Prolongation Loop Algebras for a Solitonic System of Equations

We consider an integrable system of reduced Maxwell-Bloch equations that describes the evolution of an electromagnetic field in a two-level medium that is inhomogeneously broadened. We prove that the relevant Backlund transformation preserves the reality of the n-soliton potentials and establish their pole structure with respect to the broadening parameter. The natural phase space of the model is embedded in an infinite dimensional loop algebra. The dynamical equations of the model are associated to an infinite family of higher order Hamiltonian systems that are in involution. We present the Hamiltonian functions and the Poisson brackets between the extended potentials.


Introduction
Integrable systems are closely related to the inverse scattering method that serves as a means of integrating the initial value problem, and also to infinite dimensional Lie algebras, else known as Kac-Moody Lie algebras or loop algebras. Studying the underlying loop algebra reveals encoded properties of the equations that are inherited from the integrable character of the model. The Adler-Kostant-Symes (AKS) theorem gives the Lie algebraic formulation of the dynamics of the system.
In [1], several nonlinear partial differential equations were realized as the compatibility condition of two linear matrix systems of the form, where q and r are the potentials satisfying a certain nonlinear evolution equation and A, B, C are functions of x and t. The Zakharov-Shabat dressing transformation [2] was then employed to produce the soliton solutions of the system. The matrices appearing on the right-hand-side of (1) are 2 × 2, traceless matrices that can be viewed as elements of the finite dimensional Lie algebra sl (2). sl (2) may be prolonged to an infinite dimensional Kac-Moody Lie algebra with the aid of the spectral parameter λ, and lead to an infinite number of systems in involution. This is made exact in the context of the AKS theorem. Briefly, the theorem states that if one starts with a set of commuting functions on a Lie algebra, then the corresponding Hamiltonian systems are of course trivial. However, if we project those functions to appropriate subalgebras then the resulting Hamiltonian systems need not be trivial and continue to be in involution.
In [3,4,5,6,7,8,9], several integrable equations have been studied in the context of the inverse scattering technique and the AKS theorem. For example, the Toda system and the Korteweg-de Vries equation, serving as representatives of the ordinary and partial differential evolution equations respectively, have been associated to Lax pair equations with one degree of freedom represented by the spectral parameter. The phase space of the relevant model was extended to include infinite dimensional loop algebras. The loop algebra was then decomposed into a vector space direct sum of subalgebras, and with the aid of the trace functional it was identified with its dual. In that way, the systems obtain the Kostant-Kirillov symplectic structure, and an application of the AKS theorem revealed an infinite number of integrable Hamiltonian systems in involution.
In this paper we consider a reduction of the Maxwell-Bloch equations that models the optical pulse propagation of an electric field through a two-level medium in the presence of an external constant electric field. The optical resonance line of the medium is inhomogeneously broadened. Following the terminology used by McCall and Hahn [10] and Lamb [11], we refer to inhomogeneous broadening as the phenomenon that occurs when the atoms of the medium possess different resonant frequencies due to microscopic interactions between them. In such a case, the induced electric dipole polarization is represented as a continuum, and the resulting optical resonance line is inhomogeneously broadened. In solids, such a broadening could be caused by a distribution of static crystalline electric and magnetic fields and in gases by the distribution of Doppler frequencies.
Since the late sixties and seventies with the papers [10,11,12], among others, the Maxwell-Bloch equations have undergone several treatments. Recently, various reductions of the equations have been studied both analytically and numerically. Lax pair operators, Darboux transformations and soliton solutions were constructed and analyzed [13,14,15,16] and interesting applications in crystal acoustics have emerged [17,18].
Our scope in this paper is the study of the integrable structure of a reduced Maxwell-Bloch system and the connections that arise with Kac-Moody Lie algebras. In particular, we prove that the Bäcklund transformation preserves the reality of the n-soliton potentials ∀ n ∈ N, and establish their pole structure with respect to the broadening parameter. The solitonic phase space of the model is embedded in an infinite dimensional loop algebra and an application of the AKS theorem allows us to view the system as a member of an infinite family of systems in involution. We present the higher order Hamiltonian functions and flows, as well as the Poisson brackets between the extended potentials.

Phase space
The optical equations we shall consider are the ones presented in [19]. They model the propagation of an electric field in a two-level quantized medium, where the optical resonance line of the medium has been inhomogeneously broadened. The classical wave equation of Maxwell (2) is used for the evolution of a unidirectional electric field and is coupled with the quantum mechanical Bloch equations (3)- (5), that describe the behavior of the induced polarization field, f g = ∞ ∞ f (ω)g(ω)dω, and denotes the weighted average of the function f (ω) with respect to the distribution function, .
For a physical interpretation of the model see [19]. In this paper we shall study the system (2)-(5) from a Lie algebraic point of view.
The system is completely integrable and admits a Lax pair representation. We define the differential operators L and A whose commutativity, [L, A] := LA − AL = 0, is equivalent to equations (2)-(5) where, H, F and E form a basis of the semi-simple Lie algebra su (2), and are given as follows, We call h j , f j , e j for j = 0, 1, the potentials and Q (0) , Q (1) loop elements because they can be considered as elements of an infinite dimensional loop algebra that we will define in Section 3. We note that the potentials depend on the solutions e, R ω , S ω , U ω of the inhomogeneously broadened reduced Maxwell-Bloch (ib-rMB) equations (2)- (5). The commutation of the differential operators L and A gives rise to the following Lax pair equation which is equivalent to the ib-rMB system. The Lax pair can be used to construct a Bäcklund transformation (BT) that iteratively produces the soliton solutions of equations (2)- (5).
We consider the spectral problem, and aim to find a new eigenfunction Ψ and the corresponding new loop element Q (1) that satisfy the spectral problem. The loop elements are functions of h j , f j , e j , for j = 0, 1 and will, in turn, give rise to the new solutions of the ib-rMB system via expressions (6). This transformation theory leads to an analogue of superposition formulas that allows one to construct multi-soliton solutions starting from single solitons by algebraic means [21,22,23,24,15]. We briefly describe the procedure and quote the relevant theorem from reference [19].
One begins with a constant solution to equations (2)- (5), which in turn determines potentials (6) and the corresponding loop element, call it Q 0 . We then find a simultaneous, fundamental solution Ψ 1 to the Lax pair system LΨ = 0, AΨ = 0 and define Φ 1 = φ 1 is a constant vector with c 1 1 , c 1 2 ∈ R, and the matrix N 1 as, The BT matrix function is constructed as: Applying G to Ψ 1 yields a new fundamental solution: Ψ 2 (ν 1 , c 1 ; λ) = G(ν 1 , c 1 ; λ)Ψ 1 (ν 1 , c 1 ; λ), and the procedure is iterated. The formula for the loop element after n iterations of the BT, call it Q n , in terms of the previous one Q n−1 and the matrix N n is the context of the next theorem.
We have taken the specific value of the spectral parameter to be purely imaginary, ν n = im n ∈ iR. The general form of the n-soliton loop is given by, where the upper index of e n 0 , h n 1 , f n 1 , e n 1 and the lower index of Q n , Ψ n , Φ n , N n indicate the level of the Bäcklund transform. We note that h 0 and f 0 are constant functions of space and time and are invariants of the level of the BT. To ensure the reality of the potentials and consequently of the solutions e, R ω , S ω , U ω , we compare (8) and (9) and impose the following conditions: where I is the 2×2 identity matrix. One can find that conditions (10)- (11) are satisfied if and only if We can construct φ n 1 , φ n 2 such that one of the following two cases holds: This is the context of the following proposition.
Proposition 1. At any given level of the Bäcklund transformation the reality of the n-soliton potentials can be secured by an appropriate choice of the transformation data.

Proof . A constant set of solutions to equations (2)-(5) is given by
and v 1 is an expression independent of ω, λ, ζ and τ . We choose c 1 1 ∈ R and c 1 2 ∈ iR so that φ 1 1 ∈ R, φ 1 2 ∈ iR and condition (12) holds for n = 1. The proof that the condition is satisfied at any given level of the BT lies on the following: N 2n−1 ∈ span{I, E}, and N 2n ∈ span{H, F}. This yields G n (λ) ∈ span{λI, H, F}, ∀ n ∈ N.
, and use Φ n = Ψ n c n , we obtain We aim to give an integral-free representation of Q n (λ) so that its λ-structure becomes apparent. We begin with Q 1 (λ) and then generalize the construction for Q n (λ). The general form of Q 1 (λ) is the following, The one soliton potentials are given as, sech(x 1 ), We write the last three in the following more convenient form, where β 1 , β 2 , β 3 , β 4 are functions of ζ and τ , but do not depend on ω or λ. We will use contour integration and Cauchy's integral formula to compute Q 1 . We define the complexvalued function, which has three poles in the upper half complex plane at z 1 = λ, z 2 = im 1 , and z 3 = ω 0 + iσ. We integrate around a simple contour that consists of a semicircle of radius R in the upper-half complex plane, call it C R , and the segment on the real axis from −R to R. We choose R big enough to include all the poles of h(z) that appear in the upper-half complex plane. It is not hard to see that lim R →∞ C R h(z)dz = 0, and thus using the Cauchy integral formula we obtain, which can be written as, where δ j ,δ j ,δ j , j = 1, 2, 3 are independent of λ and ω. Let S 1 = {δ 1 , δ 2 , δ 3 }, S 2 = {δ 1δ2 ,δ 3 } and The elements of each set S 1 , S 2 or S 3 are functions of the one-soliton potentials and they obey the ib-rMB equations. We will use induction to obtain the general form of the n-soliton potentials h n 1 , f n 1 , e n 1 .
Proposition 2. The pole structure with respect to the broadening parameter ω of the general n-soliton potentials of the ib-rMB system is the following: .
Proof . The proposition holds for n = 1 as can be seen by (13)- (15). We assume that the proposition holds for n − 1 and show that it holds for n. Using the induction hypothesis in Theorem 1 and the fact that N k HN −1 k ∈ span{H, F} for any k ∈ N we get, By definition Therefore, by equating the last two expressions of Q n (λ) we prove the proposition. We note that some of the coefficients γ k , δ k , ǫ k may be equal to zero. For example, for n = 1, δ 1 = 0 and ǫ 0 = 0.
The form of the n-soliton potentials can be used to identify the λ-structure of a general n-soliton loop element Q n (λ).
The proof follows the same idea that was used to derive the integral-free form (16) of Q 1 (λ) and is omitted.
The elements of each set S k , k = 1, . . . , n + 2 are functions of the m-soliton potentials for m = 1, . . . , k and they satisfy the ib-rMB equations.

Embedding
We call α k = im k , k = 1, . . . , n, α n+1 = ω 0 − iσ and α n+2 = ω 0 + iσ. Then Q n (λ) of (18) can be rewritten as where X − k , X + k are linear combinations of the basis elements of the Lie algebra su(2), {H, F, E}. We define the following infinite dimensional Lie algebra Q ext that extends loops of the form (18), We note that the coefficients h k j , f k j , e k j of (19) are not the same as those of (17) where the upper index denotes the level of the BT. In (19) X is an element that extends the natural solitonic phase space of the ib-rMB equations and the upper index indicates the relevant pole α k , whereas the lower index indicates the order of the pole α 2 k . We embed this Lie algebra into a larger one, where we have set α k+n+2 = −α k , for k = 1, . . . , n + 2. We note that for Q to be a Lie algebra a finiteness condition needs to imposed. Namely, X j = 0 and X k j = 0, ∀ k = 1, . . . , 2(n + 2) and j ≥ j 0 for some j 0 ∈ N.

Application of the Adler-Kostant-Symes theorem
Following the ideas in the theorem of Adler, Kostant and Symes [5,6,7,8], we decompose the infinite dimensional loop algebra Q into a direct sum of two subalgebras a and b, and define an ad-invariant, non-degenerate inner product on Q. The perpendicular complements a ⊥ , b ⊥ with respect to the inner product serve as another direct sum decomposition of Q. Using the Riesz representation theorem and the non-degenerate inner product one may define an isomorphism between a ⊥ and b * , the dual of the Lie subalgebra b. The canonical Lie-Poisson bracket that exists on b * is then represented on a ⊥ . If necessary, one may translate a ⊥ by any element β ∈ b ⊥ that satisfies (β, [x, y]) = 0 ∀ x, y ∈ a, so that the translated space β + a ⊥ includes the natural phase space of the relevant system [20].
We write Q as the vector space direct sum of the Lie subalgebras A non-degenerate inner product is defined on Q using the trace map. Namely, where (XY ) 0 denotes the matrix coefficient of λ 0 in the product XY . We note that the inner product is ad-invariant. That is, (X, ad Y Z) + (ad Y X, Z) = 0, where ad X Y = [X, Y ]. The perpendicular complements of the subalgebras a and b with respect to this inner product take the form, The natural phase space of the ib-rMB equations, as can be seen in (18), contains elements that belong in a ⊥ as well as terms of order λ 1 . Therefore we translate the space a ⊥ by β = λX 1 ∈ b ⊥ . We note that (β, [X, Y ]) = 0 ∀ X, Y ∈ a, which is a necessary condition for the AKS theorem. We consider the set, which includes the phase space of the ib-rMB equations, and define a Lie-Poisson bracket on a ⊥ + β. If Φ and Ψ are functions on a ⊥ + β we first compute their gradients ∇Φ, ∇Ψ in the full Lie algebra Q and then define We note that b denotes projection on the Lie subalgebra b. ∇H(X) denotes the gradient of a function H on Q and is defined as follows, We quote the AKS theorem and show how it can be applied in the case of the ib-rMB equations.
Theorem 2. If H, F are invariant functions on Q * ∼ = Q, and we denote by H proj , F proj their projection to the subspace a ⊥ +β then {H proj , F proj } = 0, and the Hamiltonian system associated with the invariant function H is given as, Remark 1. ad * is negative the dual of the ad map.

Remark 2. The definition of H being an invariant function is ad
Remark 3. The existence of an invariant inner product allows one to identify the ad-map with ad * , because on one hand by definition (ad x y, z) = −(y, ad * x z), and on the other a simple calculation using the definition of the inner product shows that (ad x y, z) = −(y, ad x z).
If a set of invariant functions is given, the theorem guarantees their commutativity in the canonical Lie-Poisson bracket and gives the form of the Hamiltonian system associated with the invariant function. To apply the theorem we construct a set of invariant functions {Φ k } k∈N via the following operator, The invariant functions are defined as Φ k (X) = 1 2 (M k (X), X). We compute the gradient of the Φ k (X). By definition, Thus ∇Φ k (X) = M k (X). According to Remarks 2 and 3, to actually demonstrate the invariance Therefore the functions Φ k , k ∈ N are invariant and can be used in the context of the AKS theorem, which reads as follows, We used the subscript k for the time variable t k to distinguish between the different dynamical evolutions of the systems associated with the Hamiltonian functions Φ k . For each k ∈ N the following systems are in involution: For k = 0 and a truncated X of the form (19) where h i j , f i j , e i j = 0 for j ≥ 2 and i = 1, . . . , n + 2, the Hamiltonian system (25) becomes which is the zero curvature representation of the Lax pair equation (7) for the ib-rMB equations.
We have thus identified the ib-rMB system as a member of the infinite family of systems (25), that commute with respect to the canonical Lie-Poisson bracket (22).

Extended f low
We consider a general element of the Lie algebra Q of the form, or equivalently, We remark that h i j , f i j , e i j in (28) are not the same as the ones in (29). The latter ones are linear combinations of the former. However, to avoid introducing yet another symbol we use h 2m+1 (29), where j indicates the order of the pole at λ 2 = α 2 i and 2m + 1 or 2m the power of λ multiplying H, F or E respectively in the numerator.
To examine the extended flow associated with the ib-rMB equations we set k = 0 in the involutive systems (25). The relevant system takes the form, where ∇Φ 0 (X) = M 0 (X) = X. We write X as follows: where P j is a polynomial in λ given as and P 0 = λ(h 0 H + f 0 F) + e 0 E. The degree of P j (λ) is given by, deg(P j (λ)) = 2j(n + 1) + 1.
Projecting ∇Φ 0 (X) to the subalgebra b is equivalent to keeping the polynomial part of X, which is P 0 . Thus the Hamiltonian flow for k = 0 takes the form, which unravels to The coefficients of the matrices H, F, and E appearing in (30) are defined as follows, We observe that the expressions multiplying H and F in (30) have no polynomial part since the degree of the numerator that equals 2j(n + 1) + 1 is strictly smaller that the degree of the denominator that equals 2j(n + 2), for j ≥ 1. However, the expression multiplying E carries a polynomial term. In particular the evolution equation (30) can be written as , . . . , n + 2}. On the other hand by the definition of X we have that, By equating the different powers of λ we disclose the system induced by the Hamiltonian function Φ 0 . As previously noted, for a fixed j ∈ N, the elements of the sets S m = {h 2m+1 j , f 2m+1 j , e 2m j } satisfy the same system of equations for any m = 0, 1, . . . , j(n + 1). Therefore it suffices to consider only one such set. Without loss of generality we choose S j(n+1) . The evolution equations take the form: .
To reveal the extended flow for the ib-rMB equations we set j=1: de 2n+2 We note that the coupling of the above system to the evolution equations satisfied by the higher order potentials that correspond to j ≥ 2, is captured in the dynamical equation for e 2n+2 1 . If h 4n+5 2 = f 4n+5 2 = 0, then the system reduces to the dynamical equations that the n-soliton potentials of the ib-rMB equations satisfy.

Hamiltonian functions and Poisson brackets
In this section we aim to write the extended flow of the ib-rMB equations given by system (31) in Section 5, in the canonical Poisson form, The Hamiltonian functions for the systems (25) described in the context of the AKS theorem in Section 4 are defined as We let k = 0 and consider a general X of the form, The Hamiltonian function is found to be: We define the following functionals, relevant to the potentials that appear in the Hamiltonian: Using definition (23) we compute the gradients of these functionals: For example, to obtain ∇h 0 (X) we consider the equation which implies that Tr(Y ∇h 0 (X)) 0 = h 0 (Y ). Therefore ∇h 0 (X) is the element of Q such that the constant term (with respect to λ) of the product Y ∇h 0 (X) has trace that equals precisely h 0 (Y ). We note that the matrices HF, FE, EH are traceless whereas H 2 = F 2 = E 2 = −I. Having that in mind, we find that ∇h 0 ( The Poisson brackets between the potentials appearing in the Hamiltonian can be computed using definition (22). For instance, Using the Poisson brackets (35), we find that the extended flow for the ib-rMB equations given in (31) can be expressed as the canonical flow (32) associated with the Hamiltonian function Φ 0 .
Higher order functionals can also be defined using a diagonal formation that gradually sweeps all the potentials. In particular, for a general X of the form (29) we define for N = 0, 1, 2, . . . , the following higher order functionals (in bold to distinguish between h 0 , f 0 , e 0 and h 0 , f 0 , e 0 ): We note that if the upper index of the potentials that appear in the sums is less than zero then the potentials are set to zero. The set S = {e 0 , h N , f N , e N : N = 0, 1, 2, . . .} includes all the dynamical quantities that enter the AKS flows (25). The gradients of these higher order functionals can be computed using definition (23). For instance, where ∇h 0 = ∇h 2n+3 1 , ∇f 0 = ∇f 2n+3 1 , ∇e 0 = ∇e 2n+2 1 , and are given in (34). Working in a similar manner as in the example for {e 0 , h 2n+3 1 }, one can find the Poisson brackets between the higher order functionals, i.e. {e 0 , h 1 }(X) = f 1 (X).

Summary
In this paper we have considered an integrable system of reduced Maxwell-Bloch equations, that is inhomogeneously broadened. We show that the relevant Bäcklund transformation preserves the reality of the n-soliton potentials ∀ n ∈ N, and establish their pole structure with respect to the broadening parameter. We obtain a representation of the relevant phase space in the spectral parameter λ, which is then embedded in a prolonged loop algebra. The equations satisfied by the n-soliton potentials are associated to an infinite family of higher order Hamiltonian involutive systems. We present the Hamiltonian functions of the higher order flows and the Poisson brackets between the extended potentials.