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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 obtained by 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′) = (dzχ)χ′ +
(−1)r χ dzχ

′, 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 −AA . (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

d̄ dφ = 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 some Δ ∈ A, we have d̄A−AA = (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-dual Yang-Mills equa-
tion, in which case (3) and (4) correspond to well-
known potential forms [1]. Having found a bidiffer-
ential calculus in terms of which 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 algebra of n×n ma-

trices of smooth functions on R
2. Let Ω = A⊗

∧
(C2)

with the exterior algebra
∧
(C2) of C

2. 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 = (q2)y , qxy = q − pyq − qpy . (9)

We refer to them as matrix-SIT equations (see sec-
tion 5), not purporting that they have a similar phys-
ical relevance as in the scalar case. The Miura 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 the Miura 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 the map of left (right) multiplication
by r. This can be used to eliminate q from the 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 = sin θ . (17)

(15) can be obtained directly 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 + r2)−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 be written
as (rx + rρ)y = r(I + r2)−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 = αP .

These are the sharp line self-induced transparency
(SIT) equations [5, 6, 7]. We note that P2 + N 2 is
conserved. Indeed, as a consequence of (14), we have
P2 +N 2 = 1. Writing P = − sin θ and N = − cos θ,
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
sin θ , (18)

py =
1
2
(1− cos θ) .

These are precisely the equations that result from the
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. IfX is invertible, the n×n matrix variable

φ = ŨY X−1Ṽ ∈ Mat (n, n,B)

solves d̄φ = (dφ)φ+dϑ with ϑ = ŨY 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=UΞ (IM + (KΞ)2)−1V ,

p= p0 −UΞKΞ (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 set of eigenvalues of S), then
the Sylvester equation (19) has a unique solution K
(for any choice of the matrices U , V ), see e.g. [9].
By a lengthy calculation [8] one can verify directly

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]), and we 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

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

S �→ MSM−1 , K �→ MKM−1 ,

V �→ MV , U �→ UM−1 .

As a consequence, we can choose S in Jordan normal
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 �→ UA−1 , Ξ �→ ABΞ .

(3) Reflexion symmetry:

S �→ −S , K �→ −K−1 , V �→ K−1V ,

U �→ UK−1 , p0 �→ p0 −UK−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† = TST−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 = TST−1, Ū = UT−1 and
V̄ = TV . Let K be a solution of (19), which can
then be chosen such that K̄ = TKT−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 =
2 s k ξ

1 + (kξ)2
π , p = p̃0 +

2 s

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 = − 2 kξ

1− (kξ)2
π ,

which is singular. But θ = −2 arctan(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 (IM + (KΞ)2)−1

)
= p0 +

(
log det

(
IM + (KΞ)2

))
x

, (22)

using (19) and the identity (detM)x = tr(MxM−1)
detM for an invertible matrix functionM . q in (20)
can be expressed as

q = 2 tr
(
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 the Miura 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 = 4 arctan
( √

β

1 +
√
1− β

)
with

β :=
(
log | det(IM + (KΞ)2)|

)
xy

(23)

solves the sine-Gordon equation θxy = sin θ in any
open set of R

2 where det(IM + (KΞ)2) �= 0.
Proof: Let p be given by (22). Due to the assumption
det(IM +(KΞ)2) ∈ R, py is real, hence (14) implies
|1 − 2py|2 = 1 − 4qy

2. It follows that qy
2 is real.

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

(
log det(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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