Bidifferential calculus, matrix SIT and sine-Gordon equations

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.

1. Introduction. The bidifferential calculus approach (see [1] and the references therein) aims to extract the essence of integrability 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) generates 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 matrix generalization of the SIT equations.
3. Dressing a bidifferential calculus. Let (Ω, d,d) be a bidifferential calculus. Replacing d z in (1) by D z :=d − A − z d with a 1-form A ∈ Ω 1 (in the expression for D z to be regarded as a multiplication operator), the resulting condition D 2 z = 0 (for all z ∈ C) can be expressed as If (2) is equivalent to a PDDE, we have a bidifferential calculus formulation for it. This requires that A depends on independent variables and the derivations d,d 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) intō (2) The second of (2) can be solved by setting A = (dg) g −1 . The first equation then reads As a consequence, if ∆ is chosen such thatd∆ = (d∆) ∆, then the two equations (2) reduce to With the choice of a suitable bidifferential calculus, (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 equation, in which case (3) and (4) correspond to well-known potential forms [1]. Having found a bidifferential 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 is a hetero-Bäcklund transformation relating solutions of the two PDDEs. Bäcklund, Darboux and binary Darboux transformations can be understood in this general framework [1], and there is a construction of an infinite set of (generalized) conservation laws. Exchanging d andd leads to what is known in the literature as 'negative flows' [3].

4.
A matrix generalization of SIT equations and its Miura-dual. A = Mat(n, n, C ∞ (R 2 )) denotes the algebra of n × n matrices of smooth functions on R 2 . Let Ω = A ⊗ (C 2 ) with the exterior algebra (C 2 ) of C 2 . In terms of coordinates x, y of R 2 , a basis ζ 1 , ζ 2 of 1 (C 2 ), and a constant n × n matrix J, maps d andd are defined as follows on A, (see also [4]). They extend in an obvious way (with dζ i =dζ i = 0) to Ω such that (Ω, d,d) becomes a bidifferential calculus. We find that (3) is equivalent to Let n = 2m and J = block-diag(I, −I), where I = I m denotes the m × m identity matrix. Decomposing φ into m × m blocks, and constraining it as follows, (7) splits into the two equations We refer to them as matrix-SIT equations (see section 5), not purporting that they have a similar physical relevance as in the scalar case. The Miura transformation (6) (with ∆ = 0) now reads Writing with m × m matrices a, b, c, d, and assuming that a and its Schur complement The last equation can be replaced by d x d −1 = a x a −1 . Invertibility of S(a) implies that d and I + r 2 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' obtained from (4). Taking (11) into account, the Miura transformation reads As a consequence, we have Furthermore, the second of (11) and the first of (13) imply a x a −1 = qr. Hence we obtain the system which may be regarded as a matrix or 'noncommutative' generalization of the sine-Gordon equation.
There are various such generalizations in the literature. The first equation has the solution q = − ∞ k=0 (−1) k r k r x r k , if the sum exists. Alternatively, we can express this as q = −(I+r L r R ) −1 (r x ), where r L (r R ) denotes the map of left (right) multiplication by r. This can be used to eliminate q from the second equation, resulting in If r = tan(θ/2) π with a constant projection π (i.e. π 2 = π) and a function θ, then (16) reduces to the sine-Gordon equation (15) can be obtained directly from (12) as follows, by setting

Sharp line SIT equations and sine-Gordon.
We consider the scalar case, i.e. m = 1. Introducing E = 2 √ αq with a positive constant α, P = 2q y , N = 2p y − 1, and new coordinates z, t via x = √ α(z − t) and y = √ αz, the system (9) is transformed into and the relation between E and P takes the form These are the sharp line self-induced transparency (SIT) equations [5][6][7]. We note that P 2 + N 2 is conserved. Indeed, as a consequence of (14), we have P 2 + N 2 = 1. Writing P = − sin θ and N = − cos θ, reduces the first two equations to E = θ t . 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, These are precisely the equations that result from the Miura transformation (10) (or from (13) 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]). Then, with Ξ = e −Sx−S −1 y and any p 0 ∈ Mat(m, m, C) (more generally x-dependent), (assuming the inverse exists) is a solution of (9).
If the matrix S satisfies the spectrum condition (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). Alternatively, one can show that these solutions actually 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 narrowed down by observing that the following transformations leave (19) and (20) invariant (see also the NLS case treated in [2]).
(1) Similarity transformation with an invertible M ∈ Mat(M, M, C) : 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) : (3) Reflexion symmetry: 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 [2,10]). The Jordan normal form can be restored afterwards via a similarity transformation.
Proposition 3 yields sufficient conditions on the matrix data for which the last two assumptions in Proposition 4 are satisfied.