e. the potentials of L and M are elements of some affine Kac-Moody algebra in our case D 1) and D ) ). The Lax equation [ ] L, M = 0 9) must hold identically with respect to λ. If the inverse scattering problem for L is uniquely solvable, then we have an effective tool not only to solve the relevant NLEE, but also a strong indication of its complete integrability. For more details in this respect in the special cases of Z h -reductions see [10 1, 18]. In fact equation 9) implies the following set of recursion relations λ [ ] : J,K = 0, λ 3 [ : J,V ) ] + [ Q,K ] = 0, λ s : λ 0 : V s) i x + [ Q,V s)] [ J,V s 1)] = 0, i Q t + i V 0) x + [ Qx,t),V 0)] = 0. where s = 1,. They can be solved with the use of recursion operators see [10, 11]), which can be used to construct the whole hierarchy of soliton equations. For the cases of D 1) and D ) those hierarchies can be found in [16, 17]. Here we will only present the first members - a set of mkdv equations. The resulting equations can be presented in Hamiltonian form as follows: 8) 10) t q i = {q i,h} 11) with a Poisson bracket given by {F,G} = ω ij x,y) δf δg dxdy, 1) R δq i δq j where we sum over repeating indexes. The Poisson structure tensor is ω ij x,y) = 1 δ ij x δx y) y δx y)). 13) RJP 61Nos. 1-), ) c) v.1.3a*

5 11 V. S. Gerdjikov et al. 5 In this case 11) reduces to t q i = x δh δq i. 1) The Hamiltonian densities can be found at the end of the corresponding sections... MKDV EQUATIONS RELATED TO D 1) The equations related to D 1) can be found in [19], but for the sake of completeness, besides the equations here we will also give the elements of the potential of L and M. We will use the Lax pair given by 8). The potential of L is parametrized by 0 q 1 q 1 q q 3 q q 0 q 1 0 q q 3 q q 1 0 q q 1 q 0 q 3 q 0 q 1 q Q = q q 3 q q q q 3 q 3 q q 0 0 q 3 q 3 q, 15) q q 1 0 q q 3 0 q q 1 q 0 q 1 q q 3 q 0 q 1 0 q q q 3 q q 1 q 1 0 and J = diag1,ω 1,ω1 5,0,0, ω5 1, ω 1, 1) with ω 1 = expπi/6). As for M, the elements V k) are given as a linear combination of the basis elements in g k) with some coefficients v k) i and K = diag a, a, a,b, b,a,a, a). Solving 10) gives for V ) we obtain: v ) 1 = ωaq 1, v ) = 0, v ) 3 = ωa + b)q 3, v ) = ωa b)q. 16) Next for V 1) we get v 1) 1 = a 1 ω 3 ) x q 1 + ω 3) ω ) + 3) q q 3 1 ω 3 ) ) q 1 q, v 1) = a x q + q 3 q 1 ) 1 + )q )) )q 3, v 1) a 3 = i ω 3 ) x q 3 ω + 3) + ω ) 3) q 1 q ω 3 ) ) q q 3, v 1) = a 1 + )q1 + q + 1 ) )q3. RJP 61Nos. 1-), ) c) v.1.3a* )

6 6 mkdv equations related to the D ) algebra 115 Note that v 1) 5 can formally be interpreted as particle density and D = aq1 1 a + b)q 3 1 ) a b)q dx 18) is an integral of motion. Finally the solution for V 0) is v 0) 1 = a xq 1 3q 1 x q ) 33a + b)q x q 3 + 3a b)q 3 x q ) 3q 1 aq a b)q3 a + b)q), v 0) = 3 3a x q1 a + b) xq3 3q aq 1 a + b)q3 a b)q), 3 a b) xq v 0) 3 = a + b) xq 3 3q 3 x q ) + 33a + b)q x q 1 + bq 1 x q ) 3q 3 aq a b)q 1 a + b)q ), v 0) = a b) xq 3q x q ) + 33a b)q 3 x q 1 bq 1 x q 3 ) 3q aq 3 a b)q a + b)q 1). 19) And finally, the equations are t q 1 = x a xq 1 3q 1 x q ) 33a + b)q x q 3 + 3a b)q 3 x q ) ) 3q 1 aq a b)q3 a + b)q), 3 3 ) t q = x 3a x q1 a + b) xq3 a b) xq 3q aq 1 a + b)q3 a b)q), t q 3 = x a + b) xq 3 3q 3 x q ) + 33a + b)q x q 1 + bq 1 x q ) 3q 3 aq a b)q1 a + b)q) ), t q = x a b) xq 3q x q ) + 33a b)q 3 x q 1 bq 1 x q 3 ) 3q aq 3 a b)q a + b)q1) ). They admit a Hamiltonian whose density is linear combination H = ah 1 +bh with RJP 61Nos. 1-), ) c) v.1.3a*

7 116 V. S. Gerdjikov et al. 7 arbitrary constants a and b of the following densities: H 1 = q 1 xq 1 1 q3 xq 3 + q xq ) q 1q + 3q 3 q ) x q 1 ) 3 + q q q1) ) x q + q q 3 3q 1 q ) x q 3 + q q 3q 1 q 3 ) x q 0) 3 q q1 1 q3 q) + q3q q1q q1q 3 ), H = 1 q3 xq 3 + q xq ) 3 + q 1 q)q 3 q) ) 1) 3 3q1 q q q 3 ) x q 3 + q q ) 3 3) x q 3q 1 q 3 q q ) x q. One can get rid of one of the constants say, a by replacing t t a ). However the effective Hamiltonian density will depend on the ratio b/a. Thus we have a one parameter family of MKdV-type Hamiltonians. This is deeply related to the fact that 3 is a double-valued exponent of D 1). 3. MKDV EQUATIONS RELATED TO D ) Again, the Lax pair is given by 8), with the potential 0 q 1 q 1 q q q 3 q 3 0 q 1 0 q q 1 q 3 q 0 q 3 q 1 q 0 q 3 q 1 0 q q 3 Q = q q 1 q q 1 q 3 q q q 3 q q 3 q 1 q, ) q 3 q 0 q 1 q 3 0 q q 1 q 3 0 q q 3 q 1 q 0 q 1 0 q 3 q 3 q q q 1 q 1 0 and J = diag1, ω 3,ω, i,i, ω, ω 3, 1). The elements V k) are given as a linear combination of the basis elements in g k) with some coefficients v k) i, and K = aj 3. Solving 10) gives us: For V ) we have v ) 1 = 1 i aq 1, v ) = aq, v 3) 1 = i 1 + aq 3. 3) RJP 61Nos. 1-), ) c) v.1.3a*

8 8 mkdv equations related to the D ) algebra 117 For V 1) we get v 1) 1 = a 1 ω 3 ) x q ω 3 ) ) q 1 q, v 1) = a v 1) ω 3) ω ) + 3) q q 3 x q + q 3 q 1 ) 1 + )q )) )q 3, a 3 =i ω 3 ) x q 3 ω + 3) + ω ) 3) q 1 q ω 3 ) ) q q 3, v 1) = a 1 + )q 1 + q + 1 )q 3 ). ) Solving for V 0) we obtain v 0) 1 = a 81 + ) xq q )q 3 ) x q +61 ) )q x q 3 + a q3 3 6q 1 q3 6qq 3 + 3q1q 3 3q 1 q), v 0) =a 8 xq )q 1 + q 3 ) x q 1 q )) )q 3 ) x q 3 a 1q 1q q 3 + q 3 + q q3 + q1q ), v 0) 3 = a + 81 ) xq )q 1 q 3 ) x q )q x q 1 ) + a q3 1 6q 1q 3 6q 1 q + 3q 1 q 3 3q q 3 ). 5) And finally, the λ-independent terms in the Lax representation provide us with RJP 61Nos. 1-), ) c) v.1.3a*

9 118 V. S. Gerdjikov et al. 9 the equations t q 1 = a x 81 + ) xq q )q 3 ) x q +61 ) )q x q 3 + a xq3 3 6q 1 q3 6qq 3 + 3q1q 3 3q 1 q), t q =a x 8 xq )q 1 + q 3 ) x q 1 q )) )q 3 ) x q 3 a x1q 1 q q 3 + q 3 + q q3 + q1q ), t q 3 = a + x 81 ) xq )q 1 q 3 ) x q )q x q 1 ) + a xq 3 1 6q 1q 3 6q 1 q + 3q 1 q 3 3q q 3 ). 6) The above equations admit a hamiltonian with the following density H = 1 16 a ) q 1 xq 1 x q 1 ) ) + 1 q 3 xq x q ) ) ) q 3 xq 3 x q 3 ) ) + + ) q x q 1 q 1 x q ) + ) q x q 3 q 3 x q ) + 6 q q 3 x q 1 q 1 x q 3 ) 7) + q 3 1q 3 6q 1q 1q 1q 3 q 1 q 3 q 6q q 3 + q 1 q 3 3 q ) ).. ON THE SPECTRAL PROPERTIES OF THE LAX OPERATORS Here we just briefly formulate the spectral properties of the Lax operators for the class of smooth potentials Qx) vanishing fast enough for x ±. Detailed derivations, concerning the construction of their fundamental analytic solutions, scattering matrix, etc. will be published elsewhere. It is well known that solving the direct and the inverse scattering problems for the Lax operator L is based on the construction of the fundamental analytic solution χ ν x,t,λ) which can be viewed as a solution to a Riemann-Hilbert problem RHP). The spectral properties of the generic n n Zakharov-Shabat system with complexvalued J have been analyzed in [18, 0, 1]. Skipping the details here we formulate the main results for constructing the FAS for the Lax operators considered above. Doing this properly requires special care treating the reduction conditions. First of all we have to establish the regions of analyticity of χ ν x,t,λ) and of ξ ν x,t,λ) = χ ν x,t,λ)e ijλx. These regions are in fact sectors of the complex λ- plane. The rays l ν which separates the two sectors Ω ν+1 and Ω ν are defined by the RJP 61Nos. 1-), ) c) v.1.3a*

10 10 mkdv equations related to the D ) algebra 119 l 5 l Ω Ω 3 l 3 Ω l Ω 1 λ l 1 l 7 l 6 Ω 6 l 5 Ω 5 Ω l Ω 3 l 3 Ω l Ω 1 λ l 1 l 6 Ω 5 Ω 0 l 0 l 8 Ω 7 Ω 0 l 0 Ω 6 Ω 11 Ω 8 Ω 16 l 7 Ω 7 Ω 8 Ω 9 Ω 10 l 11 l 9 Ω 9 Ω 10 Ω 1 l 15 l 8 l 10 l 10 Ω 11 Ω 1 Ω 13 l 1 l 9 l11 l 1 l 13 Fig. 1 The contours h 1 ν=0lν of the RHP and the analyticity sectors Ων of the FAS for the Lax operators: the case of D 1) - left panel; the case of D ) - right panel; condition: Im λαj) = 0, 8) where α is a root of the algebra g and J g 1) h. Note that J carries with itself the reduction condition; indeed, CJ) = ωj, where ω h = 1. Putting in eq. 8) λ = λ e iϕ we easily obtain for each α a set of simple linear equations involving ϕ and the arguments of the eigenvalues of J. So for each of the Lax operators we obtain that the relevant RHP can be formulated on a set of h rays l ν closing angles π/h, see Figure 1. The solutions of this simple equation allows one to find that to each line l ν l ν+6 we can relate a subset of the set of positive roots that are mutually orthogonal. The result is given in Table. Skipping the details, we briefly reformulate the general results obtained in [10, 11] for this particular Lax operators. A) First, in each of the sectors one can construct FAS χ ν x,t,λ), λ Ω ν and calculate its limits for x ± along the lines l ν ; more specifically lim x eiλjx ξ ν x,t,λ)e iλjx = S ν + t,λ), λ l lim x eiλjx ξ ν x,t,λ)e iλjx = Tν t,λ)d ν + ν e +i0, 9) λ), RJP 61Nos. 1-), ) c) v.1.3a*

11 10 V. S. Gerdjikov et al. 11 Table Subsets of positive roots related to the lines l ν l ν+6 for D 1). l 0 l 6 l 1 l 7 l l 8 l 3 l 9 l l 10 l 5 l 11 e e 3,e 1 ± e e 1 e 3 e 1 e,e 3 ± e e + e 3 e 1 + e 3,e ± e e 1 + e Table 3 Subsets of positive roots related to the lines l ν l ν+8 for D ). l 0 l 8 l 1 l 9 l l 10 l 3 l 11 e + e 3 e 1 + e,e 3 + e e 1 + e e + e,e 1 e 3 l l 1 l 5 l 13 l 6 l 1 l 7 l 15 e e 3 e 1 e,e 3 e e 1 e e e,e 1 + e 3 and lim x eiλjx ξ ν 1 x,t,λ)e iλjx = Sν t,λ), lim x eiλjx ξ ν 1 x,t,λ)e iλjx = T ν + t,λ)dν λ), λ l ν e i0, 30) where S ν ±, T ν ± and D ν ± are elements of the subgroup G ν whose Lie algebra g ν has as positive roots the subset of roots related to l ν, see Table. As a minimal set of scattering data we can use the limits of the FAS along both sides of the ray l 0 and along l 11 e i0 and l 1 e i0. The rest of the scattering data can be recovered from the minimal set by acting with the Coxeter automorphism. An important fact following from these consideration is formulated as Lemma 1 Each of the subalgebras g ν is a direct sum of sl) subalgebras. Proof 1 We will consider the subalgebra g 0 related to l 0. According to table g 0 has as positive roots e e 3, e 1 + e and e 1 e. Each of these roots generates an sl) subalgebra of D 1). It is easy to check that these roots are orthogonal to each other, which means that their sums are not roots [15]. Therefore g 0 is a direct sum of three sl) subalgebras. All other cases are considered and proved analogously. B) The RHP satisfied by FAS is formulated as: ξ ν x,t,λ) = ξ ν 1 x,t,λ)g ν x,t,λ), λ l ν G ν x,t,λ) = e iλjx Ŝ ν t,λ)s + ν t,λ)e iλjx, ν = 0,...,h 1. Similarly, we can consider the spectral properties of the Lax operator related to D ). Since the element J is different see )), then the solutions of the equation 8) is also different. Now the spectrum of L consists of 8 straight lines closing angles π/8, see the right panel of Figure 1. The roots related to each line are given in Table 3. RJP 61Nos. 1-), ) c) v.1.3a* )

12 1 mkdv equations related to the D ) algebra CONCLUSIONS We have presented the mkdv-type equations related to D 1) and D ). As was shown, the equations related to D 1) are effectively a one-parameter family of equations, which is a consequence of the fact that 3 is a double-valued exponent of D 1). However D) has only simply-valued exponents, so the relevant mkdv system does not contain arbitrary parameters. The direct and inverse spectral problems for both L can be reformulated in terms of a RHP problems. We briefly reviewed their properties and define minimal sets of scattering data. The RHP are formulated on a set of h rays l ν. We show that to each ray l ν one can relate a subalgebra of g which is direct sum of sl) subalgebras. The connection between integrable equations and infinite dimensional Lie algebras is deep and intriguing. The present paper is a small step towards the completion of the study of this connection. A more challenging task would be the study of the properties of the solutions of the equations, starting with the soliton solutions and their interactions. This next step will be done in the near future. Acknowledgements. The work is partially supported by the ICTP - SEENET-MTP project PRJ- 09. We are grateful to professor R. Constantinescu for his hospitality in Sinaia. One of us VSG) is grateful to professors A.S. Sorin, A. V. Mikhailov and V. V. Sokolov for useful discussions during his visit to JINR, Dubna, Russia under project /018. A. APPENDIX The simple Lie algebra D so8) is usually represented by a 8 8 antisymmetric matrices. In this realization the Cartan subalgebra is not diagonal, so we will use a realization for which every X D satisfies SX + SX) T = 0, 3) where the matrix S is given by S = ) This way the Cartan subalgebra is given by diagonal matrices. RJP 61Nos. 1-), ) c) v.1.3a*

13 1 V. S. Gerdjikov et al. 13 The Cartan-Weyl basis of D is given by H i = e ii e 9 i,9 i, 1 i, E αj = e j,j+1 + e 8 j,9 j, 1 j 3, E α = e 3,5 + e,6, E αj = E αj ) T, where by e ij we denote a matrix that has a one at the i th row and j th column and is zero everywhere else. The Coxeter automorphisms for different algebras are given in Table 1. For the case of D 1) the Coxeter automorphism is realized as a similarity transformation 3) C 1 X) = c 1 Xc 1 1, 35) where the matrix c is given by c 1 = ) For the case of D ) the Coxeter automorphism is given by C X) = c rxr 1 c 1, 37) where c = , r = ) RJP 61Nos. 1-), ) c) v.1.3a*

14 1 mkdv equations related to the D ) algebra 13 REFERENCES 1. C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Phys. Rev. Lett. 19, ).. P. D. Lax, Comm. Pure Appl. Math. 1, ). 3. J. E. Allen, Phys. Scr. 57, ).. L. D. Faddeev, L. A. Takhtadjan, Hamiltonian Method in the Theory of Solitons Springer Verlag, Berlin 1987). 5. M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, Studies Appl. Math. 53, ). 6. V. G. Drinfeld, V. V. Sokolov, Sov. Math. Dokl. 3, ). 7. V. V. Drinfel d, V. G. Sokolov, Sov. J. Math. 30, ). 8. A. V. Mikhailov, Physica Utrecht) 3D, ). 9. V. E. Zakharov, A. B. Shabat, Funct. Anal. Appl. 13, ). 10. V. S. Gerdjikov, A. B. Yanovski, J. Geom. Symmetry Phys. 31, ). 11. V. S. Gerdjikov, A. B. Yanovski, Stud. Appl. Math. 13, ). 1. A. B. Yanovsky, J. Geom. Symm. Phys. 3, ). 13. R. Carter, Lie Algebras of Finite and Affine Type Cambridge University Press, Cambridge, 005). 1. V. Kac, Infinite Dimensional Lie Algebras Cambridge University Press, Cambridge, 1995). 15. S. Helgasson, Differential geometry, Lie Groups and Symmetric Spaces Academic Press, New York, 1978). 16. V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, Soliton equations related to the affine Kac-Moody algebra D 1), arxiv: [nlin] 01). 17. V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev, Integrable hierarchy associated with the affine Kac-Moody algebra D ), in preparation. 18. V. S. Gerdjikov, A. B. Yanovski, J. Math. Phys. 35, ). 19. V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S.K. Varbev, in Mathematics in Industry Ed. A. Slavova, pp , Cambridge Scholar Publishing, 01). 0. P. Caudrey, Physica D D6, ). 1. R. Beals, R. Coifman, Inverse Probl. 3, ). RJP 61Nos. 1-), ) c) v.1.3a*