1. Introduction

We are going to study the auxiliary linear problem

and the theory of expansions over the adjoint solutions related to it. Above ε = ± 1 and * stands for complex conjugation. The complex valued functions u and v (the potentials) are assumed to be smooth for x∈ℝ and they satisfy the relation:

The potentials must also satisfy some asymptotic conditions when x → ±∞ to be discussed further in text.

The system (1.1) for ε = 1 was considered by Gerdjikov, Mikhailov and Valchev [6] with the sign + before S. It is a particular case of a vector system introduced independently by Golubchik and Sokolov [12]. For reasons that will become clear later we shall prefer this equivalent form (one just needs to change (u, v) to (−u, −v)).

Let us adopt the following convention: if ε appears in formulas then it will be + 1 or − 1, if it is used as an index or label it will mean either + or −. Thus we shall denote the original Gerdjikov-Mikhailov−Valchev system by GMV₊ while the general case will be referred to as GMV_ε system.

According to [6, 7] the GMV₊ system arises naturally when one looks for integrable system having a Lax representation [L, A] = 0 with L and A subject to Mikhailov-type reduction requirements, see [16–18]. This is also true for both GMV_± systems. Indeed, the Mikhailov reduction group G₀ acting on the fundamental solutions of the system (1.1) could be defined as generated by the two elements g₁ and g₂ acting in the following way:

where † denotes Hermitian conjugation. Since g₁ g₂ = g₂ g₁ and g12=g22=id , G0=ℤ2×ℤ2 . We shall call the reduction defined by g₁, g₂ for ε = 1 Hermitian while the reduction defined by g₁, g₂ for ε = −1 pseudo-Hermitian. The requirement that G₀ is a reduction group immediately gives that the coefficients of the operators L˜ and must satisfy:

One may consider a more general form of pseudo-Hermitian reduction, i.e. one with Qε1ε2=diag(1,ε1,ε2),ε12=ε22=1 . However, it is easily checked this does not give anything new compared to the pseudo-Hermitian reduction under consideration here.

As it can be checked the matrix S has constant eigenvalues. We have g⁻¹Sg = J₀, where g is of the form:

In case ε = + 1 the matrix g belongs to the group SU(3) (g^† = g⁻¹) and when ε = −1 to the group SU(2,1) (Q ₋ g^†Q ₋ = g⁻¹). Further on we shall use the general notation SU(ε) referring to both cases, i.e. it implies SU(ε) ≡ S U(3) when ε = 1 and SU(ε) ≡ SU(2, 1) when ε = −1.

Since g(x) ∈ SU(ε), the values of S(x) will be in the orbit 𝒪J0(SU(ε)) of J₀ with respect to SU(ε) (it is a submanifold of isu (ε)). Thus S(x)∈𝒪J0(SU(ε))∩𝔤1 where 𝔤1 is the space of the matrices X in sl (3,ℂ) such that HXH = − X, see (2.6) for the reason for this notation. Let us also note that conversely, if we assume that S(x)∈𝒪J0(SU(ε))∩𝔤1 then on the first place S has the form as in (1.1). Next, as easily checked, the eigenvalues of the matrix S are

But since they coincide with 0,±1 we must have ε|u|² + |v|² = 1.

Our approach to the GMV_± system will be based on the fact that it is gauge equivalent to a generalized Zakharov-Shabat auxiliary systems (GZS systems) on the algebra sl (3,ℂ) , see Sec. 3. Generalized Zakharov-Shabat systems, called Caudrey-Beals-Coifman (CBC) systems [2] when J is complex, are probably the best known auxiliary linear problems. In their most general form they are written on an arbitrary fixed simple Lie algebra 𝔤 in some finite dimensional irreducible representation and have the form:

Here, q(x) and J belong to 𝔤 in some fixed representation, ψ belongs to the corresponding group. The element J must be such that the kernel of ad_J (adJ(X)≡[J,X],X∈𝔤) is a Cartan subalgebra 𝔥J of 𝔤 . The potential q(x) belongs to the orthogonal complement 𝔥J⊥ of 𝔥 with respect to the Killing form:

It is assumed that q(x) is sufficiently smooth and it converges to 0 fast enough when x → ±∞. The system (1.7) is called GZS (CBC) system over 𝔤 in canonical gauge. When the algebra is understood from the context and as a representation is chosen the canonical one it is just called GZS (CBC) system. The system (1.7) is gauge equivalent to

Usually, it is assumed that lim_{x → ±∞}S(x) = J where the convergence is sufficiently fast. Here, 𝒪J stands for the orbit of J under the adjoint action of the group G corresponding to 𝔤:𝒪J={X˜:X˜=gJg−1,g∈G} . The concept of gauge transformation, gauge equivalent auxiliary problems and gauge equivalent soliton equations originates from the famous work [27] where it has been employed to solve an equation that is a classical analogue of equations describing waves in magnetic chains (spin 1 / 2). It has been shown that one of the nonlinear evolution equations (NLEEs) related to L˜ is the Heisenberg ferromagnet equation

being gauge equivalent to the famous nonlinear Schrödinger equation, see [4] for an extensive discussion on that issue. It should be mentioned that the soliton equations solvable through the auxiliary linear problem L˜ψ˜=0 in the case sl (3,ℂ) is related to a classical analog of the equation describing spin 1 particle chains dynamics, see [3].

Later, the results of [27] were generalized to the soliton equations hierarchies associated with L and L˜ , the conservation laws of those NLEEs, the hierarchies of their Hamiltonian structures etc. This was achieved by generalizing the so-called AKNS approach [1] (generating operators or recursion operators approach). Initially, this was done in the sl (2,ℂ) case, next in the case of GZS system on arbitrary semisimple Lie algebra [5, 9, 19]. Now, this theory is referred to as the gauge-covariant theory of the recursion operators related to the GZS (CBC) systems in canonical and pole gauge. For a detailed explanations of all these issues and more references (prior to 2008), see the monograph [8].

For GZS system in pole gauge most of the essential issues could be reformulated from the canonical gauge. Perhaps the main difficulty is to express explicitly all quantities depending on q and its derivative through S and its derivatives. While there is a clear procedure how to achieve that goal, in each particular case the details could be different. The procedure has been developed in detail in the PhD thesis [19]. In the case of sl (3,ℂ) in general position (with no reductions) it has been carried out in [20].

Regarding now the system GMV_±, it was the Hermitian case, i.e. GMV₊, that was mainly considered so far. The first paper in which GMV₊ was studied in detail for the asymptotic conditions lim_{x → ±∞} u = 0, lim_{x → ±∞} v = expiΦ_± was [7]. It contained discussion of the spectral properties of the GMV₊ system, two operators whose product play the role of a recursion operator were presented and expansions over the so-called adjoint solutions were derived. We shall also discuss these issues, however, we want to stress on the following: a) We shall be dealing with both GMV_± systems simultaneously; b) Our approach will be completely different, based on the gauge equivalence we mentioned above. Consequently, we shall be able to consider general asymptotic conditions constant limits lim_{x → ±∞} u and lim_{x → ±∞} v; c) Our point of view on the recursion operators when reductions are present is somewhat different from that adopted in [7].

2. Algebraic preliminaries

In order to proceed further, we shall need some information about the algebra sl (3,ℂ) and the involutions we are going to use which we introduce below. All facts from the theory of the semisimple Lie algebras we use are classical, see [13]. We also use the normalizations adopted in this monograph and most of its notation.

The Lie algebra sl (3,ℂ) is a simple Lie algebra of rank 2. We shall denote its Killing form tr (ad_Xad_Y) by ⟨X, Y⟩ where as usual ad_X(Y) ≔ [X, Y]. It is known that tr (ad_Xad_Y) = 6 tr (XY) which simplifies considerably the calculations. A Cartan subalgebra could be introduced using any regular element X∈sl (3,ℂ) and constructing the space 𝔥X=keradX . As it is known, the subalgebra of the diagonal matrices is a canonical choice for Cartan subalgebra. It is also equal to 𝔥=keradJ where J is any diagonal matrix diag (λ₁, λ₂, λ₃) with distinct λ_i. In that case, we shall call 𝔥 the Cartan subalgebra. For the canonical choice of the Cartan subalgebra the system of roots Δ for sl (3,ℂ) is

where ε_i are functionals acting on 𝔥 in the following way: ε_i(diag (h₁, h₂, h₃)) = h_i. Then the set of positive roots Δ₊ consists of the elements α_i:

The corresponding root vectors E_α, α ∈ Δ together with the matrices Hα1,Hα2 written below:

form the Cartan-Weil basis of sl (3,ℂ) associated with the Cartan subalgebra 𝔥 . Here as usual e_ij means a matrix whose only nonzero entry equal to one is located in the intersection of the i-th row and j-th column. The matrices Hα1,Hα2 span 𝔥 and the matrices E_α span 𝔥⊥ .

The relations (1.5) have the following Lie algebraic meaning. First, the map h: X ↦ HXH = HXH⁻¹ is involutive automorphism of sl (3,ℂ). Next, Q_ε defines a complex conjugation σ_ε of sl (3,ℂ) :

The complex conjugation σ_ε defines the real form su (3) (ε = + 1) or the real form su (2,1) (ε = − 1) of sl (3,ℂ) . In order to treat both cases simultaneously we shall adopt the notation su (ε) meaning su(3) when ε = + 1 and su (2,1) when ε = − 1. Note that the complex conjugation σ_ε commutes with the automorphism h. Now, let us introduce the spaces:

Then we shall have the splittings

The space 𝔤1 consists of all off-diagonal matrices X = (x_ij) for which x₂₃ = x₃₂ = 0 and 𝔤0 consists of all traceless matrices X = (x_ij) for which x₁₂ = x₂₁ = x₁₃ = x₃₁ = 0. Of course, S involved in the GMV_± satisfies S∈𝔤1 . Also, since h is automorphism, the spaces 𝔤0 and 𝔤1 are orthogonal with respect to the Killing form.

Below we shall see that h we already introduced (X ↦ h(X) = HXH) is closely related to the automorphism X ↦ k(X) = KXK where

So we shall need how h and k act on the Cartan-Weil basis. Note that similar to h the automorphism k is also involutive, that is k² = id. For h we get that

while for k we have

The above formulas become simpler if we introduce the action 𝒦 of k on the roots:

We observe that 𝒦 maps positive roots into negative and vice-versa. Of course, 𝒦 just as k satisfies 𝒦2=id . There is no need to introduce the action of h on the roots since this is simply the identity. With the above notation we have

where r(α) = 1 if α = ± α₁, ± α₃ and r(α) = −1 if α = ± α₂. Note that r(α)r(−α) = 1.

The invariance under the group generated by g₁, g₂ means that if ψ is the common G₀-invariant fundamental solution of the linear problem

and the linear problem of the type: we must have S∈𝔤1∩isu (ε) and

The conditions on A˜k and S coincide with (1.5), in particular, we have that h(S) = −S and σ_ε(S) = −S. This forces S to be in the form we introduced in (1.1). Also, as a direct consequence from the last relations we obtain that

Consequently, the spaces keradS=𝔥S (which obviously is a Cartan subalgebra) and its orthogonal complement 𝔥S⊥ are invariant under h. Each of them splits into two h-eigenspaces (for the eigenvalues +1 and −1 respectively):

Several consequences follow from the above:

1. (1)

   ad_S and adS−1 interchange the spaces 𝔣0 and 𝔣1 :

   adS𝔣0=𝔣1, adS𝔣1=𝔣0,adS−1𝔣0=𝔣1, adS−1𝔣1=𝔣0. (2.17)

2. (2)

   Since S∈𝔤1,S1=S2−2361∈𝔤0 , the spaces 𝔥0,𝔥1 are 1-dimensional and are spanned by S and S1=S2−231 respectively.

3. (3)

   From the previous item follows that

   Sx∈𝔤1, (S1)x=(S2)x∈𝔤0 (2.18)

   and since S_x and (S₁)_x are orthogonal to 𝔥S we have

   Sx∈𝔣1, S1x≡(S1)x=(S2)x∈𝔣0. (2.19)

Another issue we must discuss is the relation between h from one side and adS−1 and π_S from the other. Here π_S is the orthogonal projection on the space 𝔥S⊥ with respect to the Killing form and adS−1 is defined only on the space 𝔥S⊥ . As one can show, see [7, 22], adS−1 could be expressed as a polynomial p₃(ad_S) in ad_S where only odd degrees of adS are present ( adL1−1 through adL1 in the notation of [7]). From the other side, this means that π_S could be expressed as a polynomial in ad_S where enter only the even powers of ad_S. Consequently, one obtains that

and in the same way

3. The GMV_± system and its gauge equivalent

It has been pointed out [7] that the GMV₊ system is gauge equivalent to a GZS type system. For the system GMV_± the situation is the same. After the gauge transformation L˜↦L′=g−1(x)L˜g(x) with g(x) given in (1.6) we obtain the system

where q′ = ig⁻¹ g_x. After expressing g with u and v, we obtain

As it should be for GZS system J₀ is a real diagonal matrix and its diagonal elements are ordered to decrease going down along the main diagonal. This is in fact our motivation when we choose the sign in front of S. As one could see, in the above q′ is not off-diagonal as GZS requires and in [7] authors just stopped here. It turns out, however, that the issue could be easily addressed making one more gauge transformation. Indeed, due to the condition ε|u|² + |v|² = 1 the diagonal part of q′ equals i2(εuux*+vvx*)J′=b(x)J′ where J′ = diag (1, −2, 1). Moreover, the diagonal part is real, that is b(x)=i2(εuux*+vvx*) is real. Then the gauge transformation

maps the spectral problem into where

One can easily see that q is off-diagonal and since the entries of q′ decay when x → ±∞ the entries of q also decay. Now, since the diagonal elements of J₀ are distinct we conclude that GMV_± is a problem of GZS type in pole gauge on sl (3,ℂ) . We shall exploit this fact but we want to address first some other issues. The first one is whether we can be more precise about the potential q in the GZS system (3.4) to which GMV_± is equivalent. With direct calculation we are able to establish that the potential q in (3.4) has special properties, namely Q_εq^†Q_ε = q and KqK = q where K has been introduced in (2.7). We also notice that KJ₀K = − J₀ and QεJ0†Qε=J0 .

The next issue is about the asymptotic conditions S must satisfy. Passing from GZS system in canonical gauge (3.4) to GZS in pole gauge (1.9) usually involves the Jost solution ψ′0 of (3.4) such that limx→−∞ψ′0(x)=1 at λ = 0. The gauge transformation is then ψ→ψ˜=ψ0′−1ψ . In that case we obtain S=ψ′0−1J0ψ′0 and limx→−∞S(x)=J0. However, this is not our case and such a condition is not compatible with the reductions. Natural for the GMV_± system are conditions when u, v tend to some constant values. In this work we shall assume that limx→±∞u(x)=u±,limx→±∞v(x)=v± . In that case the limit of g when x → ±∞ is equal to

Now we notice the following. If we come back to how q was constructed, we shall see that the function

(for the definitions of g and b see the explanations after (3.2)) is as a matter of fact a solution to the system (3.4) for λ = 0. It satisfies: where

The properties of the fundamental solutions of the auxiliary systems play a paramount role in the spectral theory of such systems and for the theory of integration of the NLEEs associated with them. Since for the GZS systems with ℤp reductions these questions have been studied in detail, see [11] (whether or not K is Coxeter automorphism does not make any difference here) the above theorem opens the possibility to study the GMV_± system using its gauge-equivalent GZS system with ℤ2×ℤ2 reduction. In [11], it was considered much more complicated case of a CBC system and the automorphism has order p > 1 (in our case p = 2 since K² = 1). Then the complex plane is divided into p sectors by straight lines through the origin and in each sector there is a fundamental analytic solution to the corresponding CBC system. In our case the things are much simpler: we have the real line dividing ℂ into upper and lower half-plane. Then the system (3.12) possesses fundamental analytic solutions (FAS) χ^±(x, λ) of the form m^±(x, λ) exp (−i λxJ₀). The functions m⁺ (x, λ) (m⁻(x, λ)) are meromorphic in λ in the upper (lower) half plane ℂ+(ℂ−) . Naturally, χ^±(x, λ) have the same analytic properties in λ as m^±(x, λ). The points of ℂ that belong to the discrete spectrum are those at which m^±(x, λ) and their inverse have singularities with respect to λ in ℂ+(ℂ−) . It is usually considered the case when these singularities are poles, see for example [10] in the general case and [11] for the case of reductions. There is no difficulty to include the discrete spectrum for the GMV_± in our considerations but this will make all the formulas much more complicated so we shall do it elsewhere. Here we intend to explain mainly how our approach works and most of the issues are algebraic. Thus, in what follows we shall assume that there is no discrete spectrum for the GZS system gauge equivalent GMV_± and consequently no discrete spectrum for GMV_±.

Continuing with the properties of m^±(x, λ), if the potential q(x) has integrable derivatives up to the n-th order then

when |λ| → ∞ and λ remains in ℂ+ ( ℂ− respectively). The asymptotic is uniform in x∈ℝ , and the coefficients a_i(x) could be calculated through q and its x-derivatives. In particular, for absolutely integrable q we have lim|λ|→∞m±(x,λ)=1 . The functions m^± allow extensions by continuity to the real line, these extensions will be denoted by the same letters. In addition, m^± satisfy:

Since KJ₀K = −J₀, the solutions χ^± also allow extensions by continuity to the real line (denoted again by the same letters) and satisfy:

It is simple to see that in our case χ^± will also satisfy:

We build from χ^±(x, λ) fundamental analytic solutions χ˜±(x,λ) of the system (2.12) by setting:

Now we have:

4. Recursion operators for the GMV_± system

The recursion operators (Λ-operators) are theoretical tools that permit to describe [8]:

• •

  The hierarchies of the NLEEs related to the auxiliary linear problem.

• •

  The hierarchies of conservation laws for these NLEEs.

• •

  The hierarchies of compatible Hamiltonian structures of these NLEEs.

The issue about the Hamiltonian structures is closely related to a beautiful interpretation of the recursion operators originating from the work [15]. In fact, the adjoint of the recursion operators could be interpreted as Nijenhuis tensors on the infinite dimensional manifold of ‘potentials’ (the functions S(x)), see [8].

Recursion operators arise naturally when one tries to find the hierarchy of Lax pairs related to a particular auxiliary linear problem [28]. So suppose we want to find the NLEEs having Lax representation [L˜,A˜]=0 with L˜=i∂x−λS as in (2.12) and

The first thing we notice is that we must have i∂xA˜n∈𝔥S⊥ and A˜0 = const in x. Using a gauge transformation depending only on t one can achieve A˜0=0 . Then one can show that the coefficients A˜k for k = 1, 2,…n−1 are calculated recursively. In order to see it we first recall that the elements S and S1=S2−231 span 𝔥S=keradS and that

As before, denote by π_S the orthogonal projection (with respect to the Killing form) onto the space 𝔥S⊥ . Then the orthogonal projection onto 𝔥S will be 1−π_S and for X∈sl (3,ℂ) we shall have

Putting

we obtain that the coefficients A˜k for k = 1, 2,…n − 1 are obtained recursively in the following way: where and for the sake of brevity we write here and below S1x instead of (S1)x .

The operators Λ˜± are recursion (generating) operators for GMV_± system. As one can see, they do not depend on the second reduction, that is on the choice of the real form. In fact, the recursion operators for the sl (3,ℂ) -GZS system in general position were already known both in canonical and pole gauge [20, 21] and one can obtain Λ˜ for GMV from those as it was described in [22]. The expressions for the recursion operators for sl (n,ℂ) -CBC in pole gauge is also known [23]. From it one also could easily find Λ˜± . Let us note that in [7] we already mentioned what is called recursion operator is in fact Λ˜±2 and it has been introduced as a product of two operators which differ from Λ˜±. This complicates the picture (we shall discuss that a little further).

Let us continue with the calculation of the coefficients A˜k. For the coefficient A˜n−1 one has that i∂xA˜n=[S,A˜n−1] and since A˜n∈𝔥S there are scalar functions α, β such that A˜n=αS+βS1 . This gives

and therefore α and β are constants. Thus and the hierarchy of NLEEs related to L˜ in general position is

In the case of GMV_ε system A˜n−1∈isu (ε) so α and β must be real. Next, if A˜n∈𝔥1 and n = 1(mod2) because of the (4.5) and (4.6) we shall have automatically

If A˜n∈𝔥0 and n = 0(mod2) we have again (4.9). Thus the Mikhailov type reductions are compatible with the action of recursion operator and the general form of the equations related to the GMV_ε system is:

where a_i are some real constants. This is the hierarchy found in [7] although it was presented there in a different form.

Let us denote the space of rapidly decreasing functions Z˜(x) with values in 𝔣0(𝔣1) by 𝔣0[x](𝔣1[x]) . Since 𝔣0 and 𝔣1 are orthogonal with respect to the Killing form and taking into account Remark 2.1 we have

Now, let us do the following:

• •

  Put in the above expressions S = −L₁ (in order to express everything through L₁).

• •

  Take into account that what is called Killing form in [7] and denoted ⟨X, Y⟩ is not the canonical definition of the Killing form, it is in fact tr (XY) and as we know in our notation ⟨X, Y⟩ = 6 tr (XY).

• •

  Take into account that because Z˜ is orthogonal to 𝔥S we have 〈Z˜(x),Sx〉=−〈Z˜x,S(x)〉 and 〈Z˜(x),S1x〉=−〈Z˜x,S1(x)〉 .

Then one could convince himself that the operators Λ1±,Λ2± introduced in [7] are defined by the expressions standing in the right hand sides of (4.11) and (4.12), see [22]. In other words, Λ1±,Λ2± are restrictions of the operators Λ˜± on the spaces 𝔣0[x] and 𝔣1[x] respectively. If the hierarchy of the soliton equations (4.10) is written in terms of Λ1±,Λ2± it acquires the form that was presented in [7].

5. Completeness relations for the GMV_± system

We have seen that the consideration of the NLEEs hierarchy (4.10) shows that the operator ‘moving’ the equations along the hierarchy is Λ˜±2 and it does not depend on the real form. The geometric interpretation of the hierarchies and their conservation laws which is developed for the case of GMV₊ [24] also suggests that the appropriate operator we must consider is Λ˜±2 . It remains to consider the third important aspect of the recursion operators – their relations to the so-called expansions over the adjoint solutions. In that approach one considers these expansions and then finds operators, for which the adjoint solutions entering into them are eigenfunctions. The importance of the adjoint solutions is related to the fact that they form complete systems, so roughly speaking, one can expand S and its variation over them and then have the famous interpretation of the inverse scattering method as a generalized Fourier transform, see [8, 14]. Recently, the theory of the expansions over adjoint solutions has been extended to the case when there are reductions. In [11] the theory for the CBC systems with ℤp reductions was presented in full taking into account the discrete spectrum while in [25] is discussed also the gauge-covariant formulation of the expansions with ℤp reductions. The geometric aspects of the NLEEs associated with CBC systems in canonical and pole gauge with reductions were presented recently in [26].

Our approach will be the following. We start from the spectral theory of the generating operators for (3.12) (a GZS system in canonical gauge) which is very well known and from it obtain the theory of the generating operators for GMV_± (a GZS system in pole gauge). So let us sketch the theory for GZS in canonical gauge, for all the details see [8, 20]. The adjoint solutions for the system GZS system (3.12) are defined as follows:

where π₀ is the orthogonal projector on 𝔥⊥ where 𝔥 is the Cartan subalgebra of sl (3,ℂ) and χ^ stands for the matrix inverse to χ∈SL(3,ℂ) . The generating operators Λ_± then have the form:

As mentioned, the above formula together with its derivation could be found in many sources, see for example [5, 8, 10, 11]. From the asymptotic behaviour of the solutions χ^± follows that for α > 0 we have

(It is of no importance whether or not we have some reductions). Let us define now the adjoint solutions for the GMV_± in the following way:

Let us recall now that we have a reduction defined by the automorphism h. Taking into account that h commutes with π_S (see Proposition (2.1)) and the properties of ψ₀ (see (3.11)) we see that h(e˜α±)(x,λ) equals

so we have:

One proves easily that πS=Ad(ψ0−1)∘π0∘Ad(ψ0) , and then one sees that for α ∈ Δ we have that e˜α±(x,λ)=Ad(ψ0−1)eα±(x,λ) . Here Ad (r) denotes the adjoint action of the group on its algebra, that is Ad (r)X = rXr⁻¹. All the theory then develops in a similar way as in the case when we make a gauge transformation with a Jost solution which is very well known [10]. For example, one has that

As a consequence, for α ∈ Δ₊ we have

We are now ready to discuss the completeness of the adjoint solutions for the GZS system in pole gauge (which under reductions becomes our GMV_± problem). In order to simplify the formulas we shall have further, let us adopt the following notation: for the functions X(x),Y(x):ℝ→sl (3,ℂ) we put

First, we write the completeness relations for the ‘adjoint ‘ solutions of the canonical gauge GZS system (details can be found in [5, 8]). We also remind that we disregard the discrete spectrum. Then for the GZS system in canonical gauge (3.12) the following theorem holds:

Let us make some final comments about the expansions we obtained. In the case GMV₊ the families s˜±αη(x,λ) and a˜±αη(x,λ) as well as the relations (5.19), (5.20) were introduced in [7]. However, they were written in terms of the restrictions of Λ˜± on the spaces 𝔣0[x](𝔣1[x]) , namely through the operators Λ1±,Λ2± we mentioned already. This complicates their form and obscures their meaning. We believe that the form in which we cast them now and in relation to Lemma 5.1 is are much easier to understand. As about the expansions (5.15), (5.16) and about expansions in Theorem 5.3 they were not presented until now in their general form but only for the specific cases of expansions of the functions S_x and (S₁)_x. Moreover, since in [7] the theory of the GMV₊ have been developed not in parallel (though by analogy) to the corresponding system in canonical gauge, the knowledge about the GZS system in canonical gauge could not be exploited in full and everything ought to be developed from the beginning. In particular, there have been considered more restricted boundary conditions than necessary, namely the case u_± = 0, v_± = exp (iΦ_±). Consequently, all the results were obtained in less generality and with more effort.

The above expansions have big theoretical interest. They show that similar to the general position case (without reductions) the ’potential’ and its variation could be expanded over a family of functions that are eigenfunctions of the operator that appears when one recursively solves the equations for the coefficients A˜k . Of course, it is important also to know that when we have reductions the expansions could be written in a different form. Practically however, it is easier to adopt the following attitude. We could simply ’forget’ that we have reductions. In other words, the whole Gauge Covariant Theory of the Generating Operators will proceed in the usual way, with or without reductions. Basically, all the formulas will remain true, one must simply take into account the following:

• •

  Some additional symmetries of the scattering data (see the next Section) which lead to symmetries in the corresponding Riemann-Hilbert problem when one uses it for finding exact solutions.

• •

  The operators that permit to ‘move’ along the hierarchies are Λ˜±2 .

• •

  Some of the conservation laws and Poisson structures in the corresponding hierarchies trivialize.

For the first of these items one can see [7], as for the second and the third, according to the general theory (see [8] or [20] specifically for the case of sl (3,ℂ)) we have:

• •

  The NLEEs (4.10) have the following series of conservation laws:

  DB(s)=∫−∞∞[∫−∞y〈Sy,Λ˜±s(adS−1B˜y)〉dy]dx, B∈𝔥, B=const, B˜=gBg−1 (5.35)

  where s = 1,2,… Now, if k(B) = − B, then adS−1B˜x∈𝔥0 and if s is even Λ˜±s(B˜x)∈𝔥0[x] . Since Sy∈𝔥1[x] we have DB(s)=0 . This of course does not happen if s is odd. Analogously, DB(s)=0 if s is odd and k(B) = B. Thus indeed some of the conservation laws trivialize.

• •

  The NLEEs (4.10) are Hamiltonian with respect to the hierarchy of symplectic forms:

  Ω˜(p)(δS1,δS2)=∫−∞∞〈δS1,Λ˜padS−1δS2〉dx, p=0,1,2,… (5.36)

  where δ₁S, δ₂S are some variations of S and

  Λ˜=12(Λ˜++Λ˜−). (5.37)

  Now, since ⟨S, S⟩ = 12, δ₁S, δ₂S are orthogonal to S and since h(S) = −S we have h(δ₁S) = −δ₁S, h(δ₁S) = −δ₁S so finally δ1S,δ2S∈𝔣1[x] . Further, adS−1δ2S∈𝔣0[x] and therefore, if p is even Ω˜(p) is identically zero. The simplest case is of course the case p = 0 that corresponds to the Kirillov Poisson structure ad_S (restricted to the manifold of the potentials) which obviously trivializes. On the contrary, for p odd, these structures are not trivial. The symplectic structure for p = 1 corresponds to the Poisson tensor i∂_x (restricted to the manifold of the potentials) and does not trivialize, see [24] for the details.