Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type

New reductions for the multicomponent modified Korteveg-de Vries (MMKdV) equations on the symmetric spaces of {\bf DIII}-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert problem. The minimal sets of scattering data $\mathcal{T}_i$, $i=1,2$ which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on $\mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak{g}\simeq so(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.


Introduction
The modified Korteweg-de Vries equation [1] q t + q xxx + 6ǫq x q 2 (x, t) = 0, ǫ = ±1, has natural multicomponent generalizations related to the symmetric spaces [2]. They can be integrated by the ISM using the fact that they allow the following Lax representation: The analysis in [2,3,4] reveals a number of important results. These include the corresponding multicomponent generalizations of KdV equations and the generalized Miura transformations interrelating them with the generalized MMKdV equations; two of their most important reductions as well as their Hamiltonians.
Our aim in this paper is to explore new types of reductions of the MMKdV equations. To this end we make use of the reduction group introduced by Mikhailov [5,6] which allows one to impose algebraic reductions on the coefficients of Q(x, t) which will be automatically compatible with the evolution of MMKdV. Similar problems have been analyzed for the N -wave type equations related to the simple Lie algebras of rank 2 and 3 [7,8] and the multicomponent NLS equations [9,10]. Here we illustrate our analysis by the MMKdV equations related to the algebras g ≃ so(2r) which are linked to the DIII-type symmetric spaces series. Due to the fact that the dispersion law for MNLS is proportional to λ 2 while for MMKdV it is proportional to λ 3 the sets of admissible reductions for these two NLEE equations differ substantially.
In the next Section 2 we give some preliminaries on the scattering theory for L, the reduction group and graded Lie algebras. In Section 3 we construct the fundamental analytic solutions of L, formulate the corresponding Riemann-Hilbert problem and introduce the minimal sets of scattering data T i , i = 1, 2 which define uniquely both the scattering matrix and the solution of the MMKdV Q(x, t). Some of these facts have been discussed in more details in [10], others had to be modified and extended so that they adequately take into account the peculiarities of the DIII type symmetric spaces. In particular we modified the definition of the fundamental analytic solution which lead to changes in the formulation of the Riemann-Hilbert problem. In Section 4 we show that the ISM can be interpreted as a generalized Fourier [10] transform which maps the potential Q(x, t) onto the minimal sets of scattering data T i . Here we briefly outline the hierarchy of Hamiltonian structures for the generic MMKdV equations. The next Section 5 contains two classes of nontrivial reductions of the MMKdV equations related to the algebra so (8). The first class is performed with automorphisms of so(8) that preserve J; the second class uses automorphisms that map J into −J. While the reductions of first type can be applied both to MNLS and MMKdV equations, the reductions of second type can be applied only to MMKdV equations. Under them 'half' of the members of the Hamiltonian hierarchy become degenerated [11,2]. For both classes of reductions we find examples with groups of reductions isomorphic to Z 2 , Z 3 and Z 4 . We also provide the corresponding reduced Hamiltonians and symplectic forms and Poisson brackets. At the end of Section 5 we derive the effects of these reductions on the scattering matrix and on the minimal sets of scattering data. In Section 6 following [3] we analyze the classical r-matrix for the corresponding NLEE. The effect of reductions on the classical r-matrix is discussed. The last Section contains some conclusions.

Cartan-Weyl basis and Weyl group for so(2r)
Here we fix the notations and the normalization conditions for the Cartan-Weyl generators of g ≃ so(2r), see e.g. [12]. The root system ∆ of this series of simple Lie algebras consists of the roots ∆ ≡ {±(e i − e j ), ±(e i + e j )} where 1 ≤ i < j ≤ r. We introduce an ordering in ∆ by specifying the set of positive roots ∆ + ≡ {e i − e j , e i + e j } for 1 ≤ i < j ≤ r. Obviously all roots have the same length equal to 2.
We introduce the basis in the Cartan subalgebra by h k ∈ h, k = 1, . . . , r where {h k } are the Cartan elements dual to the orthonormal basis {e k } in the root space E r . Along with h k we introduce also where (α, e k ) is the scalar product in the root space E r between the root α and e k . The basis in so(2r) is completed by adding the Weyl generators E α , α ∈ ∆. The commutation relations for the elements of the Cartan-Weyl basis are given by [12] [h k , E α ] = (α, e k )E α , We will need also the typical 2r-dimensional representation of so(2r). For convenience we choose the following definition for the orthogonal algebras and groups where by 'hat' we denote the inverse matrixT ≡ T −1 and Here and below by E jk we denote a 2r × 2r matrix with just one non-vanishing and equal to 1 matrix element at j, k-th position: (E jk ) mn = δ jm δ kn . Obviously S 2 0 = 1 1. In order to have the Cartan generators represented by diagonal matrices we modified the definition of orthogonal matrix, see (2.1). Using the matrices E jk defined by equation (2.2) we get wherek = 2r + 1 − k.
We will denote by a = r k=1 e k the r-dimensional vector dual to J ∈ h; obviously J = r k=1 h k .
If the root α ∈ ∆ + is positive (negative) then (α, a) ≥ 0 ((α, a) < 0 respectively). The normalization of the basis is determined by The root system ∆ of g is invariant with respect to the Weyl reflections S α ; on the vectors y ∈ E r they act as All Weyl reflections S α form a finite group W g known as the Weyl group. On the root space this group is isomorphic to S r ⊗ (Z 2 ) r−1 where S r is the group of permutations of the basic vectors e j ∈ E r . Each of the Z 2 groups acts on E r by changing simultaneously the signs of two of the basic vectors e j . One may introduce also an action of the Weyl group on the Cartan-Weyl basis, namely [12] S The matrices A α are given (up to a factor from the Cartan subgroup) by where H A is a conveniently chosen element from the Cartan subgroup such that H 2 A = 1 1. The formula (2.3) and the explicit form of the Cartan-Weyl basis in the typical representation will be used in calculating the reduction condition following from (4.16).

Graded Lie algebras
One of the important notions in constructing integrable equations and their reductions is the one of graded Lie algebra and Kac-Moody algebras [12]. The standard construction is based on a finite order automorphism C ∈ Aut g, C N = 1 1. The eigenvalues of C are ω k , k = 0, 1, . . . , N − 1, where ω = exp(2πi/N ). To each eigenvalue there corresponds a linear subspace g (k) ⊂ g determined by g (k) ≡ X : X ∈ g, C(X) = ω k X .
Then g = N −1 ⊕ k=0 g (k) and the grading condition holds where k + n is taken modulo N . Thus to each pair {g, C} one can relate an infinite-dimensional algebra of Kac-Moody type g C whose elements are The series in (2.5) must contain only finite number of negative (positive) powers of λ and g (k+N ) ≡ g (k) . This construction is a most natural one for Lax pairs; we see that due to the grading condition (2.4) we can always impose a reduction on L(λ) and M (λ) such that both U (x, t, λ) and V (x, t, λ) ∈ g C . In the case of symmetric spaces N = 2 and C is the Cartan involution. Then one can choose the Lax operator L in such a way that as it is the case in (1.1). Here the subalgebra g (0) consists of all elements of g commuting with J.
The special choice of J = r k=1 h k taken above allows us to split the set of all positive roots ∆ + into two subsets Obviously the elements α ∈ ∆ + 1 have the property α(J) = (α, a) = 2, while the elements β ∈ ∆ + 0 have the property β(J) = (β, a) = 0. In this section we outline some of the well known facts about the spectral theory of the Lax operators of the type (1.1).

The scattering problem for L
Here we briefly outline the basic facts about the direct and the inverse scattering problems [13,14,15,16,17,18,19,20,21,22,23] for the system (1.1) for the class of potentials Q(x, t) that are smooth enough and fall off to zero fast enough for x → ±∞ for all t. In what follows we treat DIII-type symmetric spaces which means that Q(x, t) is an element of the algebra so(2r).
In the examples below we take r = 4 and g ≃ so (8).
The main tool for solving the direct and inverse scattering problems are the Jost solutions which are fundamental solutions defined by their asymptotics at x → ±∞ lim x→∞ ψ(x, λ)e iλJx = 1 1, lim Along with the Jost solutions we introduce which satisfy the following linear integral equations These are Volterra type equations which, have solutions providing one can ensure the convergence of the integrals in the right hand side. For λ real the exponential factors in (2.6) and (2.7) are just oscillating and the convergence is ensured by the fact that Q(x, t) is quickly vanishing for x → ∞. Remark 1. It is an well known fact that if the potential Q(x, t) ∈ so(2r) then the corresponding Jost solutions of equation (1.1) take values in the corresponding group, i.e. ψ(x, λ), φ(x, λ) ∈ SO(2r).
The Jost solutions as whole can not be extended for Im λ = 0. However some of their columns can be extended for λ ∈ C + , others -for λ ∈ C − . More precisely we can write down the Jost solutions ψ(x, λ) and φ(x, λ) in the following block-matrix form where the superscript + and (resp. −) shows that the corresponding r × r block-matrices allow analytic extension for λ ∈ C + (resp. λ ∈ C − ). Solving the direct scattering problem means given the potential Q(x) to find the scattering matrix T (λ). By definition T (λ) relates the two Jost solutions and has compatible block-matrix structure. In what follows we will need also the inverse of the scattering matrix The diagonal blocks of T (λ) andT (λ) allow analytic continuation off the real axis, namely a + (λ), c + (λ) are analytic functions of λ for λ ∈ C + , while a − (λ), c − (λ) are analytic functions of λ for λ ∈ C − . We introduced also ρ ± (λ) and τ ± (λ) the multicomponent generalizations of the reflection coefficients (for the scalar case, see [27,17,28]) The reflection coefficients do not have analyticity properties and are defined only for λ ∈ R.
From Remark 1 one concludes that T (λ) ∈ SO(2r), therefore it must satisfy the second of the equations in (2.1). As a result we get the following relations between c ± , d ± and a ± , b ± c + (λ) =ŝ 0 a +,T (λ)s 0 , and in addition we have Next we need also the asymptotics of the Jost solutions and the scattering matrix for λ → ∞ Now it is the collections of rows ofψ(x, λ) andφ(x, λ) that possess analytic properties in λ (2.14) Just like the Jost solutions, their inverse (2.14) are solutions to linear equations (2.12) with regular boundary conditions (2.13); therefore they can have no singularities on the real axis λ ∈ R. The same holds true also for the scattering matrix T (λ) =ψ(x, λ)φ(x, λ) and its inversê as well as are analytic for λ ∈ C ± and have no singularities for λ ∈ R. However they may become degenerate (i.e., their determinants may vanish) for some values λ ± j ∈ C ± of λ. Below we briefly analyze the structure of these degeneracies and show that they are related to discrete spectrum of L.
3 The fundamental analytic solutions and the Riemann-Hilbert problem 3

.1 The fundamental analytic solutions
The next step is to construct the fundamental analytic solutions (FAS) χ ± (x, λ) of (1.1). Here we slightly modify the definition in [10] to ensure that χ ± (x, λ) ∈ SO(2r). Thus we define where the block-triangular functions S ± (λ) and T ± (λ) are given by The matrices D ± (λ) are block-diagonal and equal The upper scripts ± here refer to their analyticity properties for λ ∈ C ± . In view of the relations (2.10) it is easy to check that all factors S ± , T ± and D ± take values in the group SO(2r). Besides, since we can view the factors S ± , T ± and D ± as generalized Gauss decompositions (see [12]) of T (λ) and its inverse. The relations between c ± (λ), d ± (λ) and a ± (λ), b ± (λ) in equation (2.9) ensure that equations (3.3) become identities. From equations (3.1), (3.2) we derive valid for λ ∈ R. Below we introduce Strictly speaking it is X ± (x, λ) that allow analytic extension for λ ∈ C ± . They have also another nice property, namely their asymptotic behavior for λ → ±∞ is given by Along with X ± (x, λ) we can use another set of FASX ± (x, λ) = X ± (x, λ)D ± , which also satisfy equation (3.6) due to the fact that lim λ→∞ D ± (λ) = 1 1.

The Riemann-Hilbert problem
The equations (3.4) and (3.5) can be written down as where Likewise the second pair of FAS satisfỹ Equation (3.7) (resp. equation (3.8)) combined with (3.6) is known in the literature [24] as a Riemann-Hilbert problem (RHP) with canonical normalization. It is well known that RHP with canonical normalization has unique regular solution; the matrix-valued solutions X + 0 (x, λ) and X − 0 (x, λ) of (3.7), (3.6) is called regular if det X ± 0 (x, λ) does not vanish for any λ ∈ C ± . Let us now apply the contour-integration method to derive the integral decompositions of X ± (x, λ). To this end we consider the contour integrals and where λ ∈ C + and the contours γ ± are shown on Fig. 1.
Each of these integrals can be evaluated by Cauchy residue theorem. The result for λ ∈ C + are The discrete sums in the right hand sides of equations (3.9) and (3.10) naturally provide the contribution from the discrete spectrum of L. For the sake of simplicity we assume that L has a finite number of simple eigenvalues λ ± j ∈ C ± ; for additional details see [10]. Let us clarify the above statement. For the 2 × 2 Zakharov-Shabat problem it is well known that the discrete eigenvalues of L are provided by the zeroes of the transmission coefficients a ± (λ), which in that case are scalar functions. For the more general 2r × 2r Zakharov-Shabat system (1.1) the situation becomes more complex because now a ± (λ) are r × r matrices. The discrete eigenvalues λ ± j now are the points at which a ± (λ) become degenerate and their inverse develop pole singularities. More precisely, we assume that in the vicinities of λ ± j a ± (λ), c ± (λ) and their inverseâ ± (λ),ĉ ± (λ) have the following decompositions in Laurent series where all the leading coefficients a ± j ,â ± j c ± j ,ĉ ± j are degenerate matrices such that In addition we have more relations such aŝ that are needed to ensure that the identitiesâ ± (λ)a ± (λ) = 1 1,ĉ ± (λ)c ± (λ) = 1 1 etc hold true for all values of λ.
The assumption that the eigenvalues are simple here means that we have considered only first order pole singularities ofâ ± j (λ) andĉ ± j (λ). After some additional considerations we find that the 'halfs' of the Jost solutions |ψ ± (x, λ) and |φ ± (x, λ) satisfy the following relationships and the additional coefficients b ± j and d ± j are constant r × r nondegenerate matrices which, as we shall see below, are also part of the minimal sets of scattering data needed to determine the potential Q(x, t).
These considerations allow us to calculate explicitly the residues in equations (3.9), (3.10) with the result where |0 stands for a collection of r columns whose components are all equal to 0. We can also evaluate J 1 (λ) and J 2 (λ) by integrating along the contours. In integrating along the infinite semi-circles of γ ±,∞ we use the asymptotic behavior of X ± (x, λ) andX ± (x, λ) for λ → ∞. The results are where in evaluating the integrands we made use of equations (2.8), (2.9), (3.7) and (3.8).
Equating the right hand sides of (3.9) and (3.11), and (3.10) and (3.12) we get the following integral decomposition for X ± (x, λ): where X ± j (x) = X ± (x, λ ± j ) and Equations (3.13), (3.14) can be viewed as a set of singular integral equations which are equivalent to the RHP. For the MNLS these were first derived in [25].
We end this section by a brief explanation of how the potential Q(x, t) can be recovered provided we have solved the RHP and know the solutions X ± (x, λ). First we take into account that X ± (x, λ) satisfy the differential equation which must hold true for all λ. From equation (3.6) and also from the integral equations (3.13), (3.14) one concludes that X ± (x, λ) and their inverseX ± (x, λ) are regular for λ → ∞ and allow asymptotic expansions of the form Inserting these into equation (3.15) and taking the limit λ → ∞ we get

The generalized Fourier transforms
It is well known that the ISM can be interpreted as a generalized Fourier [10] transform which maps the potential Q(x, t) onto the minimal sets of scattering data T i . Here we briefly formulate these results and in the next Section we will analyze how they are modified under the reduction conditions. The generalized exponentials are the 'squared solutions' which are determined by the FAS and the Cartan-Weyl basis of the corresponding algebra as follows

Expansion over the 'squared solutions'
The 'squared solutions' are complete set of functions in the phase space [10]. This allows one to expand any function over the 'squared solutions'. Let us introduce the sets of 'squared solutions' where the subscripts 'c' and 'd' refer to the continuous and discrete spectrum of L. The 'squared solutions' in bold-face Ψ + α , . . . are obtained from Ψ + α , . . . by applying the projector P 0J , i.e. Ψ + α (x, λ) = P 0J Ψ + α (x, λ). Using the Wronskian relations one can derive the expansions over the 'squared solutions' of two important functions. Skipping the calculational details we formulate the results [10]. The expansion of Q(x) over the systems {Φ ± } and {Ψ ± } takes the form The next expansion is of ad −1 J δQ(x) over the systems {Φ ± } and {Ψ ± } The expansions (4.1), (4.2) is another way to establish the one-to-one correspondence between Q(x) and each of the minimal sets of scattering data T 1 and T 2 (4.6). Likewise the expansions (4.3), (4.4) establish the one-to-one correspondence between the variation of the potential δQ(x) and the variations of the scattering data δT 1 and δT 2 .
The expansions (4.3), (4.4) have a special particular case when one considers the class of variations of Q(x, t) due to the evolution in t. Then Assuming that δt is small and keeping only the first order terms in δt we get the expansions for

The generating operators
To complete the analogy between the standard Fourier transform and the expansions over the 'squared solutions' we need the analogs of the operator D 0 = −id/dx. The operator D 0 is the one for which e iλx is an eigenfunction: D 0 e iλx = λe iλx . Therefore it is natural to introduce the generating operators Λ ± through where the generating operators Λ ± are given by The rest of the squared solutions are not eigenfunctions of neither Λ + nor Λ − i.e.,Ψ + α;j (x) andΦ + α;j (x) are adjoint eigenfunctions of Λ + and Λ − . This means that λ ± j , j = 1, . . . , N are also the discrete eigenvalues of Λ ± but the corresponding eigenspaces of Λ ± have double the dimensions of the ones of L; now they are spanned by both Ψ ± ∓α;j (x) andΨ

The minimal sets of scattering data
Obviously, given the potential Q(x) one can solve the integral equations for the Jost solutions which determine them uniquely. The Jost solutions in turn determine uniquely the scattering matrix T (λ) and its inverseT (λ). But Q(x) contains r(r − 1) independent complex-valued functions of x. Thus it is natural to expect that at most r(r − 1) of the coefficients in T (λ) for λ ∈ R will be independent; the rest must be functions of those. The set of independent coefficients of T (λ) are known as the minimal set of scattering data.
The completeness relation for the 'squared solutions' ensure that there is one-to-one correspondence between the potential Q(x, t) and its expansion coefficients. Thus we may use as minimal sets of scattering data the following two sets where the reflection coefficients ρ ± (λ) and τ ± (λ) were introduced in equation (2.9), λ ± j are (simple) discrete eigenvalues of L and ρ ± j and τ ± j characterize the norming constants of the corresponding Jost solutions.
and vice versa. In particular from (4.9) there follows that a ± (λ) and c ± (λ) are time-independent and therefore can be considered as generating functionals of integrals of motion for the NLEE.
Let us, before going into the non-trivial reductions, briefly discuss the Hamiltonian formulations for the generic (i.e., non-reduced) MMKdV type equations. It is well known (see [10] and the numerous references therein) that the class of these equations is generated by the so-called recursion operator Λ = 1/2(Λ + + Λ − ) which is defined by equation (4.5).
If no additional reduction is imposed one can write each of the equations in (4.8) in Hamiltonian form. The corresponding Hamiltonian and symplectic form for the MMKV equation are given by ) The Hamiltonian can be identified as proportional to the fourth coefficient I 4 in the asymptotic expansion of A + (λ) (5.15) over the negative powers of λ This series of integrals of motion is known as the principal one. The first three of these integrals take the form We will remind also another important result, namely that the gradient of I k is expressed through Λ as Then the Hamiltonian equations written through Ω (0) and the Hamiltonian vector field X H (0) in the form for H (0) given by (4.10) coincides with the MMKdV equation.
An alternative way to formulate Hamiltonian equations of motion is to introduce along with the Hamiltonian the Poisson brackets on the phase space M which is the space of smooth functions taking values in g (0) and vanishing fast enough for x → ±∞, see (5.1). These brackets can be introduced by Then the Hamiltonian equations of motions with the above choice for H (0) again give the MMKdV equation. Along with this standard Hamiltonian formulation there exist a whole hierarchy of them. This is a special property of the integrable NLEE. The hierarchy is generated again by the recursion operator and has the form Of course there is also a hierarchy of Poisson brackets For a fixed value of m the Poisson bracket {·, ·} (m) is dual to the symplectic form Ω (m) in the sense that combined with a given Hamiltonian they produce the same equations of motion. Note that since Λ is an integro-differential operator in general it is not easy to evaluate explicitly its negative powers. Using this duality one can avoid the necessity to evaluate negative powers of Λ. Then the analogs of (4.11) and (4.12) take the form: where the hierarchy of Hamiltonians is given by: The equations (4.13) and (4.14) with the Hamiltonian H (m) given by (4.15) will produce the NLEE (4.8) with dispersion law f (λ) = k f k λ k for any value of m.
Remark 3. It is a separate issue to prove that the hierarchies of symplectic structures and Poisson brackets have all the necessary properties. This is done using the spectral decompositions of the recursion operators Λ ± which are known also as the expansions over the 'squared solutions' of L. We refer the reader to the review papers [26,10] where he/she can find the proof of the completeness relation for the 'squared solutions' along with the proof that any two of the symplectic forms introduced above are compatible.

The reduction group of Mikhailov
The reduction group G R is a finite group which preserves the Lax representation (1.1), i.e. it ensures that the reduction constraints are automatically compatible with the evolution. G R must have two realizations: i) G R ⊂ Aut g and ii) G R ⊂ Conf C, i.e. as conformal mappings of the complex λ-plane. To each g k ∈ G R we relate a reduction condition for the Lax pair as follows [5] C k (L(Γ k (λ))) = η k L(λ), C k (M (Γ k (λ))) = η k M (λ), (4.16) where C k ∈ Aut g and Γ k (λ) ∈ Conf C are the images of g k and η k = 1 or −1 depending on the choice of C k . Since G R is a finite group then for each g k there exist an integer N k such that g N k k = 1 1.

Finite order reductions of MMKdV equations
In order that the potential Q(x, t) be relevant for a DIII type symmetric space it must be of the form In the next two subsections we display new reductions of the MMKdV equations.

Class A Reductions preserving J
The class A reductions can be applied also to the MMKdV type equations. The corresponding automorphisms C preserve J, i.e. C −1 JC = J and are of the form where J is an element of the Cartan subalgebra dual to the vector e 1 + e 2 + e 3 + e 4 . In the typical representation of so(8) U (x, λ) takes the form Remark 4. The automorphisms that satisfy C −1 JC = J naturally preserve the eigensubspaces of ad J ; in other words their action on the root space maps the subsets of roots ∆ ± 1 onto themselves: C∆ ± 1 = ∆ ± 1 .
We list here several inequivalent reductions of the Zakharov-Shabat system. In the first one we choose C = C 0 to be an element of the Cartan subgroup where s k take the values 0 and 1. This condition means that C 2 0 = 1 1, so this will be a Z 2reduction, or involution. Then the first example of Z 2 -reduction is or in components Obviously ǫ j takes values ±1 depending on whether s j equals 0 or 1.
The next examples of Z 2 -reduction correspond to several choices of C as elements of the Weyl group eventually combined with the Cartan subgroup element C 0 where S e i −e j is the Weyl reflection related to the root e i − e j . Again we have a Z 2 -reduction, or an involution Written in components it takes the form (ǫ 12 = ǫ 34 = 1) p 12 = −q * 12 , p 24 = −ǫ 23 q * 13 , p 23 = −ǫ 13 q * 14 , p 14 = −ǫ 13 q * 23 , p 13 = −ǫ 23 q * 24 , p 34 = −q * 34 .
As a consequence we get , Next we consider a Z 3 -reduction generated by C 3 = S e 1 −e 2 S e 2 −e 3 which also maps J into J. It splits each of the sets ∆ ± 1 into two orbits which are In order to be more efficient we make use of the following basis in g (0) where ω = exp(2πi/3) and α takes values e 1 + e 2 and e 1 + e 4 . Obviously In addition, since ω * = ω −1 we get (E for k = 1, 2. Then we introduce the potential In view of equation ( Similarly the reduction (4.18) leads to , i.e. in this case we get two decoupled mKdV equations.
The Again we make use of a convenient basis in g (0) where α takes values e 1 + e 2 and e 1 + e 3 . Obviously and in addition, (E for k = 1, 2, 3. Then we introduce the potential In view of equation (5.5) the reduction condition (4.17) leads to the following relations between the coefficients for k = 1, 2, 3. Here p α , q α coincide with p 12 , q 12 (resp. p 13 , q 13 ) for α = e 1 +e 2 (resp. α = e 1 +e 3 ). Analogously the reduction (4.18) gives for k = 1, 2, 3. Both conditions (5.6), (5.7) lead to p 12 . We provide below slightly more general formulae for the corresponding Hamiltonian and symplectic form which are obtained by imposing the constraints (5.6) or (5.7); again for simplicity of notations we skip the upper zeroes in q (0) ij and p (0) ij and replace q (2) ij and p (2) ij byq ij andp ij , Now we get two specially coupled mKdV-type equations. Now we get four specially coupled mKdV-type equations given by where we use for simplicity q 12 = q 0 ,q 12 = q 2 , p 12 = p 0 ,p 12 = p 2 and with second reduction (5.7) p 0 = q * 0 and p 2 = −q * 2 we have Obviously in the system (5.8) we can put both q 0 , q 2 real with the result The reduction (4.18) means that q

Class B Reductions mapping J into −J
The class B reductions of the Zakharov-Shabat system change the sign of J, i.e. C −1 JC = −J; therefore we must have also λ → −λ.
Remark 5. Note that ad J has three eigensubspaces W a , a = 0, ±1 corresponding to the eigenvalues 0 and ±2. The automorphisms that satisfy C −1 JC = −J naturally preserve the eigensubspace W 0 but map W −1 onto W 1 and vice versa. In other words their action on the root space maps the subset of roots ∆ + 1 onto ∆ − 1 and vice versa: Here we first consider Obviously the product of the the above 4 Weyl reflections will change the sign of J. Its effect on Q in components reads i.e. some of the components of Q become purely imaginary, others may become real depending on the choice of the signs ǫ j . There is no Z 3 reduction for the so(8) MMKdV that maps J into −J. So we go directly to the Z 4 -reduction generated by C 6 = S e 1 +e 2 S e 2 +e 3 S e 3 +e 4 which maps J into −J. Again we make use of a convenient basis in g (0) where α takes values e 1 + e 2 and e 1 + e 3 . Obviously where again (E for k = 1, 2, 3. Then we introduce the potential In view of equation (5.9) the reduction condition (4.17) leads to the following relations between the coefficients while the reduction (4.18) gives 12 . We provide below slightly more general formulae for the corresponding Hamiltonian and symplectic form which are obtained by imposing the constraints (5.10) or (5.11) and again for simplicity we skip the upper zeroes in q ij and p ij and replace q , Now we get two specially coupled mKdV-type equations.
The class B reductions also render all the symplectic forms and Hamiltonians in the hierarchy real-valued. They allow to render the corresponding systems of MMKdV equations into ones involving only real-valued fields. 'Half' of the Hamiltonian structures do not survive these reductions and become degenerate. This holds true for all symplectic forms Ω (2m) and integrals of motion I 2m with even indices. However the other 'half' of the hierarchy with Ω (2m+1) and integrals of motion I 2m+1 remains and provides Hamiltonian properties of the MMKdV. Now we get four specially coupled mKdV-type equations given by where we use for simplicity q 12 = q 0 ,q 12 = q 2 , p 12 = p 0 ,p 12 = p 2 and with second reduction (5.10) p 0 = q * 0 and p 2 = −q * 2 we have Obviously in the system (5.12) we can put both q 0 , q 2 real with the result The reduction (5.11) means that q The class B reductions also render all the symplectic forms and Hamiltonians in the hierarchy real-valued. They allow to render the corresponding systems of MMKdV equations into ones involving only real-valued fields. 'Half' of the Hamiltonian structures do not survive these reductions and become degenerate. This holds true for all symplectic forms Ω (2m) and integrals of motion I 2m with even indices. However the other 'half' of the hierarchy with Ω (2m+1) and integrals of motion I 2m+1 remains and provides Hamiltonian properties of the MMKdV.
Then Plemelj-Sokhotsky formulae allows us to recover A ± (λ) and C ± (λ) where A(λ) = A + (λ) for λ ∈ C + and A(λ) = −C − (λ) for λ ∈ C − . In deriving (5.16) we have also assumed that λ ± j are simple zeroes of A ± (λ) and C ± (λ). Let us consider a reduction condition (4.17) with C 1 from the Cartan subgroup: C 1 = diag (B + , B − ) where the diagonal matrices B ± are such that B 2 ± = 1 1. Then we get the following constraints on the sets T 1,2 From the general theory of RHP [24] one may conclude that (5.13), (5.14) allow unique solutions provided the number and types of the zeroes λ ± j are properly chosen. Thus we can outline a procedure which allows one to reconstruct not only T (λ) andT (λ) and the corresponding potential Q(x) from each of the sets T i , i = 1, 2: i) Given T 2 (resp. T 1 ) solve the RHP (5.13) (resp. (5.14)) and construct a ± (λ) and c ± (λ) for λ ∈ C ± .
ii) Given T 1 we determine b ± (λ) and d ± (λ) as iii) The potential Q(x) can be recovered from T 1 by solving the RHP (3.7) and using equation (3.16).
Another method for reconstructing Q(x) from T j uses the interpretation of the ISM as generalized Fourier transform, see [27,28,29].
Let in this subsection all automorphisms C i are of class A. Therefore acting on the root space they preserve the vector r k=1 e k which is dual to J, and as a consequence, the corresponding Weyl group elements map the subset of roots ∆ + 1 onto itself. Remark 7. An important consequence of this is that C i will map block-upper-triangular (resp. block-lower-triangular) matrices like in equation (3.2) into matrices with the same block structure. The block-diagonal matrices will be mapped again into block-diagonal ones. Using equation (5.17) and C i (J) = J one finds that: It remains to take into account that the reductions (4.17)-(4.20) for the potentials of L lead to the following constraints on the scattering matrix T (λ) These results along with Remark 7 lead to the following results for the generalized Gauss factors of T (λ) and

Effects of class B reductions on the scattering data
In this subsection all automorphisms C i are of class B. Therefore acting on the root space they map the vector r k=1 e k dual to J into − r k=1 e k . As a consequence, the corresponding Weyl group elements map the subset of roots ∆ + 1 onto ∆ − 1 ≡ −∆ + 1 . Remark 8. An important consequence of this is that C i will map block-upper-triangular into block-lower-triangular matrices like in equation (3.2) and vice versa. The block-diagonal matrices will be mapped again into block-diagonal ones.
The reductions (4.17)-(4.20) for the potentials of L lead to the following constraints on the scattering matrix T (λ): Then along with Remark 8 we find the following results for the generalized Gauss factors of T (λ) a) C 1 (S ±, † (−λ * )) =Ŝ ± (λ), 6 The classical r-matrix and the NLEE of MMKdV type One of the definitions of the classical r-matrix is based on the Lax representation for the corresponding NLEE. We will start from this definition, but before to state it will introduce the following notation which is an abbreviated record for the Poisson bracket between all matrix elements of U (x, λ) and U (y, µ) In particular, if U (x, λ) is of the form and the matrix elements of Q(x, t) satisfy canonical Poisson brackets The classical r-matrix can be defined through the relation [19] U (x, λ) ⊗ ′ U (y, µ) = i [r(λ − µ), U (x, λ) ⊗ 1 1 + 1 1 ⊗ U (y, µ)] δ(x − y), (6.1) which can be understood as a system of N 2 equation for the N 2 matrix elements of r(λ − µ). However, these relations must hold identically with respect to λ and µ, i.e., (6.1) is an overdetermined system of algebraic equations for the matrix elements of r. It is far from obvious whether such r(λ − µ) exists, still less obvious is that it depends only on the difference λ − µ. In other words far from any choice for U (x, λ) and for the Poisson brackets between its matrix elements allow r-matrix description. Our system (6.1) allows an r-matrix given by The matrix P possesses the following special properties [P, X ⊗ 1 1 + 1 1 ⊗ X] = 0 ∀ X ∈ g.
By using these properties of P we are getting i.e., the r.h.s. of (6.1) does not contain Q(x, t). Besides: where we used the commutation relations between the elements of the Cartan-Weyl basis. The comparison between (6.3), (6.4) and (6.1) leads us to the result, that r(λ − µ) (6.2) indeed satisfies the definition (6.1).

Remark 9.
It is easy to prove that equation ( Let us now show, that the classical r-matrix is a very effective tool for calculating the Poisson brackets between the matrix elements of T (λ). It will be more convenient here to consider periodic boundary conditions on the interval [−L, L], i.e. Q(x − L) = Q(x + L) and to use the fundamental solution T (x, y, λ) defined by i dT (x, y, λ) dx + U (x, λ)T (x, y, λ) = 0, T (x, x, λ) = 1 1.
Skipping the details we just formulate the following relation for the Poisson brackets between the matrix elements of T (x, y, λ) [19] T (x, y, λ) ⊗ ′ T (x, y, µ) = [r(λ − µ), T (x, y, λ) ⊗ T (x, y, µ)] . (6.5) The corresponding monodromy matrix T L (λ) describes the transition from −L to L and T L (λ) = T (−L, L, λ). The Poisson brackets between the matrix elements of T L (λ) follow directly from equation (6.5) and are given by An important property of the integrals I L,k and J L,k is their locality, i.e. their densities depend only on Q and its x-derivatives.
The simplest consequence of the relation (6.5) is the involutivity of I L,k , J L,k . Indeed, taking the trace of both sides of (6.5) shows that {tr T L (λ), tr T L (µ)} = 0. We can also multiply both sides of (6.5) by C ⊗ C and then take the trace using equation (6.3); this proves {tr T L (λ)C, tr T L (µ)C} = 0.
In particular, for C = 1 1 + J and C = 1 1 − J we get the involutivity of tr a + L (λ), tr a + L (µ) = 0, tr a − L (λ), tr a − L (µ) = 0, tr c + L (λ), tr c + L (µ) = 0, tr c − L (λ), tr c − L (µ) = 0. Equation (6.5) was derived for the typical representation V (1) of G ≃ SO(2r), but it holds true also for any other finite-dimensional representation of G. Let us denote by V (k) ≃ ∧ k V (1) the k-th fundamental representation of G; then the element T L (λ) will be represented in V (k) by ∧ k T L (λ) -the k-th wedge power of T L (λ), see [12]. In particular, if we consider equation (6.5) in the representation V (n) and sandwich it between the highest and lowest weight vectors in V (n) we get [30] {det a + L (λ), det a + L (µ)} = 0, Somewhat more general analysis along this lines allows one to see that only the eigenvalues of a ± L (λ) and c ± L (λ) produce integrals of motion in involution. Taking the limit L → ∞ we are able to transfer these results also for the case of potentials with zero boundary conditions. Indeed, let us multiply (6.5) by E(y, λ) ⊗ E(y, µ) on the right and by E −1 (x, λ) ⊗ E −1 (x, µ) on the left, where E(x, λ) = exp(−iλJx) and take the limit for x → ∞, y → −∞. Since lim x→±∞ e ix(λ−µ) λ − µ = ±iπδ(λ − µ), we get where Π 0J is defined by Analogously we prove that i) the integrals I k = lim

Conclusions
We showed that the interpretation of the ISM as a generalized Fourier transform holds true also for the generalized Zakharov-Shabat systems related to the symmetric spaces DIII. The expansions over the 'squared solutions' are natural tool to derive the fundamental properties not only of the MNLS type equations, but also of the NLEE with generic dispersion laws, in particular MMKdV equations. Some of these equations, besides the intriguing properties as dynamical systems allowing for boomerons, trappons etc., may also have interesting physical applications.
Another interesting area for further investigations is to study and classify the reductions of these NLEE. For results along this line for the MNLS equations see the reports [10] and [9]; reductions of other types of NLEE have been considered in [26,7,8,6].
One can also treat generalized Zakharov-Shabat systems related to other symmetric spaces. The expansions over the 'squared solutions' can be closely related to the graded Lie algebras, and to the reduction group and provide an effective tool to derive and analyze new soliton equations. For more details and further reading see [11,26].
In conclusion, we have considered the reduced multicomponent MKdV equations associated with the DIII-type symmetric spaces using A.V. Mikhailov reduction group. Several examples of such nontrivial reductions leading to new MMKdV systems related to the so(8) Lie algebra are given. In particular we provide examples with reduction groups isomorphic to Z 2 , Z 3 , Z 4 and derive their effects on the scattering matrix, the minimal sets of scattering data and on the hierarchy of Hamiltonian structures. These results can be generalized also for other types of symmetric spaces.