The strict AKNS hierarchy: its structure and solutions

In this paper we discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space of ${\rm sl}_{2}$ than that in the AKNS-case and whose Lax equations are based on a different decomposition of this loop space. We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations. We present a linearization of the system and conclude by giving a wide construction of solutions of this hierarchy.


Introduction
Integrable hierarchies often occur as the evolution equations of the generators of a deformation of a commutative subalgebra inside some Lie algebra g. The deformation and the evolution equations are determined by a splitting of g in the direct sum of two Lie subalgebras, like in the Adler-Kostant-Symes Theorem (see [1]). This gives then rise to a compatible set of Lax equations, a so-called hierarchy, and the simplest nontrivial equation often determines the name of the hierarchy. The hierarchy we discuss here corresponds to a somewhat different splitting of an algebra of sl 2 loops that with respect to the usual decomposition yields the AKNS hierarchy. Following the terminology used in similar situations (see [2]), we use the name strict AKNS hierarchy for the system of Lax equations corresponding to this different decomposition.
The letters AKNS refer to the following fact: Ablowitz et al. showed (see [3]) that the initial value problem of the following system of equations for two complex functions and depending on the variables and ) . ( A direct computation shows that the following identities hold for them: ) , ) . ( By combining them, one sees that the AKNS equations are equivalent to 2 Advances in Mathematical Physics consequently carries the name AKNS hierarchy (see [4]).
Besides that, we introduce here also another system of Lax equations for a more general deformation of the basic generator of the commutative algebra. It relates to a different decomposition of the relevant Lie algebra. Since its analogue for the KP hierarchy is called the strict KP hierarchy, we use in the present context the term strict AKNS hierarchy. We show first of all that this new system is compatible and equivalent with a set of zero curvature relations. Further, we describe suitable linearizations that can be used to construct solutions of both hierarchies. We conclude by constructing solutions of both hierarchies.

The AKNS Hierarchy and Its Strict Version
We present here an algebraic description of the AKNS hierarchy and its strict version that underlines the deformation character of these hierarchies as pointed at in Introduction. At (4), one has to deal with 2×2-matrices whose coefficients were polynomial expressions of a number of complex functions and their derivatives with respect to some parameters. The same is true for the other equations in both hierarchies. We formalize this as follows: our starting point is a commutative complex algebra that should be seen as the source from which the coefficients of the 2 × 2-matrices are taken. We will work in the Lie algebra sl 2 ( )[ , −1 ) consisting of all elements and the bracket We will also make use of the slightly more general Lie algebra gl 2 ( )[ , −1 ), where the coefficients in the -series from (5) are taken from gl 2 ( ) instead of sl 2 ( ) and the bracket is given by the same formula. In the Lie algebra gl 2 ( )[ , −1 ), we decompose elements in two ways. The first is as follows: (7) and this induces the splitting where the two Lie subalgebras gl 2 ( )[ , −1 ) ⩾0 and gl 2 ( )[ , −1 ) <0 are given by By restriction, it leads to a similar decomposition for sl 2 ( )[ , −1 ), which is relevant for the AKNS hierarchy. The second way to decompose elements of This yields the splitting By restricting it to sl 2 ( )[ , −1 ), we get a similar decomposition for this Lie algebra, which relates, as we will see further on, to the strict version of the AKNS hierarchy. The Lie subalgebras gl 2 ( )[ , −1 ) >0 and gl 2 ( )[ , −1 ) ⩽0 in (11) are defined in a similar way as the first two Lie subalgebras Both hierarchies correspond to evolution equations for deformations of the generators of a commutative Lie algebra in the first component. Denote the matrix ( − 0 0 ) by 0 . Inside sl 2 ( )[ , −1 ) ⩾0 , this commutative complex Lie subalgebra 0 is chosen to be the Lie subalgebra with the basis { 0 | ⩾ 0} and in sl 2 ( )[ , −1 ) >0 our choice will be the Lie subalgebra In the first case that we work with 0 , we assume that the algebra possesses a set { | ⩾ 0} of commuting C-linear derivations : → , where each should be seen as the derivation corresponding to the flow generated by 0 . The data ( , { | ⩾ 0}) is also called a setting for the AKNS hierarchy. In the case that we work with the decomposition (11), we merely need a set of commuting C-linear derivations : → , ⩾ 1, with the same interpretation. Staying in the same line of terminology, we call the data ( , { | ⩾ 1}) a setting for the strict AKNS hierarchy. We let each derivation occurring in some setting act coefficient-wise on the matrices from gl 2 ( ) and that defines then a derivation of this algebra. The same holds for the extension to gl 2 ( )[ , −1 ) defined by We have now sufficient ingredients to discuss the AKNS hierarchy and its strict version. In the AKNS case, our interest is in certain deformations of the Lie algebra 0 obtained by conjugating with elements of the group corresponding to gl 2 ( )[ , −1 ) <0 . At the strict version, we are interested in certain deformations of the Lie algebra 1 obtained by conjugating with elements from a group linked to gl 2 ( )[ , −1 ) ⩽0 .
Note that for each ∈ gl 2 ( )[ , −1 ) <0 the exponential map yields a well-defined element of the form and with the formula for the logarithm one retrieves back from . One verifies directly that the elements of the form (14) form a group with respect to multiplication and this we see as the group <0 corresponding to gl 2 ( )[ , −1 ) <0 . In the case of the Lie subalgebra gl 2 ( )[ , −1 ) ⩽0 , one cannot move back and forth between the Lie algebra and its group. Nevertheless, one can assign a proper group to this Lie algebra. A priori, the exponential exp( ) of an element ∈ gl 2 ( )[ , −1 ) ⩽0 does not have to define an element in gl 2 ( )[ , −1 ) ⩽0 . That requires convergence conditions. However, if it does, then it belongs to where gl 2 ( ) * denotes the elements in gl 2 ( ) that have a multiplicative inverse in gl 2 ( ). It is a direct verification that ⩽0 is a group and we see it as a proper group corresponding to the Lie algebra gl 2 ( )[ , −1 ) ⩽0 . In fact, ⩽0 is isomorphic to the semidirect product of <0 and gl 2 ( ) * . Now there holds the following. Proof. Take first any ∈ <0 . Then, there is an ∈ gl 2 ( )[ , −1 ) <0 such that = exp( ). Now there holds for every ∈ and this shows that the coefficients for the different powers of in this expression are commutators of elements of gl 2 ( ) and sl 2 ( ) and that proves the claim for elements from <0 . Since conjugation with an element from gl 2 ( ) * maps sl 2 ( ) to itself, the same holds for sl 2 ( )[ , −1 ). This proves the full claim.
Next we have a look at the different deformations. The deformations of 0 by elements of <0 are determined by that of 0 . Therefore, we focus on that element and we consider for of 0 . From this formula, we see directly that if each = ( − ), = 1, 2, then In particular, we get in this way that 11 = − ( /2) and 22 = ( /2). Since is the deformation of 0 and Id is central, the deformation of each 0 , ∈ Z, is given by . The deformation of the Lie algebra 1 by elements from ⩽0 is basically determined by that of the element 0 . So we focus on the deformation of this element. Using the same notations as at the deformation of 0 by <0 , we get that the deformation of 0 by a ∈ ⩽0 , with ∈ gl 2 ( ) * and ∈ <0 , looks like Consequently, the corresponding deformation of each 0 , ⩾ 1, is −1 .

Advances in Mathematical Physics
Now we are looking for deformations of the form (17) such that the evolution with respect to the { } satisfies the following: for all ⩾ 0, where the second identity follows from the fact that all { } commute. Equations (20) are called the Lax equations of the AKNS hierarchy and the deformation satisfying these equations is called a solution of the hierarchy. Note that = 0 is a solution of the AKNS hierarchy and it is called the trivial one. AKNS equation (4) occurs among the compatibility equations of this system, as we will see further on. Note that (20) for = 0 is simply 0 ( ) = [ 0 , ]. Therefore, if 0 = / 0 and both − 0 and 0 belong to the algebra of matrix coefficients, then we can introduce the loop̂∈ sl 2 ( )[ , −1 ) given bŷ which is easily seen to satisfy 0 (̂) = 0. This handles then the dependence of of 0 . Among the deformations of the form (19), we look for such that their evolution with respect to { } satisfies the following: for all ⩾ 1, where the second identity follows from the fact that all { −1 } commute. Since (22) correspond to the strict cutoff (10), they are called the Lax equations of the strict AKNS hierarchy and the deformation is called a solution of this hierarchy. Again there is at least one solution = 0 . It is called the trivial solution of the hierarchy.

Proposition 3. Both sets of Lax equations
corresponding to a solution to (22) satisfy the zero curvature relations Proof. The idea is to show that the left-hand side of (23) and (24), respectively, belongs to and thus has to be zero. We give the proof for { }; that for { } is similar and is left to the reader. The inclusion in the first factor is clear as both and ( ) belong to the Lie subalgebra sl 2 ( )[ , −1 ) >0 . To show the other one, we use Lax equations (22). Note that the same Lax equations hold for all the { | ⩾ 0}: and Taking into account the second identity in (22), we see that the left-hand side of (24) is equal to This element belongs to the Lie subalgebra sl 2 ( )[ , −1 ) ⩽0 and that proves the claim.
Reversely, we have the following.   Proof. Again we prove the statement for ; that for is shown in a similar way. So, assume that there is one Lax equation (22) that does not hold. Then, there is a ⩾ 1 such that Since both ( ) and [ , ] are of order smaller than or equal to one in , we know that ( ) ⩽ 1. Further, we can say that for all ⩾ 0 and we see by letting go to infinity that the right-hand side can obtain any sufficiently large order in . By the zero curvature relation for and , we get for the left-hand side and this last expression is of order smaller or equal to in . This contradicts the unlimited growth in orders of of the right-hand side. Hence, all Lax equations (22) have to hold for .
Because of the equivalence between Lax equations (20) for and the zero curvature relations (23) for { }, we call this last set of equations also the zero curvature form of the AKNS hierarchy. Similarly, the zero curvature relations (24) for { } are called the zero curvature form of the strict AKNS hierarchy.
Besides the zero curvature relations for the cut-off 's { } and { }, respectively, corresponding to, respectively, a solution of the AKNS hierarchy and a solution of the strict AKNS hierarchy, also other parts satisfy such relations. Define Then, we have the following result.

Corollary 5. The following relations hold:
(i) The parts { | ⩾ 0} of a solution of the AKNS hierarchy satisfy (ii) The parts { | ⩾ 0} of a solution of the strict AKNS hierarchy satisfy Proof. Again we show the result only in the strict case. Note that { −1 } satisfy Lax equations similar to : Now we substitute in the zero curvature relations for { } everywhere the relation = + −1 and use the Lax equations above and the fact that all { −1 } commute. This gives the desired result.
To clarify the link with the AKNS equation, consider relation (23) for 1 = 2 and 2 = 1: Then, this identity reduces in sl 2 ( )[ , −1 ) ⩾0 , since 0 is constant, to the following two equalities: The first gives an expression of the off-diagonal terms of 2 in the coefficients and of 1 ; that is, and the second equation becomes AKNS equations (4), if one has 1 = / and 2 = / .

The Linearization of Both Hierarchies
The zero curvature form of both hierarchies points at the possible existence of a linear system of which the zero curvature equations form the compatibility conditions. We present here such a system for each hierarchy. For the AKNS hierarchy, this system, the linearization of the AKNS hierarchy, is as follows: take a potential solution of the form (17) and consider the system Likewise, for a potential solution of the strict AKNS hierarchy of the form (19), the linearization of the strict AKNS hierarchy is given by Before specifying and , we show the manipulations needed to derive the Lax equations of the hierarchy from their linearization. We do this for the strict AKNS hierarchy; for the Assuming that each derivation equals / , one arrives for (43) at the solution where is the short hand notation for { | ⩾ 0}, and for (44) it leads to In general, one needs in the linearizations ( ) and ( ), respectively, a left action with elements like , and , from gl 2 ( )[ , −1 ), respectively, an action of all the , and a right action of 0 and 0 , respectively. This can all be realized in suitable gl 2 ( )[ , −1 ) modules that are perturbations of the trivial solutions 0 and 0 . Consider in a AKNS setting and in a strict AKNS setting We define the right-hand action of 0 and 0 , respectively, by and the action of each by Analogous to the terminology in the scalar case (see [5]), we call the elements of M 0 and M 1 oscillating matrices. Note that both M 0 and M 1 are free gl 2 ( )[ , −1 ) modules with generators 0 and 0 , respectively, because for each ℎ( ) ∈ gl 2 ( )[ , −1 ) we have Hence, in order to be able to perform legally the scratching of = {ℎ( )} 0 or = {ℎ( )} 0 , it is enough to find oscillating matrices such that ℎ( ) is invertible in gl 2 ( )[ , −1 ). We will now introduce a collection of such elements that will occur at the construction of solutions of both hierarchies. For = ( 1 , 2 ) ∈ Z 2 , let ( ) ∈ gl 2 ( )[ , −1 ) be given by so that also here the wave matrix fully determines the solution.
For both hierarchies, there is a milder condition that is to be satisfied by oscillating matrices of a certain type, in order that they become a wave matrix of that hierarchy. (a) If there exists for each ⩾ 0 an element ∈ gl 2 ( )[ , −1 ) ⩾0 such that then each = ( ) ⩾0 and is a wave matrix of type ( ) for the AKNS hierarchy.
then each = ( −1 ) >0 and is a wave matrix of type ( ) for the strict AKNS hierarchy.
Proof. We give the proof again for the strict AKNS case; the other one is similar. By using the fact that M 1 is a free gl 2 ( )[ , −1 ) module with generator 0 we can translate the relations ( ) = into equations in gl 2 ( )[ , −1 ). This yields Projecting the right-hand side on gl 2 ( )[ , −1 ) >0 gives us the identity we are looking for.
In the next section, we produce a context where we can construct wave matrices for both hierarchies in which the product is real.

The Construction of the Solutions
In this section, we will show how to construct a wide class of solutions of both hierarchies. This is done in the style of [6]. We first describe the group of loops we will work with. For each 0 < < 1, let be the annulus Following [7], we use the notation an GL 2 (C) for the collection of holomorphic maps from some annulus into GL 2 (C). It is a group with respect to pointwise multiplication and contains in a natural way GL 2 (C) as a subgroup as the collection of constant maps into GL 2 (C). Other examples of elements in an GL 2 (C) are the elements of Δ. However, an GL 2 (C) is more than just a group; it is an infinite dimensional Lie group. Its manifold structure can be read off from its Lie algebra an gl 2 (C) consisting of all holomorphic maps : → gl 2 (C), where is an open neighborhood of some annulus , 0 < < 1. Since gl 2 (C) is a Lie algebra, the space an gl 2 (C) becomes a Lie algebra with respect to the pointwise commutator. Topologically, the space an gl 2 (C) is the direct limit of all the spaces an, gl 2 (C), where this last space consists of all corresponding to the fixed annulus . One gives each an, gl 2 (C) the topology of uniform convergence and with that topology it becomes a Banach space. In this way, an gl 2 (C) becomes a Fréchet space. The pointwise exponential map defines a local diffeomorphism around zero in an gl 2 (C) (see, e.g., [8]). Now each ∈ an gl 2 (C) possesses an expansion in a Fourier series that converges absolutely on the annulus : This Fourier expansion is used to make two different decompositions of the Lie algebra an gl 2 (C). The first is the analogue for the present Lie algebra an gl 2 (C) of decomposition (8) Both are Lie subalgebras of an gl 2 (C) and their direct sum equals the whole Lie algebra. The first Lie algebra consists of the elements in an gl 2 (C) that extend to holomorphic maps defined on some disk { ∈ C | | | ⩽ 1/ }, 0 < < 1, and the second Lie algebra corresponds to the maps in an gl 2 (C) that have a holomorphic extension towards some disk around infinity { ∈ P 1 (C) | | | ⩾ }, 0 < < 1, and that are zero at infinity. To each of the two Lie subalgebras belongs a subgroup of an GL 2 (C). The pointwise exponential map applied to elements of an gl 2 (C) <0 yields elements of and the exponential map applied to elements of an gl 2 (C) ⩾0 maps them into Both − and + are easily seen to be subgroups of an GL 2 (C) and since the direct sum of their Lie algebras is an gl 2 (C), their product is open in an GL 2 (C) and is called, like in the finite dimensional case, the big cell with respect to − and + . The second decomposition is the analogue of the splitting (11) of gl 2 ( )[ , −1 ) that led to the Lax equations of the strict AKNS hierarchy. Now we consider the subspaces Both are Lie subalgebras of an gl 2 (C) and their direct sum equals the whole Lie algebra. an gl 2 (C) >0 consists of maps whose holomorphic extension to = 0 equals zero in that point and an gl 2 (C) ⩽0 is the set of maps that extend homomorphically to a neighborhood of infinity. To each of them belongs a subgroup of an GL 2 (C). The pointwise exponential map applied to elements of an gl 2 (C) >0 yields elements of and the exponential map applied to elements of an gl 2 (C) ⩽0 maps them into − = { | ∈ an gl 2 (C) , = 0 Both − and + are easily seen to be subgroups of an GL 2 (C).
Since the direct sum of their Lie algebras is an gl 2 (C), their product − + is open and because + = GL 2 (C) + = + GL 2 (C) and − = GL 2 (C) − = − GL 2 (C), this gives the equality the big cell in an GL 2 (C). The next two subgroups of an GL 2 (C) correspond to the exponential factors in both linearizations. The group of commuting flows relevant for the AKNS hierarchy is the group and, similarly, for the strict AKNS hierarchy, we use the commuting flows from Besides the groups Δ, Γ 0 , and Γ 1 , there are more subgroups in an GL 2 (C) that commute with those three groups and they are in a sense complimentary to Δ, Γ 0 and Δ, Γ 1 , respectively. That is why we use the following notations for them: in − there is and in − we have We have now all ingredients to describe the construction of the solutions to each hierarchy and we start with the AKNS hierarchy. Take inside the product an GL 2 (C) × Δ the collection 0 of pairs ( , ( )) such that there exists a 0 ( ) ∈ Γ 0 satisfying For such a pair ( , ( )), we take the collection Γ 0 ( , ( )) of all 0 ( ) satisfying the condition (77). This is an open nonempty subset of Γ 0 . Let 0 ( , ( )) be the algebra of analytic functions Γ 0 ( , ( )) → C. This is the algebra of Advances in Mathematical Physics 9 functions that we associate with the point ( , ( )) ∈ 0 and for the commuting derivations of 0 ( , ( )) we choose fl / , ⩾ 0. By property (77), we have for all 0 ( ) ∈ Γ 0 ( , ( )) with − ( , ( )) ∈ − , + ( , ( )) ∈ + . Then, all the matrix coefficients in the Fourier expansions of the elements − ( , ( )) and + ( , ( )) belong to the algebra 0 ( , ( )). It is convenient to write relation (78) in the form with + ( , ( )) ∈ + . Clearly, Ψ , ( ) is an oscillating matrix of type ( ) in M 0 , where all the products in the decomposition − ( , ( )) ( ) 0 ( ) are no longer formal but real. This is our potential wave matrix for the AKNS hierarchy.