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Acta Polytechnica Vol. 51 No. 1/2011

Bidifferential Calculus, Matrix SIT and Sine-Gordon Equations

A. Dimakis, N. Kanning, F. Müller-Hoissen

Abstract

We express a matrix version of the self-induced transparency (SIT) equations in the bidifferential calculus framework.
An infinite family of exact solutions is then obtainedby application of a general result that generates exact solutions from
solutions of a linear system of arbitrary matrix size. A side result is a solution formula for the sine-Gordon equation.

Keywords: bidifferential calculus, integrable system, self-induced transparency, sine-Gordon.

1 Introduction
The bidifferential calculus approach (see [1] and the
references therein) aims to extract the essence of in-
tegrability aspects of integrable partial differential or
difference equations (PDDEs) and to express them,
and relations between them, in a universal way, i.e.
resolved from specific examples. A powerful, though
simple to prove, result [1, 2, 3] (see section 6) gener-
ates families of exact solutions from a matrix linear
system. In the following we briefly recall the basic
framework and then apply the latter result to a ma-
trix generalization of the SIT equations.

2 Bidifferential calculus
A graded algebra is an associative algebra Ω over C
with a direct sum decomposition Ω =

⊕
r≥0
Ωr into a

subalgebra A := Ω0 and A-bimodules Ωr, such that
Ωr Ωs ⊆ Ωr+s. A bidifferential calculus (or bidif-
ferential graded algebra) is a unital graded algebra
Ω equipped with two (C-linear) graded derivations
d, d̄ : Ω → Ω of degree one (hence dΩr ⊆ Ωr+1,
d̄Ωr ⊆ Ωr+1), with the properties

d2z =0 ∀z ∈ C , where dz := d̄− z d , (1)

and the graded Leibniz rule dz(χ χ
′) = (dz χ)χ

′ +
(−1)r χdzχ′, for all χ ∈ Ωr and χ′ ∈ Ω.

3 Dressing a bidifferential
calculus

Let (Ω,d, d̄) be a bidifferential calculus. Replacing
dz in (1) by Dz := d̄− A − z d with a 1-form A ∈ Ω1
(in the expression for Dz to be regarded as a multi-
plication operator), the resulting condition D2z = 0
(for all z ∈ C) can be expressed as

dA =0= d̄A − A A . (2)

If (2) is equivalent to a PDDE, we have a bidiffer-
ential calculus formulation for it. This requires that
A depends on independent variables and the deriva-
tions d, d̄ involve differential or difference operators.
Several ways exist to reduce the two equations (2) to
a single one:
(1) We can solve the first of (2) by setting A = dφ.
This converts the second of (2) into

d̄dφ =dφ dφ . (3)

(2) The second of (2) can be solved by setting A =
(d̄g)g−1. The first equation then reads

d
(
(d̄g)g−1

)
=0 . (4)

(3) More generally, setting A = [d̄g − (dg)Δ] g−1,
with someΔ ∈ A, wehave d̄A−A A =(dA)gΔg−1+
(dg)(d̄Δ − (dΔ)Δ)g−1. As a consequence, if Δ is
chosen such that d̄Δ = (dΔ)Δ, then the two equa-
tions (2) reduce to

d
(
[d̄g − (dg)Δ] g−1

)
=0 . (5)

With the choice of a suitable bidifferential calcu-
lus, (3) and (4), or more generally (5), have been
shown to reproduce quite a number of integrable
PDDEs. This includes the self-dualYang-Mills equa-
tion, in which case (3) and (4) correspond to well-
known potential forms [1]. Having found a bidiffer-
ential calculus in terms ofwhich e.g. (3) is equivalent
to a certain PDDE, it is not in general guaranteed
that also (4) represents a decent PDDE. Then the
generalization (5) has a chance to work (cf. [1]). In
such a case, the Miura transformation

[d̄g − (dg)Δ] g−1 =dφ (6)

is a hetero-Bäcklund transformation relating solu-
tions of the two PDDEs.
Bäcklund, Darboux and binary Darboux trans-

formations can be understood in this general frame-
work [1], and there is a construction of an infinite

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Acta Polytechnica Vol. 51 No. 1/2011

set of (generalized) conservation laws. Exchanging

d and d̄ leads to what is known in the literature as
‘negative flows’ [3].

4 A matrix generalization of
SIT equations and its
Miura-dual

Let A =Mat(n, n, C∞(R2)), the algebraof n×n ma-
trices of smooth functions on R2. LetΩ= A⊗

∧
(C2)

with the exterior algebra
∧
(C2) of C2. In terms of

coordinates x, y of R2, a basis ζ1, ζ2 of
1∧
(C2), and a

constant n × n matrix J, maps d and d̄ are defined
as follows on A,

df =
1
2
[J, f]⊗ ζ1 + fy ⊗ ζ2 ,

d̄f = fx ⊗ ζ1 +
1
2
[J, f]⊗ ζ2

(see also [4]). They extend in an obvious way (with
dζi = d̄ζi = 0) to Ω such that (Ω,d, d̄) becomes a
bidifferential calculus. We find that (3) is equivalent
to

φxy =
1
2

[
[J, φ], φy −

1
2
J

]
. (7)

Let n =2m and J =block-diag(I, −I),where I = Im
denotes the m × m identity matrix. Decomposing φ
into m × m blocks, and constraining it as follows,

φ =

(
p q

q −p

)
, (8)

(7) splits into the two equations

pxy =(q
2)y , qxy = q − pyq − qpy . (9)

We refer to them as matrix-SIT equations (see sec-
tion 5), not purporting that they havea similar phys-
ical relevance as in the scalar case. TheMiura trans-
formation (6) (with Δ=0) now reads

gx g
−1 =

1
2
[J, φ] ,

1
2
[J, g]g−1 = φy . (10)

Writing

g =

(
a b

c d

)
,

with m × m matrices a, b, c, d, and assuming that a
and its Schur complement S(a)= d−c a−1b is invert-
ible (which implies that g is invertible), (10)with (8)
requires

b = −c a−1d ,
ax = −cx a−1c , (11)
dx = −cx a−1c a−1d .

The last equation can be replaced by dx d
−1 =

ax a
−1. Invertibility of S(a) implies that d and I+r2

are invertible, where r := c a−1. The conditions (11)
are necessary in order that theMiura transformation
relates solutions of (9) to solutions of its ‘dual’

(gx g
−1)y =

1
4
[gJg−1, J] , (12)

obtained from (4). Taking (11) into account, the
Miura transformation reads

q = −cx a−1 = −rx − r ax a−1 ,
qy = −r(I + r2)−1 , (13)
py = I − (I + r2)−1 .

As a consequence, we have

qy
2 + py

2 = py . (14)

Furthermore, the second of (11) and the first of (13)
imply axa

−1 = qr. Hence we obtain the system

rx = −q − r q r , qy = −r(I + r2)−1 , (15)

which may be regarded as a matrix or ‘noncommu-
tative’ generalization of the sine-Gordon equation.
There are various such generalizations in the lit-
erature. The first equation has the solution q =

−
∞∑

k=0

(−1)k rk rx rk, if the sum exists. Alternatively,

we can express this as q = −(I +rLrR)−1(rx), where
rL (rR) denotes themap of left (right)multiplication
by r. This canbe used to eliminate q fromthe second
equation, resulting in(

(I + rLrR)
−1(rx)

)
y
= r (I + r2)−1 . (16)

If r = tan(θ/2)π with a constant projection π (i.e.
π2 = π) and a function θ, then (16) reduces to the
sine-Gordon equation

θxy =sinθ . (17)

(15) canbe obtaineddirectly from(12) as follows,
by setting

g =

(
a −c
c a

)
=

(
I −r
r I

)
a ,

hence

g−1 = a−1
(

I r

−r I

)
(I + r2)−1 .

This leads to(
(rx r + rρ r + ρ)(I + r

2)−1
)
y
= 0 ,(

(rx + rρ − ρ r)(I + r2)−1
)
y
= r(I + r2)−1 ,

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Acta Polytechnica Vol. 51 No. 1/2011

where ρ := axa
−1. Setting an integration ‘con-

stant’ to zero, the first equation integrates to ρ =
−rxr−rρ r. With its help, the second can bewritten
as (rx +rρ)y = r(I +r

2)−1. Since q = −(ra)x a−1 =
−rx − r ρ, this is the second of (15). The first follows
noting that qr = ρ.

5 Sharp line SIT equations
and sine-Gordon

We consider the scalar case, i.e. m = 1. In-
troducing E = 2

√
αq with a positive constant α,

P = 2qy, N = 2py − 1, and new coordinates z, t
via x =

√
α(z − t) and y =

√
αz, the system (9) is

transformed into

Pt = E N , Nt = −E P ,

and the relation between E and P takes the form

Ez +Et = α P .

These are the sharp line self-induced transparency
(SIT) equations [5, 6, 7]. We note that P2 + N2 is
conserved. Indeed, as a consequence of (14), we have
P2 +N2 =1. Writing P = −sinθ and N = −cosθ,
reduces the first two equations to E = θt. Expressed
in the coordinates x, y, the third then becomes the
sine-Gordon equation (17) (cf. [6]). As a consequence
of the above relations, q and p depend as follows on
θ,

q = −
1
2
θx ,

qy = −
1
2
sinθ , (18)

py =
1
2
(1−cosθ) .

These areprecisely the equations that result fromthe
Miura transformation (10) (or from (13)), choosing

g =

⎛
⎜⎜⎝ cos

θ

2
−sin

θ

2

sin
θ

2
cos

θ

2

⎞
⎟⎟⎠ ,

and (12) becomes the sine-Gordon equation (17).
The conditions (11) are identically satisfied as a con-
sequence of the form of g.

6 A universal method of
generating solutions from a
matrix linear system

Theorem 1 Let (Ω,d, d̄) be a bidifferential calculus
with Ω= A ⊗

∧
(C2), where A is the algebra of ma-

trices with entries in some algebra B (where the prod-
uct of two matrices is defined to be zero if the sizes

of the two matrices do not match). For fixed N, N ′,
let X ∈ Mat(N, N, B) and Y ∈ Mat(N ′, N, B) be
solutions of the linear equations

d̄X = (dX)P ,

d̄Y = (dY )P ,

R X − X P = −Q Y ,

with d-constant and d̄-constant matrices P , R ∈
Mat(N, N, B), and Q = Ṽ Ũ, where Ũ ∈
Mat(n, N ′, B) and Ṽ ∈ Mat(N, n, B) are d- and d̄-
constant. If X is invertible, the n×n matrix variable

φ = Ũ Y X−1Ṽ ∈ Mat(n, n, B)

solves d̄φ =(dφ)φ+dϑ with ϑ = Ũ Y X−1RṼ , hence
(by application of d) also (3). �

There is a similar result for (5) [3]. The Miura
transformation is a corresponding bridge.

7 Solutions of the matrix SIT
equations

From Theorem 1 we can deduce the following result,
using straightforward calculations [8], analogous to
those in [2] (see also [3]).
Proposition 2 Let S ∈ Mat(M, M, C) be invert-
ible, U ∈ Mat(m, M, C), V ∈ Mat(M, m, C), and
K ∈ Mat(M, M, C) a solution of the Sylvester equa-
tion

SK + KS = V U . (19)

Then, with Ξ = e−Sx−S
−1

y and any p0 ∈
Mat(m, m, C) (more generally x-dependent),

q =U Ξ (IM +(KΞ)
2)−1V ,

p=p0 − U ΞKΞ (IM +(KΞ)2)−1V
(20)

(assuming the inverse exists) is a solution of (9). �

If the matrix S satisfies the spectrum condition

σ(S)∩ σ(−S)= ∅ (21)

(where σ(S)denotes the setof eigenvaluesof S), then
the Sylvester equation (19) has a unique solution K
(for any choice of the matrices U , V ), see e.g. [9].
Bya lengthycalculation [8] one canverifydirectly

that the solutions in Proposition 2 satisfy (14). Al-
ternatively, one can show that these solutions actu-
ally determine solutions of the Miura transformation
(cf. [3]), andwe have seen that (14) is a consequence.
There is a certain redundancy in the matrix data

that determine the solutions (20) of (9). This can be

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Acta Polytechnica Vol. 51 No. 1/2011

narroweddownbyobserving that the following trans-
formations leave (19) and (20) invariant (see also the
NLS case treated in [2]).
(1)Similarity transformation with an invertible M ∈
Mat(M, M, C):

S �→ M SM −1 , K �→ M KM −1 ,
V �→ M V , U �→ U M −1 .

As a consequence, we can choose S in Jordannormal
form without restriction of generality.
(2)Reparametrization transformation with invertible
A, B ∈ Mat(M, M, C):

S �→ S , K �→ B−1KA−1 , V �→ B−1V ,
U �→ U A−1 , Ξ �→ ABΞ .

(3) Reflexion symmetry:

S �→ −S , K �→ −K−1 , V �→ K−1V ,
U �→ U K−1 , p0 �→ p0 − U K−1V .

This requires that K is invertible. More generally,
such a reflexion can be applied to any Jordan block
of S and then changes the sign of its eigenvalue [8]
(see also [10, 2]). The Jordan normal form can be
restored afterwards via a similarity transformation.
The following result is easily verified [8].

Proposition 3 Let S, U , V be as in Proposition 2
and T ∈ Mat(M, M, C) invertible.
(1) Let T be Hermitian (i.e. T † = T) and such
that S† = T ST −1, U = V †T . Let K be a so-
lution of (19), which can then be chosen such that
K† = T KT −1. Then q and p given by (20) with
p
†
0 = p0 are both Hermitian and thus solve the Her-
mitian reduction of (9).
(2) Let T̄ = T −1 (where the bar means complex
conjugation) and S̄ = T ST −1, Ū = U T −1 and
V̄ = T V . Let K be a solution of (19), which can
then be chosen such that K̄ = T KT −1. Then q and
p given by (20) with p̄0 = p0 satisfy q̄ = q and p̄ = p,
and thus solve the complex conjugation reduction of
(9). �

8 Rank one solutions
Let M = 1. We write S = s, U = u, V = vT,
K = k (where T means the transpose) and Ξ = ξ =

e−sx−s
−1y. Then (19) yields k = (vTu)/(2s). From

(20) we obtain

q =
2s k ξ
1+(kξ)2

π , p = p̃0 +
2s

1+(kξ)2
π ,

p̃0 := p0 −2s π , π :=
uvT

vTu
.

The Miura transformation (13) implies r = −qy (I −
py)

−1, and we obtain

r = −
2kξ

1− (kξ)2
π ,

which is singular. But θ = −2arctan(2kξ/[1−(kξ)2])
is the single kink solution of the sine-Gordon equa-
tion (17).

9 Solutions of the scalar
(sharp line) SIT equations

We rewrite p in (20), where now m =1, as follows,

p = p0 − tr
(
(SK + KS)ΞKΞ (IM +(KΞ)

2)−1
)

= p0 +tr
(
(IM +(KΞ)

2)x (I M +(KΞ)
2)−1

)
= p0 +

(
logdet

(
IM +(KΞ)

2
))

x
, (22)

using (19) and the identity (detM)x =tr(M xM
−1)

detM for an invertiblematrix function M. q in (20)
can be expressed as

q =2tr
(
SKΞ (IM +(KΞ)

2)−1
)

.

In particular, if S is diagonal with eigenvalues si,
i =1, . . . , M, and satisfies (21), then the solution K
of the Sylvester equation (19), which now amounts
to rank(SK + KS) = 1, is the Cauchy-type ma-
trix with components Kij = vi uj/(si + sj), where
ui, vi ∈ C. Figs. 1 and 2 show plots of two examples
from the above family of solutions.

Fig. 1: A scalar 2-soliton solution with S = diag(1,2)
and ui = vi =1

Fig. 2: A scalar breather solution with S = diag(1+ i,
1− i) and ui = vi =1

10 A family of solutions of the
real sine-Gordon equation

Via theMiura transformation (18), Proposition 2 de-
termines a family of sine-Gordon solutions (see also

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Acta Polytechnica Vol. 51 No. 1/2011

e.g. [6, 11, 12, 13, 14, 15, 16] for related results ob-
tained by different methods).
Proposition 4 Let S ∈ Mat(M, M, C) be invert-
ible and K ∈ Mat(M, M, C) such that rank(SK +
KS) = 1, det(IM + (KΞ)

2) ∈ R with Ξ =
e−Sx−S

−1
y, and tr(SKΞ (IM + (KΞ)

2)−1) �∈ iR
(where i is the imaginary unit). Then

θ =4arctan
( √β
1+

√
1− β

)
with

β :=
(
log |det(IM +(KΞ)2)|

)
xy

(23)

solves the sine-Gordon equation θxy = sinθ in any
open set of R2 where det(IM +(KΞ)

2) �=0.
Proof: Let p be givenby (22). Due to the assumption
det(IM +(KΞ)

2) ∈ R, py is real, hence (14) implies
|1 − 2py|2 = 1 − 4qy2. It follows that qy2 is real.
Since another of our assumptions excludes that qy is
imaginary, it follows that |1 − 2py| ≤ 1. Hence the
equation cosθ = 1 − 2py (second of (18)) has a real
solution θ. Inserting expression (22) for p, we arrive
at cosθ =1−2

(
logdet(IM +(KΞ)

2)
)

xy
. Moreover,

(14) shows that py ≥ 0 and thus 0 ≤ py ≤ 1. Using
identities for the inverse trigonometric functions, we
find (23), where β = py. �

Proposition 3 yields sufficient conditions on the
matrix data for which the last two assumptions in
Proposition 4 are satisfied.

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Aristophanes Dimakis
E-mail: dimakis@aegean.gr
Department of Financial
and Management Engineering
University of the Aegean, 41 Kountourioti Str.
GR-82100 Chios, Greece

Nils Kanning
E-mail: nils.kanning@ds.mpg.de
Max-Planck-Institute for Dynamics
and Self-Organization
Bunsenstrasse 10, D-37073 Göttingen, Germany

Folkert Müller-Hoissen
E-mail: folkert.mueller-hoissen@ds.mpg.de
Max-Planck-Institute for Dynamics
and Self-Organization
Bunsenstrasse 10, D-37073 Göttingen, Germany

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