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.Comment: 7 pages, 2 figures, 19th International Colloquium on Integrable
  Systems and Quantum Symmetries (ISQS19), Prague, Czech Republic, June 201

Dimakis, Aristophanes

Kanning, Nils

Mueller-Hoissen, Folkert

English

arXiv

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

Müller-Hoissen, Folkert

Max Planck Society eDoc Server

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

Dimakis, A.

Kanning, N.

Müller-Hoissen, F.

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

Acta Polytechnica Vol. 51 No. 1/2011Bidiﬀerential Calculus, Matrix SIT and Sine-Gordon EquationsA. Dimakis, N. Kanning, F. Mu¨ller-HoissenAbstractWe 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 fromsolutions 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 IntroductionThe bidiﬀerential calculus approach (see [1] and thereferences therein) aims to extract the essence of in-tegrability aspects of integrable partial diﬀerential ordiﬀerence equations (PDDEs) and to express them,and relations between them, in a universal way, i.e.resolved from speciﬁc examples. A powerful, thoughsimple to prove, result [1, 2, 3] (see section 6) gener-ates families of exact solutions from a matrix linearsystem. In the following we brieﬂy recall the basicframework and then apply the latter result to a ma-trix generalization of the SIT equations.2 Bidiﬀerential calculusA graded algebra is an associative algebra Ω over Cwith a direct sum decomposition Ω =⊕r≥0Ωr into asubalgebra A := Ω0 and A-bimodules Ωr, such thatΩr Ωs ⊆ Ωr+s. A bidiﬀerential calculus (or bidif-ferential graded algebra) is a unital graded algebraΩ equipped with two (C-linear) graded derivationsd, d¯ : Ω → Ω of degree one (hence dΩr ⊆ Ωr+1,d¯Ωr ⊆ Ωr+1), with the propertiesd2z = 0 ∀z ∈ C , where dz := d¯− z d , (1)and the graded Leibniz rule dz(χχ′) = (dzχ)χ′ +(−1)r χ dzχ′, for all χ ∈ Ωr and χ′ ∈ Ω.3 Dressing a bidiﬀerentialcalculusLet (Ω, d, d¯) be a bidiﬀerential calculus. Replacingdz in (1) by Dz := d¯−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 asdA = 0 = d¯A−AA . (2)If (2) is equivalent to a PDDE, we have a bidiﬀer-ential calculus formulation for it. This requires thatA depends on independent variables and the deriva-tions d, d¯ involve diﬀerential or diﬀerence operators.Several ways exist to reduce the two equations (2) toa single one:(1) We can solve the ﬁrst of (2) by setting A = dφ.This converts the second of (2) intod¯ dφ = dφ dφ . (3)(2) The second of (2) can be solved by setting A =(d¯g) g−1. The ﬁrst equation then readsd((d¯g) g−1)= 0 . (4)(3) More generally, setting A = [d¯g − (dg)Δ] g−1,with some Δ ∈ A, we have d¯A−AA = (dA) gΔg−1+(dg) (d¯Δ − (dΔ)Δ) g−1. As a consequence, if Δ ischosen such that d¯Δ = (dΔ)Δ, then the two equa-tions (2) reduce tod([d¯g − (dg)Δ] g−1)= 0 . (5)With the choice of a suitable bidiﬀerential calcu-lus, (3) and (4), or more generally (5), have beenshown to reproduce quite a number of integrablePDDEs. 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 equivalentto a certain PDDE, it is not in general guaranteedthat also (4) represents a decent PDDE. Then thegeneralization (5) has a chance to work (cf. [1]). Insuch a case, the Miura transformation[d¯g − (dg)Δ] g−1 = dφ (6)is a hetero-Ba¨cklund transformation relating solu-tions of the two PDDEs.Ba¨cklund, Darboux and binary Darboux trans-formations can be understood in this general frame-work [1], and there is a construction of an inﬁnite33Acta Polytechnica Vol. 51 No. 1/2011set of (generalized) conservation laws. Exchangingd and d¯ leads to what is known in the literature as‘negative ﬂows’ [3].4 A matrix generalization ofSIT equations and itsMiura-dualLet A = Mat (n, n, C∞(R2)), the algebra of n×n ma-trices of smooth functions on R2. Let Ω = A⊗∧(C2)with the exterior algebra∧(C2) of C2. In terms ofcoordinates x, y of R2, a basis ζ1, ζ2 of1∧(C2), and aconstant n × n matrix J , maps d and d¯ are deﬁnedas follows on A,df =12[J, f ]⊗ ζ1 + fy ⊗ ζ2 ,d¯f = fx ⊗ ζ1 + 12[J, f ]⊗ ζ2(see also [4]). They extend in an obvious way (withdζi = d¯ζi = 0) to Ω such that (Ω, d, d¯) becomes abidiﬀerential calculus. We ﬁnd that (3) is equivalenttoφxy =12[[J, φ], φy − 12J]. (7)Let n = 2m and J = block-diag(I,−I), where I = Imdenotes the m×m identity matrix. Decomposing φinto m×m blocks, and constraining it as follows,φ =(p qq −p), (8)(7) splits into the two equationspxy = (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 readsgx g−1 =12[J, φ] ,12[J, g] g−1 = φy . (10)Writingg =(a bc d),with m ×m matrices a, b, c, d, and assuming that aand its Schur complement S(a) = d−c a−1b is invert-ible (which implies that g is invertible), (10) with (8)requiresb = −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+r2are invertible, where r := c a−1. The conditions (11)are necessary in order that the Miura transformationrelates solutions of (9) to solutions of its ‘dual’(gx g−1)y =14[gJg−1, J ] , (12)obtained from (4). Taking (11) into account, theMiura transformation readsq = −cx a−1 = −rx − r ax a−1 ,qy = −r (I + r2)−1 , (13)py = I − (I + r2)−1 .As a consequence, we haveqy2 + py2 = py . (14)Furthermore, the second of (11) and the ﬁrst of (13)imply axa−1 = qr. Hence we obtain the systemrx = −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), whererL (rR) denotes the map of left (right) multiplicationby r. This can be used to eliminate q from the secondequation, 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 thesine-Gordon equationθxy = sin θ . (17)(15) can be obtained directly from (12) as follows,by settingg =(a −cc a)=(I −rr I)a ,henceg−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 ,34Acta Polytechnica Vol. 51 No. 1/2011where 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 writtenas (rx + rρ)y = r(I + r2)−1. Since q = −(ra)x a−1 =−rx− r ρ, this is the second of (15). The ﬁrst followsnoting that qr = ρ.5 Sharp line SIT equationsand sine-GordonWe 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, tvia x =√α(z − t) and y = √αz, the system (9) istransformed intoPt = E N , Nt = −E P ,and the relation between E and P takes the formEz + Et = αP .These are the sharp line self-induced transparency(SIT) equations [5, 6, 7]. We note that P2 + N 2 isconserved. Indeed, as a consequence of (14), we haveP2 +N 2 = 1. Writing P = − sin θ and N = − cos θ,reduces the ﬁrst two equations to E = θt. Expressedin the coordinates x, y, the third then becomes thesine-Gordon equation (17) (cf. [6]). As a consequenceof the above relations, q and p depend as follows onθ,q = −12θx ,qy = −12 sin θ , (18)py =12(1− cos θ) .These are precisely the equations that result from theMiura transformation (10) (or from (13)), choosingg =⎛⎜⎜⎝ cosθ2− sin θ2sinθ2cosθ2⎞⎟⎟⎠ ,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 ofgenerating solutions from amatrix linear systemTheorem 1 Let (Ω, d, d¯) be a bidiﬀerential calculuswith 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 sizesof the two matrices do not match). For ﬁxed N,N ′,let X ∈ Mat (N,N,B) and Y ∈ Mat (N ′, N,B) besolutions of the linear equationsd¯X = (dX)P ,d¯Y = (dY )P ,RX −X P = −QY ,with d-constant and d¯-constant matrices P ,R ∈Mat (N,N,B), and Q = V˜ U˜ , where U˜ ∈Mat (n,N ′,B) and V˜ ∈ Mat (N,n,B) are d- and d¯-constant. IfX is invertible, the n×n matrix variableφ = U˜Y X−1V˜ ∈ Mat (n, n,B)solves d¯φ = (dφ)φ+dϑ with ϑ = U˜Y X−1RV˜ , hence(by application of d) also (3). There is a similar result for (5) [3]. The Miuratransformation is a corresponding bridge.7 Solutions of the matrix SITequationsFrom Theorem 1 we can deduce the following result,using straightforward calculations [8], analogous tothose 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), andK ∈ Mat (M,M,C) a solution of the Sylvester equa-tionSK +KS = V U . (19)Then, with Ξ = e−Sx−S−1y 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 conditionσ(S) ∩ σ(−S) = ∅ (21)(where σ(S) denotes the set of eigenvalues of S), thenthe 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 directlythat 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 datathat determine the solutions (20) of (9). This can be35Acta Polytechnica Vol. 51 No. 1/2011narrowed down by observing that the following trans-formations leave (19) and (20) invariant (see also theNLS 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 normalform without restriction of generality.(2) Reparametrization transformation with invertibleA,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 blockof S and then changes the sign of its eigenvalue [8](see also [10, 2]). The Jordan normal form can berestored afterwards via a similarity transformation.The following result is easily veriﬁed [8].Proposition 3 Let S,U ,V be as in Proposition 2and T ∈Mat (M,M,C) invertible.(1) Let T be Hermitian (i.e. T † = T ) and suchthat S† = TST−1, U = V †T . Let K be a so-lution of (19), which can then be chosen such thatK† = TKT−1. Then q and p given by (20) withp†0 = p0 are both Hermitian and thus solve the Her-mitian reduction of (9).(2) Let T¯ = T−1 (where the bar means complexconjugation) and S¯ = TST−1, U¯ = UT−1 andV¯ = TV . Let K be a solution of (19), which canthen be chosen such that K¯ = TKT−1. Then q andp given by (20) with p¯0 = p0 satisfy q¯ = q and p¯ = p,and thus solve the complex conjugation reduction of(9). 8 Rank one solutionsLet 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 obtainq =2 s k ξ1 + (kξ)2π , p = p˜0 +2 s1 + (kξ)2π ,p˜0 := p0 − 2sπ , π := uvTvTu.The Miura transformation (13) implies r = −qy (I −py)−1, and we obtainr = − 2 kξ1− (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 equationsWe 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 asq = 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 Kof the Sylvester equation (19), which now amountsto rank (SK + KS) = 1, is the Cauchy-type ma-trix with components Kij = vi uj/(si + sj), whereui, vi ∈ C. Figs. 1 and 2 show plots of two examplesfrom the above family of solutions.Fig. 1: A scalar 2-soliton solution with S = diag (1, 2)and ui = vi = 1Fig. 2: A scalar breather solution with S = diag (1 + i,1− i) and ui = vi = 110 A family of solutions of thereal sine-Gordon equationVia the Miura transformation (18), Proposition 2 de-termines a family of sine-Gordon solutions (see also36Acta Polytechnica Vol. 51 No. 1/2011e.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−1y, and tr (SKΞ (IM + (KΞ)2)−1) ∈ iR(where i is the imaginary unit). Thenθ = 4 arctan( √β1 +√1− β)withβ :=(log | det(IM + (KΞ)2)|)xy(23)solves the sine-Gordon equation θxy = sin θ in anyopen set of R2 where det(IM + (KΞ)2) = 0.Proof: Let p be given by (22). Due to the assumptiondet(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 isimaginary, it follows that |1 − 2py| ≤ 1. Hence theequation cos θ = 1 − 2py (second of (18)) has a realsolution θ. Inserting expression (22) for p, we arriveat cos θ = 1−2 (log det(IM + (KΞ)2))xy. Moreover,(14) shows that py ≥ 0 and thus 0 ≤ py ≤ 1. Usingidentities for the inverse trigonometric functions, weﬁnd (23), where β = py. Proposition 3 yields suﬃcient conditions on thematrix data for which the last two assumptions inProposition 4 are satisﬁed.References[1] Dimakis, A., Mu¨ller-Hoissen, F.: Bidiﬀeren-tial graded algebras and integrable systems.Discr. Cont. Dyn. Systems Suppl., 2009, 2009,p. 208–219.[2] Dimakis, A., Mu¨ller-Hoissen, F.: Solutions ofmatrix NLS systems and their discretizations: auniﬁed treatment. Inverse Problems, 26, 2010,095007.[3] Dimakis, A., Mu¨ller-Hoissen, F.: Bidiﬀerentialcalculus approach to AKNS hierarchies and theirsolutions. SIGMA, 6, 2010, 2010055.[4] Grisaru, M., Penati, S.: An integrable non-commutative version of the sine-Gordon system.Nucl. Phys. B, 655, 2003, p. 250–276.[5] Lamb, G.: Analytical descriptions of ultrashortoptical pulse propagation in a resonant medium.Rev. Mod. Phys., 43, 1971, p. 99–124.[6] Caudrey, P., Gibbon, J., Eilbeck, J., Bul-lough, R.: Exact multi-soliton solutions ofthe self-induced transparency and sine-Gordonequations. Phys. Rev. Lett., 30, 1973,p. 237–238.[7] Bullough, R., Caudrey, P., Eilbeck, J., Gib-bon, J.: A general theory of self-induced trans-parency. Opto-electronics, 6, 1974, p. 121–140.[8] Kanning, N.: Integrable Systeme in der All-gemeinen Relativita¨tstheorie: ein Bidiﬀerential-kalku¨l-Zugang. Diploma thesis. Go¨ttingen : Uni-versity of Go¨ttingen, 2010.[9] Horn, R., Johnson, C.: Topics in Matrix Analy-sis. Cambridge : Cambridge Univ. Press, 1991.[10] Aktosun, T., Busse, T., Demontis, F., van derMee, C.: Symmetries for exact solutions tothe nonlinear Schro¨dinger equation. J. Phys. A:Theor. Math., 43, 2010, 025202.[11] Hirota, R.: Exact solution of the sine-Gordonequation for multiple collisions of solitons.J. Phys. Soc. Japan, 33, 1972, p. 1 459–1 463.[12] Ablowitz, M., Kaup, D., Newell, A., Segur, H.:Method for solving the sine-Gordon equation.Phys. Rev. Lett., 30, 1973, p. 1 262–1264.[13] Po¨ppe, C.: Construction of solutions of the sine-Gordon equation by means of Fredholm determi-nants. Physica D, 9, 1983, p. 103–139.[14] Zheng, W.: The sine-Gordon equation and thetrace method. J. Phys. A: Math. Gen., 19, 1986,p. L485–L489.[15] Schiebold, C.: Noncommutative AKNS systemand multisoliton solutions to the matrix sine-Gordon equation. Discr. Cont. Dyn. SystemsSuppl., 2009, 2009, p. 678–690.[16] Aktosun, T., Demontis, F., van der Mee, C.:Exact solutions to the sine-Gordon equation.arXiv:1003.2453, 2010.Aristophanes DimakisE-mail: dimakis@aegean.grDepartment of Financialand Management EngineeringUniversity of the Aegean, 41 Kountourioti Str.GR-82100 Chios, GreeceNils KanningE-mail: nils.kanning@ds.mpg.deMax-Planck-Institute for Dynamicsand Self-OrganizationBunsenstrasse 10, D-37073 Go¨ttingen, GermanyFolkert Mu¨ller-HoissenE-mail: folkert.mueller-hoissen@ds.mpg.deMax-Planck-Institute for Dynamicsand Self-OrganizationBunsenstrasse 10, D-37073 Go¨ttingen, Germany37

Bidifferential calculus, matrix SIT and sine-Gordon equations

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