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 Bäcklund transformation preserves the reality of the n-soliton potentials and establish the...
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irk-123456789-1461062019-02-08T01:23:34Z Prolongation Loop Algebras for a Solitonic System of Equations Agrotis, M.A. 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 Bäcklund 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. 2006 Article Prolongation Loop Algebras for a Solitonic System of Equations / M.A. Agrotis // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 24 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 37N20; 35A30; 35Q60; 78A60 http://dspace.nbuv.gov.ua/handle/123456789/146106 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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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 Bäcklund 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. |
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Agrotis, M.A. |
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Agrotis, M.A. Prolongation Loop Algebras for a Solitonic System of Equations Symmetry, Integrability and Geometry: Methods and Applications |
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Agrotis, M.A. |
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Agrotis, M.A. |
| title |
Prolongation Loop Algebras for a Solitonic System of Equations |
| title_short |
Prolongation Loop Algebras for a Solitonic System of Equations |
| title_full |
Prolongation Loop Algebras for a Solitonic System of Equations |
| title_fullStr |
Prolongation Loop Algebras for a Solitonic System of Equations |
| title_full_unstemmed |
Prolongation Loop Algebras for a Solitonic System of Equations |
| title_sort |
prolongation loop algebras for a solitonic system of equations |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/146106 |
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Prolongation Loop Algebras for a Solitonic System of Equations / M.A. Agrotis // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 24 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT agrotisma prolongationloopalgebrasforasolitonicsystemofequations |
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2025-07-10T23:10:50Z |
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2025-07-10T23:10:50Z |
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1837303408635150336 |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 075, 15 pages
Prolongation Loop Algebras
for a Solitonic System of Equations
Maria A. AGROTIS
Department of Mathematics and Statistics, University of Cyprus, Nicosia 1678, Cyprus
E-mail: agrotis@ucy.ac.cy
Received September 13, 2006, in final form November 01, 2006; Published online November 08, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper075/
Abstract. 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 inhomoge-
neously broadened. We prove that the relevant Bäcklund 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.
Key words: loop algebras; Bäcklund transformation; soliton solutions
2000 Mathematics Subject Classification: 37K10; 37N20; 35A30; 35Q60; 78A60
1 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 con-
dition of two linear matrix systems of the form,
~vx =
(
−iλ q(x, t)
r(x, t) iλ
)
~v, ~vt =
(
A B
C −A
)
~v, (1)
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 invo-
lution. 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
mailto:agrotis@ucy.ac.cy
http://www.emis.de/journals/SIGMA/2006/Paper075/
2 M.A. Agrotis
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 iden-
tified 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 op-
tical 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 equa-
tions have been studied both analytically and numerically. Lax pair operators, Darboux trans-
formations 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.
2 Phase space
The optical equations we shall consider are the ones presented in [19]. They model the propa-
gation 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,
∂e
∂ζ
+
∂e
∂τ
= 〈ωSω〉g , (2)
∂Rω
∂τ
= (β − γe)Sω, (3)
∂Sω
∂τ
= −(β − γe)Rω +
1
2
ωUω, (4)
∂Uω
∂τ
= −2ωSω. (5)
Prolongation Loop Algebras for a Solitonic System of Equations 3
〈f〉g =
∫∞
∞ f(ω)g(ω)dω, and denotes the weighted average of the function f(ω) with respect to
the distribution function,
g(ω) =
σ
π((ω − ω0)2 + σ2)
.
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)
A = −∂τ + Q(0), L = ∂ζ + Q(1),
where,
Q(0) = λ(h0H+ f0F) + e0E ,
Q(1) = λ(h0H+ f0F) + e0E +
∫ ∞
−∞
1
(ω2 − λ2)
[λ(h1H+ f1F) + e1E ]dω,
and
h0 = 1
2 , h1 = −1
2γωg(ω)Rω,
f0 = 0, f1 = −1
2γωg(ω)Sω, (6)
e0 = −1
2(β − γe), e1 = 1
4γω2g(ω)Uω.
H, F and E form a basis of the semi-simple Lie algebra su(2), and are given as follows,
H =
(
i 0
0 −i
)
, F =
(
0 1
−1 0
)
, E =
(
0 i
i 0
)
.
We call hj , fj , ej 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, Rω, Sω, Uω of the inhomogeneously broad-
ened 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
∂Q(0)
∂ζ
+
∂Q(1)
∂τ
= [Q(0), Q(1)], (7)
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,
∂τΨ = Q(0)Ψ, ∂ζΨ = −Q(1)Ψ,
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 hj , fj , ej , 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].
4 M.A. Agrotis
One begins with a constant solution to equations (2)–(5), which in turn determines poten-
tials (6) and the corresponding loop element, call it Q0. We then find a simultaneous, fundamen-
tal solution Ψ1 to the Lax pair system LΨ = 0, AΨ = 0 and define ~Φ1 =
(
φ1
1
φ1
2
)
:= Ψ1(λ = ν1)~c1,
where ~c1 =
(
c1
1
ic1
2
)
is a constant vector with c1
1, c
1
2 ∈ R, and the matrix N1 as,
N1 =
(
φ1
1 −φ1
2
φ1
2 φ1
1
)
.
The BT matrix function is constructed as:
G(ν1, ~c1;λ) = N1
(
λ− ν1 0
0 λ− ν1
)
N−1
1 .
Applying G to Ψ1 yields a new fundamental solution: Ψ2(ν1, ~c1;λ) = G(ν1, ~c1;λ)Ψ1(ν1, ~c1;λ),
and the procedure is iterated. The formula for the loop element after n iterations of the BT, call
it Qn, in terms of the previous one Qn−1 and the matrix Nn is the context of the next theorem.
Theorem 1.
Qn(λ) = λhn−1
0 H+ mnhn−1
0 [H, NnHN−1
n ] + en−1
0 E +
∫ ∞
−∞
1
(ω2 − λ2)
1
(ω2 + m2
n)
×
{
λ
[
ω2(hn−1
1 H+ fn−1
1 F)− (mn)2(NnHN−1
n )(hn−1
1 H+ fn−1
1 F)(NnHN−1
n )
+ mnen−1
1 [E , NnHN−1
n ]
]
+ mnω2
(
hn−1
1 [H, NnHN−1
n ] + fn−1
1 [F , NnHN−1
n ]
)
+ ω2en−1
1 E − (mn)2en−1
1 (NnHN−1
n )E(NnHN−1
n )
}
dω. (8)
We have taken the specific value of the spectral parameter to be purely imaginary, νn =
imn ∈ iR. The general form of the n-soliton loop is given by,
Qn(λ) = λ(h0H+ f0F) + en
0E +
∫ ∞
−∞
1
(ω2 − λ2)
[λ(hn
1H+ fn
1 F) + en
1E ]dω, (9)
where the upper index of en
0 , hn
1 , fn
1 , en
1 and the lower index of Qn, Ψn, ~Φn, Nn indicate the level
of the Bäcklund transform. We note that h0 and f0 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, Rω, Sω, Uω, we compare (8) and (9) and impose the following conditions:
a) [H, NnHN−1
n ], [F , NnHN−1
n ], (NnHN−1
n )E(NnHN−1
n ) ∈ span {E}, (10)
b) (NnHN−1
n )H(NnHN−1
n ), (NnHN−1
n )F(NnHN−1
n ), [E , NnHN−1
n ] ∈ span {H,F}. (11)
By definition
Nn = Re(φn
1 )I + Im(φn
1 )H− Re(φn
2 )F + Im(φn
2 )E ,
where I is the 2×2 identity matrix. One can find that conditions (10)–(11) are satisfied if and
only if
Im(φn
1 )Im(φn
2 ) + Re(φn
1 )Re(φn
2 ) = 0. (12)
We can construct φn
1 , φn
2 such that one of the following two cases holds: a) φn
1 ∈ R, φn
2 ∈ iR, or
b) φn
1 ∈ iR, φn
2 ∈ R. This is the context of the following proposition.
Prolongation Loop Algebras for a Solitonic System of Equations 5
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 e = β
γ , Sω = 0, Uω = 0
and Rω = Rinit a nonzero constant. The corresponding potentials become h0 = 1
2 , h1 =
−1
2γωg(ω)Rinit, f0 = f1 = e0 = e1 = 0. Following the procedure of the Bäcklund transform we
find, φ1
1 = Re(c1
1)e
x1 + iIm(c1
1)e
x1 , φ1
2 = Re(c1
2)e
−x1 + iIm(c1
2)e
−x1 , where x1 = 2((v1 +m1h0)ζ−
m1h0τ) ∈ R, and v1 is an expression independent of ω, λ, ζ and τ . We choose c1
1 ∈ R and c1
2 ∈ iR
so that φ1
1 ∈ R, φ1
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:
N2n−1 ∈ span{I, E}, and N2n ∈ span{H,F}. This yields Gn(λ) ∈ span{λI,H,F}, ∀n ∈ N.
Using the definition Ψn = Gn(λ = imn)Ψn−1(λ = imn) we deduce that Ψ2n−1 ∈
{(
R iR
iR R
)}
,
and Ψ2n ∈
{(
iR R
R iR
)}
. Therefore, if we choose ~cn ∈
{(
R
iR
)}
, and use ~Φn = ~Ψn~cn, we obtain
that ~Φ2n−1 ∈
{(
R
iR
)}
, and ~Φ2n ∈
{(
iR
R
)}
as desired. �
We aim to give an integral-free representation of Qn(λ) so that its λ-structure becomes
apparent. We begin with Q1(λ) and then generalize the construction for Qn(λ). The general
form of Q1(λ) is the following,
Q1(λ) = λ(h0H+ f0F) + e1
0E +
∫ ∞
−∞
1
(ω2 − λ2)
[
λ(h1
1H+ f1
1F) + e1
1E
]
dω.
The one soliton potentials are given as,
e1
0 = 2m1h0 sech(x1), e1
1 =
2m1ω
2h1
(ω2 + m2
1)
sech(x1),
f1
1 =
2m2
1h1
(ω2 + m2
1)
sech(x1) tanh(x1), h1
1 =
h1
(ω2 + m2
1)
[
ω2 + m2
1 − 2m2
1 sech2(x1)
]
,
where x1 = 2((v1+m1h0)ζ−m1h0τ)+ln(c1
1/c1
2), h1 = −1
2γωg(ω)Rinit and v1 = −1
2
γRinitm1ω0
((m1+σ)2+ω2
0)
.
We write the last three in the following more convenient form,
e1
1 =
β1ω
3
(ω2 + m2
1)((ω − ω0)2 + σ2)
, (13)
f1
1 =
β2ω
(ω2 + m2
1)((ω − ω0)2 + σ2)
, (14)
h1
1 =
β3ω
3 + β4ω
(ω2 + m2
1)((ω − ω0)2 + σ2)
, (15)
where β1, β2, β3, β4 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 Q1. We define the complex-
valued function,
h(z) =
z
(z2 − λ2)(z2 + m2
1)((z − ω0)2 + σ2)
{
λ[(β3z
2 + β4)H+ β2F ] + β1z
2E
}
,
which has three poles in the upper half complex plane at z1 = λ, z2 = im1, and z3 = ω0 + iσ.
We integrate around a simple contour that consists of a semicircle of radius R in the upper-half
complex plane, call it CR, and the segment on the real axis from −R to R. We choose R big
6 M.A. Agrotis
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 7→∞
∫
CR
h(z)dz = 0, and thus using the Cauchy integral formula we obtain,
Q1 = λ(h0H+ f0F) + e1
0E + 2πi
3∑
k=1
Resz=zk
h(z),
which can be written as,
Q1(λ) = λ(h0H+ f0F) + e1
0E +
1
(λ2 + m2
1)
[λ(δ1H+ δ2F) + δ3E ]
+
λ(δ̃1H+ δ̃2F) + δ̃3E + λ3( ˜̃
δ1H+ ˜̃
δ2F) + λ2 ˜̃
δ3E
((ω2
0 − σ2 − λ2)2 + 4ω2
0σ
2)
, (16)
where δj , δ̃j ,
˜̃
δj , j = 1, 2, 3 are independent of λ and ω. Let S1 = {δ1, δ2, δ3}, S2 = {δ̃1δ̃2, δ̃3} and
S3 = { ˜̃
δ1
˜̃
δ2,
˜̃
δ3}. The elements of each set S1, S2 or S3 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 hn
1 , fn
1 , en
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:
hn
1 =
ω
n∑
k=0
γkω
2k
((ω − ω0)2 + σ2)
n∏
k=1
(ω2 + m2
k)
, fn
1 =
ω
n∑
k=0
δkω
2k
((ω − ω0)2 + σ2)
n∏
k=1
(ω2 + m2
k)
,
en
1 =
ω
n∑
k=0
εkω
2k
((ω − ω0)2 + σ2)
n∏
k=1
(ω2 + m2
k)
.
We note that γk, δk, εk, k = 1, . . . , n, are analytic functions of ζ and τ, and do not depend on ω.
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 NkHN−1
k ∈ span{H,F} for any k ∈ N we get,
Qn(λ) = λ(h0H+ f0F) + en
0E +
∫ ∞
−∞
ω
(ω2 − λ2)((ω − ω0)2 + σ2)
n∏
k=1
(ω2 + m2
k)
×
{
λ
[
n∑
k=0
γkω
2kH+
n∑
k=0
δkω
2kF
]
+
n∑
k=0
εkω
2kE
}
dω.
By definition
Qn(λ) = λ(h0H+ f0F) + en
0E +
∫ ∞
−∞
1
(ω2 − λ2)
[λ(hn
1H+ fn
1 F) + en
1E ]dω. (17)
Therefore, by equating the last two expressions of Qn(λ) we prove the proposition. We note
that some of the coefficients γk, δk, εk may be equal to zero. For example, for n = 1, δ1 = 0
and ε0 = 0. �
Prolongation Loop Algebras for a Solitonic System of Equations 7
The form of the n-soliton potentials can be used to identify the λ-structure of a general n-soliton
loop element Qn(λ).
Proposition 3.
Qn(λ) = λ(h0H+ f0F) + en
0E +
λ(δ̃1H+ δ̃2F) + δ̃3E + λ3( ˜̃
δ1H+ ˜̃
δ2F) + λ2 ˜̃
δ3E
((ω2
0 − σ2 − λ2)2 + 4ω2
0σ
2)
+
n∑
k=1
1
(λ2 + m2
k)
[λ(δk
1H+ δk
2F) + δk
3E ]. (18)
The proof follows the same idea that was used to derive the integral-free form (16) of Q1(λ) and
is omitted.
Let Sk = {δk
1 , δk
2 , δk
3}, k = 1, . . . , n, Sn+1 = {δ̃1, δ̃2, δ̃3} and Sn+2 = { ˜̃
δ1,
˜̃
δ2,
˜̃
δ3}. The elements
of each set Sk, k = 1, . . . , n + 2 are functions of the m-soliton potentials for m = 1, . . . , k and
they satisfy the ib-rMB equations.
3 Embedding
We call αk = imk, k = 1, . . . , n, αn+1 = ω0 − iσ and αn+2 = ω0 + iσ. Then Qn(λ) of (18) can
be rewritten as
Qn(λ) = λ(h0H+ f0F) + e0E +
n+2∑
k=1
X−
k
λ− αk
+
X+
k
λ + αk
,
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 Qext that extends loops of the
form (18),
Qext =
X : X =
∞∑
j=0
n+2∑
k=1
1
(λ2 − α2
k)
j
[λ(hk
jH+ fk
j F) + ek
jE ]
. (19)
We note that the coefficients hk
j , fk
j , ek
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 αk, whereas
the lower index indicates the order of the pole α2
k.
We embed this Lie algebra into a larger one,
Q =
X : X =
∞∑
j=1
Xj−1λ
j−1 +
2(n+2)∑
k=1
Xk
j
(λ− αk)j
, (20)
where we have set αk+n+2 = −αk, for k = 1, . . . , n + 2. We note that for Q to be a Lie algebra
a finiteness condition needs to imposed. Namely, Xj = 0 and Xk
j = 0, ∀ k = 1, . . . , 2(n + 2) and
j ≥ j0 for some j0 ∈ N.
4 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
8 M.A. Agrotis
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 =
X ∈ Q : X =
∞∑
j=1
2(n+2)∑
k=1
Xk
j
(λ− αk)j
, b =
X ∈ Q : X =
∞∑
j=0
Xjλ
j
.
A non-degenerate inner product is defined on Q using the trace map. Namely,
Q×Q −→ C, (X, Y ) 7→ Tr(XY )0,
where (XY )0 denotes the matrix coefficient of λ0 in the product XY . We note that the inner
product is ad-invariant. That is, (X, adY Z) + (adY X, Z) = 0, where adXY = [X, Y ]. The
perpendicular complements of the subalgebras a and b with respect to this inner product take
the form,
a⊥ =
X ∈ Q : X = X0 +
∞∑
j=1
2(n+2)∑
k=1
Xk
j
(λ− αk)j
, b⊥ =
X ∈ Q : X =
∞∑
j=1
Xjλ
j
.
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 λ1. Therefore we translate the space a⊥ by β = λX1 ∈ b⊥.
We note that (β, [X, Y ]) = 0 ∀X, Y ∈ a, which is a necessary condition for the AKS theorem.
We consider the set,
a⊥ + β =
X ∈ Q : X = X0 + λX1 +
∞∑
j=1
2(n+2)∑
k=1
Xk
j
(λ− αk)j
, (21)
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
{Φ,Ψ}(X) = −
(
X,
[∏
b
∇Φ(X),
∏
b
∇Ψ(X)
])
, ∀X ∈ a⊥ + β. (22)
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,
(Y,∇H(X)) = lim
ε→0
H(X + εY )−H(X)
ε
. (23)
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 Hproj, F proj their
projection to the subspace a⊥+β then {Hproj, F proj} = 0, and the Hamiltonian system associated
with the invariant function H is given as,
d
dt
(β + α) = −ad∗∏
a∇H(β+α)(β + α), ∀β + α ∈ β + a⊥. (24)
Prolongation Loop Algebras for a Solitonic System of Equations 9
Remark 1. ad∗ is negative the dual of the ad map.
Remark 2. The definition of H being an invariant function is ad∗∇H(x)x = 0, ∀x ∈ Q.
Remark 3. The existence of an invariant inner product allows one to identify the ad-map
with ad∗, because on one hand by definition (adxy, z) = −(y, ad∗xz), and on the other a simple
calculation using the definition of the inner product shows that (adxy, z) = −(y, adxz).
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 {Φk}k∈N via
the following operator,
Mk(X) =
2(n+2)∏
i=1
(λ− αi)kX.
The invariant functions are defined as Φk(X) = 1
2(Mk(X), X). We compute the gradient of the
Φk(X). By definition,
(Y,∇Φk(X)) = lim
ε→0
Φk(X + εY )− Φk(X)
ε
=
2(n+2)∏
i=1
(λ− αi)kX, Y
= (Y, Mk(X)).
Thus ∇Φk(X) = Mk(X). According to Remarks 2 and 3, to actually demonstrate the invariance
of Φk we must show that [∇Φk(X), X] = 0. We have that [∇Φk(X), X]=
2(n+2)∏
i=1
(λ−αi)k[X, X]=0.
Therefore the functions Φk, k ∈ N are invariant and can be used in the context of the AKS
theorem, which reads as follows,
dX
dtk
= −
[∏
a
∇Φk(X), X
]
=
[∏
b
∇Φk(X), X
]
,
since 0 = [∇Φk(X), X] = [
∏
a∇Φk(X), X] + [
∏
b∇Φk(X), X]. We used the subscript k for
the time variable tk to distinguish between the different dynamical evolutions of the systems
associated with the Hamiltonian functions Φk. For each k ∈ N the following systems are in
involution:
dX
dtk
=
[∏
b
Mk(X), X
]
. (25)
For k = 0 and a truncated X of the form (19) where hi
j , f
i
j , e
i
j = 0 for j ≥ 2 and i = 1, . . . , n+2,
X = λ(h0H+ f0F) + e0E +
n+2∑
i=1
1
(λ2 − α2
i )
[λ(hi
1H+ f i
1F) + ei
1E ], (26)
the Hamiltonian system (25) becomes
dQ(1)
dt0
= [Q(0), Q(1)], (27)
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).
10 M.A. Agrotis
5 Extended flow
We consider a general element of the Lie algebra Q of the form,
X = λ(h0H+ f0F) + e0E +
∞∑
j=1
n+2∑
i=1
1
(λ2 − α2
i )j
[λ(hi
jH+ f i
jF) + ei
jE ], (28)
or equivalently,
X = λ(h0H+ f0F) + e0E +
∞∑
j=1
j(n+1)∑
m=0
λ2m+1(h2m+1
j H+ f2m+1
j F) + λ2me2m
j E
n+2∏
i=1
(λ2 − α2
i )j
. (29)
We remark that hi
j , f i
j , ei
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 h2m+1
j ,
f2m+1
j , e2m
j in (29), where j indicates the order of the pole at λ2 = α2
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,
dX
dt0
=
[∏
b
∇Φ0(X), X
]
,
where ∇Φ0(X) = M0(X) = X. We write X as follows:
X =
∞∑
j=0
Pj
n+2∏
i=1
(λ2 − α2
i )j
,
where Pj is a polynomial in λ given as
Pj(λ) =
j(n+1)∑
m=0
λ2m+1(h2m+1
j H+ f2m+1
j F) + λ2me2m
j E , j = 1, 2, . . . ,
and P0 = λ(h0H + f0F) + e0E . The degree of Pj(λ) is given by, deg(Pj(λ)) = 2j(n + 1) + 1.
Projecting ∇Φ0(X) to the subalgebra b is equivalent to keeping the polynomial part of X, which
is P0. Thus the Hamiltonian flow for k = 0 takes the form,
dX
dt0
=
∞∑
j=1
P0,
Pj
n+2∏
i=1
(λ2 − α2
i )j
,
which unravels to
dX
dt0
=
∞∑
j=1
j(n+1)∑
m=0
λ2m+1(dhm
0,jH+ dfm
0,jF) + λ2m+2dem
0,jE
/(n+2∏
i=1
(λ2 − α2
i )
j
)
. (30)
The coefficients of the matrices H, F , and E appearing in (30) are defined as follows,
dhm
0,j = 2
(
f0e
2m
j − e0f
2m+1
j
)
, dfm
0,j = 2
(
e0h
2m+1
j − h0e
2m
j
)
,
Prolongation Loop Algebras for a Solitonic System of Equations 11
dem
0,j = 2
(
h0f
2m+1
j − f0h
2m+1
j
)
.
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
dX
dt0
=
∞∑
j=1
j(n+1)∑
m=0
λ2m+1(dhm
0,jH+ dfm
0,jF) +
j(n+1)∑
m=1
λ2mdem−1
0,j E
+
j(n+1)∑
m=(j−1)(n+1)
Cm−(j−1)(n+1)λ
2mde
j(n+1)
0,j E
/(n+2∏
i=1
(λ2 − α2
i )
j
)
+
∞∑
j=2
λ2(j−1)(n+1)de
j(n+1)
0,j E
/(n+2∏
i=1
(λ2 − α2
i )
j−1
)
+ den+1
0,1 E ,
where Cn+1−m = (−1)m
n+2∑
il1 6=il2
α2
i1
· · ·α2
im+1
, ilp ∈ {1, . . . , n + 2}. On the other hand by the
definition of X we have that,
dX
dt0
= λ(
dh0
dt0
H+
df0
dt0
F) +
de0
dt0
E
+
∞∑
j=1
(
λ2m+1(
dh2m+1
j
dt0
H+
df2m+1
j
dt0
F) + λ2m
de2m
j
dt0
E
)/(n+2∏
i=1
(λ2 − α2
i )
j
)
.
By equating the different powers of λ we disclose the system induced by the Hamiltonian func-
tion Φ0. As previously noted, for a fixed j ∈ N, the elements of the sets Sm = {h2m+1
j , f2m+1
j , e2m
j }
satisfy the same system of equations for any m = 0, 1, . . . , j(n + 1). Therefore it suffices to con-
sider only one such set. Without loss of generality we choose Sj(n+1). The evolution equations
take the form:
dh0
dt0
=
df0
dt0
= 0,
de0
dt0
= 2
(
h0f
2n+3
1 − f0h
2n+3
1
)
,
dh
2j(n+1)+1
j
dt0
= 2
(
f0e
2j(n+1)
j − e0f
2j(n+1)+1
j
)
,
df
2j(n+1)+1
j
dt0
= 2
(
e0h
2j(n+1)+1
j − h0e
2j(n+1)
j
)
,
de
2j(n+1)
j
dt0
= 2
(
h0f
2j(n+1)−1
j − f0h
2j(n+1)−1
j
)
+ 2
(
n+2∑
i=1
a2
i
)(
h0f
2j(n+1)+1
j − f0h
2j(n+1)+1
j
)
+ 2
(
h0f
2(j+1)(n+1)+1
j+1 − f0h
2(j+1)(n+1)+1
j+1
)
.
To reveal the extended flow for the ib-rMB equations we set j=1:
dh0
dt0
=
df0
dt0
= 0,
de0
dt0
= 2
(
h0f
2n+3
1 − f0h
2n+3
1
)
,
dh2n+3
1
dt0
= 2
(
f0e
2n+2
1 − e0f
2n+3
1
)
,
df2n+3
1
dt0
= 2
(
e0h
2n+3
1 − h0e
2n+2
1
)
, (31)
12 M.A. Agrotis
de2n+2
1
dt0
= 2
(
h0f
2n+1
1 − f0h
2n+1
1
)
+ 2
(
n+2∑
i=1
a2
i
)(
h0f
2n+3
1 − f0h
2n+3
1
)
+ 2
(
h0f
4n+5
2 − f0h
4n+5
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 e2n+2
1 .
If h4n+5
2 = f4n+5
2 = 0, then the system reduces to the dynamical equations that the n-soliton
potentials of the ib-rMB equations satisfy.
6 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,
∂e0
∂t
= {e0,Φ0},
∂h2n+3
1
∂t
= {h2n+3
1 ,Φ0}, (32)
∂f2n+3
1
∂t
= {f2n+3
1 ,Φ0},
∂e2n+2
1
∂t
= {e2n+2
1 ,Φ0}.
The Hamiltonian functions for the systems (25) described in the context of the AKS theorem
in Section 4 are defined as
Φk(X) =
1
2
(Mk(X), X) =
1
2
Tr
2(n+2)∏
i=1
(λ− αi)kX2
0
.
We let k = 0 and consider a general X of the form,
X = λ(h0H+ f0F) + e0E
+
∞∑
j=1
j(n+1)∑
m=0
λ2m+1(h2m+1
j H+ f2m+1
j F) + λ2me2m
j E
n+2∏
i=1
(λ2 − α2
i )j
.
The Hamiltonian function is found to be:
Φ0(X) =
1
2
Tr
(
X2
)
0
= e2
0 + 2
(
h0h
2n+3
1 + f0f
2n+3
1
)
. (33)
We define the following functionals, relevant to the potentials that appear in the Hamiltonian:
h0(X) = h0 coefficient of λH, h2n+3
1 (X) = h2n+3
1 coefficient of
λ2n+3
n+2∏
i=1
(λ2 − α2
i )
H,
f0(X) = f0 coefficient of λF , f2n+3
1 (X) = f2n+3
1 coefficient of
λ2n+3
n+2∏
i=1
(λ2 − α2
i )
F ,
e0(X) = e0 coefficient of E , e2n+2
1 (X) = e2n+2
1 coefficient of
λ2n+2
n+2∏
i=1
(λ2 − α2
i )
E .
Prolongation Loop Algebras for a Solitonic System of Equations 13
Using definition (23) we compute the gradients of these functionals:
∇h0(X) = −1
2
λ2n+3
n+2∏
i=1
(λ2 − α2
i )
H, ∇h2n+3
1 (X) = −1
2
λH,
∇f0(X) = −1
2
λ2n+3
n+2∏
i=1
(λ2 − α2
i )
F , ∇f2n+3
1 (X) = −1
2
λF , (34)
∇e0(X) = −1
2
E , ∇e2n+2
1 (X) = −1
2
λ2E .
For example, to obtain ∇h0(X) we consider the equation
(Y,∇h0(X)) = lim
ε→0
h0(X + εY )− h0(X)
ε
,
which implies that Tr(Y∇h0(X))0 = h0(Y ). Therefore ∇h0(X) is the element of Q such that the
constant term (with respect to λ) of the product Y∇h0(X) has trace that equals precisely h0(Y ).
We note that the matrices HF , FE , EH are traceless whereas H2 = F2 = E2 = −I. Having
that in mind, we find that ∇h0(X) = −1
2
λ2n+3
n+2∏
i=1
(λ2−α2
i )
H.
The Poisson brackets between the potentials appearing in the Hamiltonian can be computed
using definition (22). For instance,
{
e0, h
2n+3
1
}
(X) = −
(
X,
[∏
b
∇e0(X),
∏
b
∇h2n+3
1 (X)
])
= −
(
X,
1
2
λF
)
= −Tr
(
1
2
λXF
)
0
= f2n+3
1 .
In a similar fashion we obtain the rest of the Poisson brackets,{
e0, h
2n+3
1
}
= f2n+3
1 ,
{
f2n+3
1 , e0
}
= h2n+3
1 ,
{
h2n+3
1 , f2n+3
1
}
= e2n+2
1 ,{
e2n+2
1 , h2n+3
1
}
= f2n+1
1 +
(
n+2∑
i=1
a2
i
)
f2n+3
1 + f4n+5
2 , (35)
{
f2n+3
1 , e2n+2
1
}
= h2n+1
1 +
(
n+2∑
i=1
a2
i
)
h2n+3
1 + h4n+5
2 .
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 Φ0.
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 h0, f0, e0 and h0, f0, e0):
hN (X) =
N+1∑
j=1
h
2j(n+2)−2N−1
j , fN (X) =
N+1∑
j=1
f
2j(n+2)−2N−1
j ,
eN (X) =
N+1∑
j=1
e
2j(n+2)−2N−2
j .
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 = {e0,hN ,fN , eN : N = 0, 1, 2, . . .} includes all
14 M.A. Agrotis
the dynamical quantities that enter the AKS flows (25). The gradients of these higher order
functionals can be computed using definition (23). For instance,
∇h1(X) = −1
2
λ3H−
(
n+2∑
i=1
α2
i
)
∇h0, ∇f1(X) = −1
2
λ3F −
(
n+2∑
i=1
α2
i
)
∇f0,
∇e1(X) = −1
2
λ4E −
(
n+2∑
i=1
α2
i
)
∇e0,
where ∇h0 = ∇h2n+3
1 , ∇f0 = ∇f2n+3
1 , ∇e0 = ∇e2n+2
1 , and are given in (34).
Working in a similar manner as in the example for {e0, h
2n+3
1 }, one can find the Poisson
brackets between the higher order functionals, i.e. {e0,h1}(X) = f1(X).
7 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.
Acknowledgements
The author would like to thank P. Shipman for useful discussions and the Cyprus Research
Promotion Foundation for support through the grant CRPF0504/03.
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http://arxiv.org/abs/nlin.SI/0512068
1 Introduction
2 Phase space
3 Embedding
4 Application of the Adler-Kostant-Symes theorem
5 Extended flow
6 Hamiltonian functions and Poisson brackets
7 Summary
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