Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems

A mixed initial-boundary value problem for nonlinear Maxwell{Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann{Hilbert (RH) problem. We use transformation operators whose existence is closely related with the Gours...

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Дата:	2017
Автори:	Filipkovska, M.S., Kotlyarov, V.P., Melamedova, E.A.
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Опубліковано:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2017
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Цитувати:	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems / M.S. Filipkovska, V.P. Kotlyarov, E.A. Melamedova // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 2. — С. 119-153. — Бібліогр.: 31 назв. — англ.

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spelling	irk-123456789-1405682018-07-11T01:23:23Z Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems Filipkovska, M.S. Kotlyarov, V.P. Melamedova, E.A. A mixed initial-boundary value problem for nonlinear Maxwell{Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann{Hilbert (RH) problem. We use transformation operators whose existence is closely related with the Goursat problems with nontrivial characteristics. We also use a gauge transformation which allows us to obtain Goursat problems of the canonical type with rectilinear characteristics, the solvability of which is known. The transformation operators and a gauge transformation are used to obtain the Jost type solutions of the Ablowitz-Kaup-Newel-Segur equations with well-controlled asymptotic behavior by the spectral parameter near singular points. A well posed regular matrix RH problem in the sense of the feasibility of the Schwartz symmetry principle is obtained. The matrix RH problem generates the solution of the mixed problem for MB equations. 2017 Article Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems / M.S. Filipkovska, V.P. Kotlyarov, E.A. Melamedova // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 2. — С. 119-153. — Бібліогр.: 31 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.02.119 Mathematics Subject Classification 2000: 34L25, 34M50, 35F31, 35Q15, 35Q51 http://dspace.nbuv.gov.ua/handle/123456789/140568 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A mixed initial-boundary value problem for nonlinear Maxwell{Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann{Hilbert (RH) problem. We use transformation operators whose existence is closely related with the Goursat problems with nontrivial characteristics. We also use a gauge transformation which allows us to obtain Goursat problems of the canonical type with rectilinear characteristics, the solvability of which is known. The transformation operators and a gauge transformation are used to obtain the Jost type solutions of the Ablowitz-Kaup-Newel-Segur equations with well-controlled asymptotic behavior by the spectral parameter near singular points. A well posed regular matrix RH problem in the sense of the feasibility of the Schwartz symmetry principle is obtained. The matrix RH problem generates the solution of the mixed problem for MB equations.
format	Article
author	Filipkovska, M.S. Kotlyarov, V.P. Melamedova, E.A.
spellingShingle	Filipkovska, M.S. Kotlyarov, V.P. Melamedova, E.A. Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems Журнал математической физики, анализа, геометрии
author_facet	Filipkovska, M.S. Kotlyarov, V.P. Melamedova, E.A.
author_sort	Filipkovska, M.S.
title	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems
title_short	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems
title_full	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems
title_fullStr	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems
title_full_unstemmed	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems
title_sort	maxwell-bloch equations without spectral broadening: gauge equivalence, transformation operators and matrix riemann-hilbert problems
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/140568
citation_txt	Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems / M.S. Filipkovska, V.P. Kotlyarov, E.A. Melamedova // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 2. — С. 119-153. — Бібліогр.: 31 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT filipkovskams maxwellblochequationswithoutspectralbroadeninggaugeequivalencetransformationoperatorsandmatrixriemannhilbertproblems AT kotlyarovvp maxwellblochequationswithoutspectralbroadeninggaugeequivalencetransformationoperatorsandmatrixriemannhilbertproblems AT melamedovaea maxwellblochequationswithoutspectralbroadeninggaugeequivalencetransformationoperatorsandmatrixriemannhilbertproblems
first_indexed	2025-07-10T10:45:40Z
last_indexed	2025-07-10T10:45:40Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2017, Vol. 13, No. 2, pp. 119–153 doi: 10.15407/mag13.02.119 Maxwell–Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann–Hilbert Problems M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine 47 Nauky Ave., Kharkiv, 61103, Ukraine E-mail: filipkovskaya@ilt.kharkov.ua kotlyarov@ilt.kharkov ua melamedova@ilt.kharkov.ua Received January 26, 2017, revised March 26, 2017 A mixed initial-boundary value problem for nonlinear Maxwell–Bloch (MB) equations without spectral broadening is studied by using the inverse scattering transform in the form of the matrix Riemann–Hilbert (RH) prob- lem. We use transformation operators whose existence is closely related with the Goursat problems with nontrivial characteristics. We also use a gauge transformation which allows us to obtain Goursat problems of the canoni- cal type with rectilinear characteristics, the solvability of which is known. The transformation operators and a gauge transformation are used to obtain the Jost type solutions of the Ablowitz–Kaup–Newel–Segur equations with well-controlled asymptotic behavior by the spectral parameter near singular points. A well posed regular matrix RH problem in the sense of the feasibil- ity of the Schwartz symmetry principle is obtained. The matrix RH problem generates the solution of the mixed problem for MB equations. Key words: Maxwell–Bloch equations, gauge equivalence, transformation operators, matrix Riemann–Hilbert problems. Mathematics Subject Classification 2010: 34L25, 34M50, 35F31, 35Q15, 35Q51. c© M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko), 2017 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) The paper is dedicated to the 95th anniversary of Vladimir Aleksandrovich Marchenko 1. Introduction Special integral operators that transform solutions of differential equations with constant coefficients into solutions of equations with variable coefficients are a distinctive feature of the Marchenko’s papers. In this paper, we propose a systematic use of the transformation operators for the construction of matrix Riemann–Hilbert problems which lead to the solution of the initial-boundary value problem for nonlinear Maxwell–Bloch equations without spectral broaden- ing. The transformation operators proposed are closely related to the Goursat problems with nontrivial characteristics. The use of a gauge transformation al- lows one to obtain the Goursat problems of the canonical type with rectilinear characteristics as well as their solvability. We use the same transformation to establish a gauge equivalence between two pairs of the Ablowitz–Kaup–Newel– Segur (AKNS) equations to construct their Jost type solutions with the well- controlled asymptotic behavior by a spectral parameter on the complex plane near the singular points. As a result, the well-posed regular matrix RH problem, which generates the solution of the mixed problem for MB equations, is obtained. The Maxwell–Bloch equations in the integrable case have the following form (sf. [16]): ∂E ∂t + ∂E ∂x = 〈ρ〉, (1) ∂ρ ∂t + 2iλρ = NE , (2) ∂N ∂t = −1 2 ( Eρ+ Eρ ) . (3) Here the symbol ¯ denotes a complex conjugation, E = E(t, x) is a complex valued function of the space variable x and the time t, ρ = ρ(t, x, λ), and N (t, x, λ) are the complex valued and real functions of t, x and a spectral parameter λ. The angular brackets 〈〉 mean the averaging by λ with the given ”weight” function n(λ), 〈ρ〉 = ∞∫ −∞ ρ(t, x, λ)n(λ)dλ, ∞∫ −∞ n(λ)dλ = ±1. (4) If n(λ) > 0, then an unstable medium is considered (the so-called two-level laser amplifier). If n(λ) < 0, then a stable medium is considered (the so-called attenuator). Equations (1)–(4) have appeared in many physical models. However, first they were studied in [22–25]. The most important is a model of the propagation 120 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems of electromagnetic waves in a medium with distributed two-level atoms. For example, there are models of self-induced transparency [1,2,14–17], and two-level laser amplifiers [14–16, 27–30]. For these models, E(t, x) is the complex valued envelope of electromagnetic wave of fixed polarization, N (t, x, λ) and ρ(t, x, λ) are the entries of a density matrix of the atomic subsystem ρ̂(t, x, λ) = ( N (t, x, λ) ρ(t, x, λ) ρ(t, x, λ) −N (t, x, λ) ) . The parameter λ denotes a deviation of the transition frequency from its mean value. The weight function n(λ) characterizes the inhomogeneous broadening which is the difference between the initial population of the upper and lower levels. Short reviews on the Maxwell–Bloch equations and applying to them of the inverse scattering transform (IST) method can be found in [1, 2, 16]. We restrict our study to the case where n(λ) = δ(λ), i.e., without spectral broadening. Then 〈ρ〉 = ρ and the system (1)–(4) is written as ∂E ∂t + ∂E ∂x = ρ, ∂ρ ∂t = NE , ∂N ∂t = −1 2 ( Eρ+ Eρ ) . (5) These equations are simpler than (1)–(4). However, applying of the IST method to (5) is somewhat complicated. The matrix Riemann–Hilbert problem for MB equations (1)–(4) was studied in [19]. The main goal of this paper is to study a mixed problem for the Maxwell–Bloch equations which is defined by the initial and boundary conditions: E(0, x) = E0(x), ρ(0, x) = ρ0(x), N (0, x) = N0(x), E(t, 0) = E1(t), (6) where x ∈ (0, l) (l ≤ ∞) and t ∈ R+. The function E1(t) is a Schwartz-type function (smooth and fast decreasing at infinity). The functions E0(x), ρ0(x), 1− N0(x) are smooth if x ∈ [0, l] or of Schwartz type if x ∈ R+. The functions ρ(t, x), N (t, x) are not independent. Indeed, equations (2), (3) give ∂ ∂t ( \|ρ(t, x)\|2 +N (t, x)2 ) = 0, and we put \|ρ(t, x)\|2 +N (t, x)2 ≡ 1. If we define ρ(0, x), then N (0, x) = ∓ √ 1− \|ρ(0, x)\|2. If one chooses the sign ”minus”, then the problem is considered in a stable medium, in the so-called attenuator (for example, a model of self-induced trans- parency). The matrix Riemann–Hilbert problems were studied for this case in [18, 20]. If the sign ”plus” is chosen, then the problem is considered in an Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 121 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) unstable medium (for example, a model of a two-level laser amplifier), which is the subject of our study. We suppose that the functions E(t, x), ρ(t, x) and N (t, x) satisfy the MB equations (5) in the domain x, t ∈ (0, l) × (0,∞). We develop the IST method in the form of the matrix Riemann–Hilbert problem in a complex z-plane. The method is based on using the transformation operators for constructing the Jost type solutions of the AKNS equations. The RH problem is defined by spectral functions which, in turn, are defined through the given initial and boundary conditions for the MB equations. The RH problem is meromorphic and simple in some sense: it is deduced by using standard approaches to the inverse scattering transform for the quarter of the xt-plane. Unfortunately, this RH problem has an essential deficiency because it may have multiple eigenvalues and spectral singularities. Therefore we deduce a new regular matrix RH problem, free from the mentioned deficiency, which has the unique solution. Then we prove that this regular RH problem generates a system of compatible differential equations, which is the AKNS system of linear equations [2] for the MB equations without broadening. Thus the RH problem generates a solution to the MB equations. Our approach differs from those considered for the mixed problem to the MB equations given in [1,16,18,27,28]. We develop an approach to the simultaneous spectral analysis proposed in [10–13] and in [3–7,21] for other nonlinear equations and prove that the mixed problem for the MB equations is completely linearizable by the appropriate matrix RH problem. 2. Gauge Transformation of the Ablowitz–Kaup–Newel–Segur Equations The Ablowitz–Kaup–Newel–Segur equations for the Maxwell–Bloch equations without spectral broadening have the form: Φt =U(t, x, λ)Φ, U(t, x, λ) = −(iλσ3 +H(t, x)), (7) Φx =V (t, x, λ)Φ, V (t, x, λ) = iλσ3 +H(t, x) + iF (t, x) 4λ , (8) where H(t, x) = 1 2 ( 0 E(t, x) −E(t, x) 0 ) , F (t, x) = ( N (t, x) ρ(t, x) ρ(t, x) −N (t, x) ) , and σ3 = ( 1 0 0 −1 ) is the Pauli matrix. It is well known [2] that the over-determined system of differential equations (7), (8) is compatible if and only if the compati- bility condition Ux − Vt + [U, V ] = 0 (9) holds. Equation (9) is equivalent to the system of nonlinear equations ∂H ∂t + ∂H ∂x = 1 4 [σ3, F ], ∂F ∂t = [F,H], (10) 122 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems which are the matrix form of MB equations (5). Two systems of the AKNS equations, Φt =U(t, x, λ)Φ, Φx =V (t, x, λ)Φ and Ψt =Û(t, x, λ)Ψ, (11) Ψx =V̂ (t, x, λ)Ψ, (12) are called gauge equivalent [9] if their compatible solutions are related by Φ(t, x, λ) = g(t, x)Ψ(t, x, λ), where the unitary matrix g(t, x) does not depend on λ. Obviously, the matrices (Û , V̂ ) and (U , V ) are connected by the relations: Û =g−1Ug − g−1gt, (13) V̂ =g−1V g − g−1gx. (14) The nonlinear equations, defined by the corresponding compatibility conditions Ux − Vt + [U, V ] = 0, Ûx − V̂t + [Û , V̂ ] = 0, are also called gauge equivalent [9]. Among all gauge transformations we are interested only in those which make the matrix F (t, x) be diagonal, i.e., F (t, x) = g(t, x)σ3g −1(t, x), g−1(t, x) = g∗(t, x), (15) where ∗ means the Hermitian conjugation. This equality does not define the unitary matrix g(t, x) uniquely. It is defined up to the unitary diagonal matrix g(t, x) = D(t, x)eχ(t,x)σ3 , (16) where χ = χ(t, x) is an imaginary scalar function. It is convenient to chose the matrix D(t, x) in the form D(t, x) = 1√ 2(1 +N(t, x)) ( 1 +N(t, x) −ρ(t, x) ρ̄(t, x) 1 +N(t, x) ) , (17) where the root is its arithmetic value. It is easy to verify that detD(t, x) ≡ 1 and D−1(t, x) = D∗(t, x). Then for the matrix Û (13) we obtain Û = −iλS(t, x) + Ĥ(t, x), Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 123 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) where S(t, x) :=g−1(t, x)σ3g(t, x), Ĥ(t, x) :=g−1(t, x)H(t, x)g(t, x) + g−1(t, x)ġt(t, x). The matrix Ĥ(t, x) = e−χ(t,x)σ3(D−1HD +D−1Ḋt + χ̇tσ3)eχ(t,x)σ3 . It will be identically equal to zero if the matrix D̂ = D−1HD+D−1Ḋt commutes with σ3. Indeed, in this case, the matrix D̂ is proportional to σ3, D̂ = f(t, x)σ3. (18) Then putting χ̇t = −f(t, x), one finds that Ĥ(t, x) ≡ 0. In order to prove (18), we use the second equation from (10). Since F = gσ3g −1 = Deχσ3σ3e−χσ3D−1 = Dσ3D −1, then Ḟt − [F,H] = D[D−1Ḋt, σ3]D−1 − [Dσ3D −1, H] = = D[D−1Ḋt +D−1HD,σ3]D−1 = D[D̂, σ3]D−1 = 0 It means that [D̂, σ3] = 0 and hence D̂ = f(t, x)σ3 + β(t, x)I, where f = f(t, x) and β = β(t, x) are arbitrary scalars. Further, since tr D̂ = trH + trD−1Ḋt = trD−1Ḋt = 2NṄt + ρ̇tρ̄+ ρ̄̇ρt ≡ 0, one finds β = 0. Finally, in view of χ̇t = −f(t, x), f(t, x) = (D−1HD +D−1Ḋt)11, where (·)11 means (11) element of the matrix, we have that Ĥ(t, x) ≡ 0, and the matrix Û is equal to Û(t, x, λ) = −iλS(t, x), S(t, x) = ( ν(t, x) p(t, x) p̄(t, x) −ν(t, x) ) , where S(t, x) = g−1(t, x)σ3g(t, x) = S∗(t, x) and S2 ≡ I. For the matrix V̂ = V̂ (t, x, λ), we have V̂ = g−1V g − g−1g′x = iλS(t, x) +R(t, x) + iσ3 4λ , where R = g−1Hg− g−1g′x = e−χσ3(D−1HD−D−1D′x−χ′xσ3)eχσ3 . Since trR = 0, then the matrix R takes the form R = (h(t, x)− χ′x)σ3 + σ3 2 [σ3, g −1Hg − g−1g′x], where h(t, x) = (D−1HD −D−1D′x)11. By putting χ′x = h(t, x), we find that R(t, x) = σ3 2 [σ3, e −χσ3(D−1HD −D−1D′x)eχσ3 ] = ( 0 r(t, x) −r̄(t, x) 0 ) . 124 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems All written above is true if the equations χ̇t = −f(t, x), f(t, x) = ρ̄ 4 ( E − ρ̇t 1 +N ) − ρ 4 ( Ē − ¯̇ρt 1 +N ) , χ̇x = h(t, x), h(t, x) = ρ̄ 4 ( E + ρ′x 1 +N ) − ρ 4 ( Ē + ρ̄′x 1 +N ) are compatible. Indeed, these simple equations are compatible if and only if ∂h ∂t + ∂f ∂x = 0. By using the Maxwell–Bloch equations (5), this condition can be verified by routine calculations. Then χ(t, x) is defined as an integral χ(t, x) = (t,x)∫ (0,0) h(t, s)ds− f(τ, x)dτ + χ(0, 0), which does not depend on a path of integration. The free parameter χ(0, 0) will be used in Sec. 5. The gauge Eqs. (11) and (12) with Û = −iλS(t, x), V̂ = iλS(t, x) +R(t, x) + iσ3 4λ are also compatible Ûx − V̂t + [Û , V̂ ] = 0. The last equation is equivalent to the system of nonlinear equation ∂S ∂t + ∂S ∂x = [R,S], Rt = 1 4 [S, σ3]. Thus we have proved the theorem below. Theorem 1. The Maxwell–Bloch Eqs. (5) are gauge equivalent to the equations ∂ν ∂t + ∂ν ∂x = pr̄ + p̄r, ∂p ∂t + ∂p ∂x = −2νr, ∂r ∂t = −1 2 p, where ν = ν(t, x) is real, and p = p(t, x), r = r(t, x) are complex val- ued functions which constitute the matrices S = ( ν(t, x) p(t, x) p̄(t, x) −ν(t, x) ) and R =( 0 r(t, x) r̄(t, x) 0 ) . Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 125 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) R e m a r k 1. The second matrix equation Rt = 1 4 [S, σ3] gives S = νσ3 + [Rt, σ3] = νσ3 + 2Rtσ3 = ( ν −2rt −2r̄t −ν ) . Then the first matrix equation ∂S ∂t + ∂S ∂x = [R,S] is read as ∂ν ∂t + ∂ν ∂x + 2 ∂\|r\|2 ∂t = 0, ∂2r ∂t2 + ∂2r ∂x∂t = νr. The gauge AKNS equations (11) and (12) will be used below for the trans- formation of some Goursat problems and also for the construction of compatible solutions of the AKNS equations (7) and (8) which have a well-controlled asymp- totic behavior as z → 0. 3. Basic Solutions of the Ablowitz–Kaup–Newel–Segur Linear Equations We suppose here that the solution (E(t, x), N (t, x), ρ(t, x)) of the mixed problem (5), (6) for the Maxwell–Bloch equations in the domain t ∈ R+, 0 ≤ x ≤ l ≤ ∞ exists, and it is unique and smooth. Then the AKNS linear equations (7) and (8) are compatible. To construct their solutions we use the following lemma. Lemma 1. Let Eqs. (7) and (8) be compatible for all t, x, λ ∈ R. Let Φ(t, x, λ) be a matrix satisfying the t-equation (7) for all x (the x-equation (8) for all t). Assume that Φ(t0, x, λ) satisfies the x-equation (8) for some t = t0 ≤ ≤ ∞ (the t-equation (7) for some x = x0 ≤ ∞). Then Φ(t, x, λ) satisfies the x-equation (8) for all t (satisfies the t-equation (7) for all x). P r o o f. The proof can be found in [3] (Lemma 2.1). Let Y (t, x, λ) be a product of the matrices Y (t, x, λ) = W (t, x, λ)Φ(t, λ), (19) where W (t, x, λ) satisfies the x-equation (8) for all t and W (t, 0, λ) = I, and Φ(t, λ) satisfies the t-equation (7) for x = 0 under the initial condition lim t→∞ Φ(t, λ)eiλtσ3 = I. Let Z(t, x, λ) be a product of the matrices Z(t, x, λ) = Ψ(t, x, λ)w(x, λ), (20) 126 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems where Ψ(t, x, λ) satisfies the t-equation (7) for all x and Ψ(0, x, λ) = I, and w(x, λ) satisfies the x-equation (8) for t = 0 under the initial condition w(l, λ) = eilµ(λ)σ3 , where µ(λ) = λ+ 1 4λ . If l =∞, the initial condition takes the form lim x→∞ w(x, λ)e−ixµ(λ)σ3 = I. It is easy to see that due to Lemma 1, the matrices Y (t, x, λ) and Z(t, x, λ) are compatible solutions of the AKNS system of Eqs. (7), (8). Lemma 2. Let E(t, 0) = E1(t) be smooth and fast decreasing as t → ∞. Then for Imλ = 0 there exists the Jost solution Φ(t, λ) of the t-equation (7) with x = 0 represented by the transformation operator Φ(t, λ) = e−iλtσ3 + ∞∫ t K(t, τ)e−iλτσ3dτ, Imλ = 0. (21) The kernel K(t, τ) satisfies the symmetry condition K(t, τ) = ΛK(t, τ)Λ−1 with matrix Λ = ( 0 1 −1 0 ) and it is defined by the Goursat problem: σ3 ∂K(t, τ) ∂t + ∂K(t, τ) ∂τ σ3 = H(t, 0)σ3K(t, τ), σ3K(t, t)−K(t, t)σ3 = σ3H(t, 0), lim t+τ→+∞ K(t, τ) = 0. The kernel K(t, τ) is smooth and fast decreasing as t+ τ →∞. The proof uses the Goursat problem and the corresponding integral equations which allow one to prove their unique solvability and thus to prove the inte- gral representation (21) (sf. [9]). Due to (21), the vector columns Φ[1](t, λ) and Φ[2](t, λ) of the matrix Φ(t, λ) = (Φ[1](t, λ),Φ[2](t, λ)) have analytic continua- tions Φ[1](t, z) and Φ[2](t, z) to the lower half-plane C− and the upper half-plane C+ of the complex z-plane, respectively. Thus the vector columns Φ[1](t, z), Φ[2](t, z) are analytic in C−, C+, respectively, continuous in C− ∪R, C+ ∪R and have the following asymptotics: Φ[1](t, z)eizt = ( 1 0 ) + O(z−1), Im z ≤ 0, z →∞; Φ[2](t, z)e−izt = ( 0 1 ) + O(z−1), Im z ≥ 0, z →∞. The symbol O(.) means a matrix whose entries have the indicated order. Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 127 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) Lemma 3. Let E(t, x), N (t, x), ρ(t, x) be smooth. Then for any x and Imλ = 0 there exists the solution Ψ(t, x, λ) of the t-equation (7) represented by the transformation operator Ψ(t, x, λ) = e−iλtσ3 + t∫ −t K0(t, τ, x)e−iλτσ3dτ, Imλ = 0. (22) The kernel K0(t, τ, x) is smooth, it satisfies the symmetry condition K0(t, τ, x) = ΛK0(t, τ, x)Λ−1 with matrix Λ = ( 0 1 −1 0 ) and is defined by the Goursat problem: σ3 ∂K0(t, τ, x) ∂t + ∂K0(t, τ, x) ∂τ σ3 = H(t, x)σ3K0(t, τ, x), σ3K0(t, t, x)−K0(t, t, x)σ3 =H(t, x)σ3, σ3K0(t,−t, x) +K0(t,−t, x)σ3 =0. The proof of the lemma can be found in [3]. The integral representation (22) gives the analyticity of the solution Ψ(t, x, z) for z ∈ C and its asymptotic behavior as z →∞: Ψ(t, x, z)eizt = ( 1 0 0 e2izt ) + O(z−1) + O(e2iztz−1), z →∞; Ψ(t, x, z)e−izt = ( e−2izt 0 0 1 ) + O(z−1) + O(e−2iztz−1), z →∞. Since t is positive, these asymptotics mean that Ψ(t, x, z)eizt is bounded in C+, and Ψ(t, x, z)e−izt is bounded in C− as z → ∞. For any fixed t and x, they are bounded on any compact set of the complex plane. Now we pass to the construction of the solutions of the x-equation (8) which has two singular points ∞ and 0 (t-equation (7) had the singular point at ∞ only). First, we consider the Jost solution of (8) which has a good behavior as z →∞. With this purpose, we represent the x-equation (8) in the form Wx = [ iµ(λ)σ3 +H(t, x) + i 4λ (F (t, x)− σ3) ] W, µ(λ) = λ+ 1 4λ . (23) If H(t, x) ≡ 0 and F (t, x) ≡ σ3, the x-equation has the exact solution eixµ(λ)σ3 . Lemma 4. Let E(t, x), N (t, x), ρ(t, x) be the smooth functions for 0 ≤ x ≤ l, and N 2(t, x) + \|ρ(t, x)\|2 ≡ 1. 128 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems Then for any t and Imµ(λ) = 0 there exists the solution W (t, x, λ) of the x- equation (23) represented by the transformation operators W (t, x, λ) = eixµ(λ)σ3 + x∫ −x L(x, y, t)eiyµ(λ)σ3dy+ 1 iλ x∫ −x M(x, y, t)eiyµ(λ)σ3dy. (24) The kernels L(x, y, t) and M(x, y, t) are smooth, they satisfy the symmetry con- dition L(x, y, t) = ΛL(x, y, t)Λ−1, M(x, y.t) = ΛM(x, y, t)Λ−1 with matrix Λ =( 0 1 −1 0 ) , and are defined by the formulas L(x, y, t) = L̂(x, y, t) and M(x, y, t) = g(t, x)M̂(x, y, t), where the unitary matrix g(t, x) is the same as in Section 2. The matrices L̂(x, y, t) and M̂(x, y, t) are the unique solution of the Goursat problem: σ3 ∂L̂ ∂x + ∂L̂ ∂y σ3 =σ3H(t, x)L̂+ σ3[σ3, g(t, x)]M̂, σ3 ∂M̂ ∂x + ∂M̂ ∂y σ3 =σ3R(t, x)M̂ + σ3 4 [g−1(t, x), σ3]L̂, σ3L̂(x, x, t)− L̂(x, x, t)σ3 =σ3H(t, x), σ3M̂(x, x, t)− M̂(x, x, t)σ3 = 1 4 [σ3, g −1(t, x)]σ3, (25) σ3L̂(x,−x, t) + L̂(x,−x, t)σ3 =0, σ3M̂(x,−x, t) + M̂(x,−x, t)σ3 =0, where R(t, x) = g−1(t, x)H(t, x)g(t, x)− g−1(t, x)g′x(t, x). P r o o f. Substituting (24) into equation (23) and integrating by parts, we get the Goursat problem (−x < y < x): ∂L ∂x + σ3 ∂L ∂y σ3 =H(t, x)L+ (σ3 − F (t, x))M, ∂M ∂x + F (t, x) ∂M ∂y σ3 =H(t, x)M + 1 4 (σ3 − F (t, x))M, σ3L(x, x, t)− L(x, x, t)σ3 =σ3H(t, x), F (t, x)M(x, x, t)−M(x, x, t)σ3 =− 1 4 (σ3 − F (t, x))σ3, σ3L(x,−x, t) + L(x,−x, t)σ3 =0, F (t, x)M(x,−x, t) +M(x,−x, t)σ3 =0. Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 129 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) To integrate the summand with the factor 1 λ2 we have to use the identity 1 λ2 = 4µλ − 4. We have obtained the Goursat problem with variable coefficients at deriva- tives. To reduce this problem to the standard form with constant matrices at derivatives, we use the same gauge transformation as in the previous section. We put F (t, x) = g(t, x)σ3g −1(t, x) where the unitary matrix g(t, x) is the same as in Section 2. Further we put L(x, y, t) = L̂(x, y, t) and M(x, y, t) = g(t, x)M̂(x, y, t). Taking into account that ∂M ∂x = g(t, x)∂M̂∂x + g′x(t, x)M̂ and ∂M ∂y = g(t, x)∂M̂∂y , we find that the Goursat problem reduces to the form: ∂L̂ ∂x + σ3 ∂L̂ ∂y σ3 =H(t, x)L̂+ [σ3, g(t, x)]M̂, ∂M̂ ∂x + σ3 ∂M̂ ∂y σ3 =R(t, x)M̂ + 1 4 [g−1(t, x), σ3]L̂, σ3L̂(x, x, t)− L̂(x, x, t)σ3 =σ3H(t, x), σ3M̂(x, x, t)− M̂(x, x, t)σ3 = 1 4 [σ3, g −1(t, x)]σ3, σ3L̂(x,−x, t) + L̂(x,−x, t)σ3 =0, σ3M̂(x,−x, t) + M̂(x,−x, t)σ3 =0. This problem coincides with (25). Thus we obtain the classical Goursat problem which in turn gives the existence of representation (24). The integral representation (24) gives the analyticity of the solution W (t, x, z) for z ∈ C \ {0} and its asymptotic behavior as z →∞ and z → 0: W (t, x, z)e−ixµ(z) =  ( 1 0 0 e−2ixµ(z) ) + O(z−1) + O(e−2ixµ(z)z−1), z →∞; O(1) + O(e−2ixµ(z)), z → 0; W (t, x, z)eixµ(z) =  ( e2ixµ(z) 0 0 1 ) + O(z−1) + O(e2ixµ(z)z−1), z →∞; O(1) + O(e2ixµ(z)), z → 0. Taking into account that Imµ(z) = (1 − 1 4\|z\|2 ) Im z, and since x > 0, the above asymptotics mean that W (t, x, z)e−ixµ(z) is bounded in the domain {z ∈ C : Imµ(z) ≤ 0}, and W (t, x, z)eixµ(z) is bounded in the domain {z ∈ C : Imµ(z) ≥ 0}. 130 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems Lemma 5. Let the initial functions E0(x), ρ0(x), N0(x) from (6) be smooth or of Schwartz type if x ∈ R+, i.e., l =∞, and N 2 0 (x) + \|ρ0(x)\|2 ≡ 1. Then the Jost solution w(x, λ) can be represented in the form (Imµ(λ) = 0), w(x, λ) = eixµ(λ)σ3 + 2l−x∫ x L(x, y)eiyµ(λ)σ3dy + 1 iλ 2l−x∫ x M(x, y)eiyµ(λ)σ3dy. (26) The kernels L(x, y) and M(x, y) satisfy the symmetry conditions L(x, y) = ΛL(x, y)Λ−1, M(x, y) = ΛM(x, y)Λ−1 with matrix Λ = ( 0 1 −1 0 ) and they are defined by the formulas L(x, y) = L̂(x, y) and M(x, y) = g(0, x)M̂(x, y), where the unitary matrix g(0, x) is the same as in Section 2 with t = 0. The matrices L̂(x, y) and M̂(x, y) are the unique solution of the Goursat problem: σ3 ∂L̂ ∂x + ∂L̂ ∂y σ3 =σ3H(0, x)L̂+ σ3[σ3, g(0, x)]M̂, σ3 ∂M̂ ∂x + ∂M̂ ∂y σ3 =σ3R(0, x)M̂ + σ3 4 [g−1(0, x), σ3]L̂, σ3L̂(x, x)− L̂(x, x)σ3 =H(0, x)σ3, σ3M̂(x, x)− M̂(x, x)σ3 = 1 4 [g−1(0, x), σ3]σ3, σ3L̂(x, 2l − x) + L̂(x, 2l − x)σ3 =0, σ3M̂(x, 2l − x) + M̂(x, 2l − x)σ3 =0, or lim x+y→+∞ L̂(x, y) = lim x+y→+∞ M̂(x, y) =0, if l =∞, where R(0, x) = g−1(0, x)H(0, x)g(0, x) − g−1(0, x)g′x(0, x). The kernels L(x, y) and M(x, y) are smooth and fast decreasing as x+ y →∞ if l =∞. P r o o f. By substituting (26) into equation (23) and integrating by parts, we get the Goursat problem (x < y < 2l − x): ∂L ∂x + σ3 ∂L ∂y σ3 =H(0, x)L+ (σ3 − F (0, x))M, Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 131 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) ∂M ∂x + F (0, x) ∂M ∂y σ3 =H(0, x)M + 1 4 (σ3 − F (0, x))M, σ3L(x, x)− L(x, x)σ3 =H(0, x)σ3, F (0, x)M(x, x)−M(x, x)σ3 = 1 4 (σ3 − F (0, x))σ3, σ3L(x, 2l − x) + L(x, 2l − x)σ3 =0, F (0, x)M(x, 2l − x) +M(x, 2l − x)σ3 =0. If l =∞, then the last two conditions (for y = 2l − x) are changed with lim x+y→+∞ L(x, y) = lim x+y→+∞ M(x, y) = 0, if l =∞. Further we put L(x, y) = L̂(x, y) and M(x, y) = g(0, x)M̂(x, y), and we finish the proof in the same way as in Lemma (4). Introduce the notations: Ω± = {z ∈ C± ∣∣ \|z\| > 1 2}, D± = {z ∈ C± ∣∣ \|z\| < 1 2}, Σ = R ∪ Cup ∪ Clow, where the semicircles Cup and Clow are: Cup = {z ∈ C ∣∣ \|z\| = 1 2 , arg z ∈ (0, π)} and Clow = {z ∈C ∣∣ \|z\| = 1 2 , arg z ∈ (π, 2π)}. Let Ω± and D± be the closures of the domains Ω± and D±, respectively. The contour Σ is the set where Imµ(λ) = 0: Σ = {λ ∈ C : Im ( λ+ 1 4λ ) = 0} = R ∪ Cup ∪ Clow. The orientation on Σ is depicted in Figure (1). Fig. 1. The domains Ω±, D± and the oriented contour Σ. 132 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems The vector columns of the matrix w(x, λ) have analytic continuations w[1](x, z) and w[2](x, z) in the domains Ω+ ∪ D− and Ω− ∪ D+, respectively. These vectors have the asymptotics: w[1](x, z)e−ixµ(z) =  ( 1 0 ) + O(z−1), z ∈ Ω+, z →∞, O(1), z ∈ D− \ {0}, z → 0. (27) w[2](x, z)eixµ(z) =  ( 0 1 ) + O(z−1), z ∈ Ω−, z →∞, O(1), z ∈ D+ \ {0}, z → 0. (28) Formula (19), Lemmas 2, 4 and Eqs. (27), (28) imply the following properties of Y (t, x, λ) = (Y [1](t, x, λ) Y [2](t, x, λ)): 1) Y (t, x, λ) (λ 6= 0) satisfies the t− and x−equations (7), (8); 2) Y (t, x, λ) = ΛY (t, x, λ)Λ−1, λ ∈ R \ {0}, where Λ = ( 0 1 −1 0 ) ; 3) detY (t, x, λ) ≡ 1, λ ∈ R \ {0}; 4) the map (x, t) 7−→ Y (t, x, λ) (λ 6= 0) is smooth in t and x; 5) the vector functions Y [1](t, x, z)eizt−iµ(z)x, Y [1](t, x, z)e−izt+iµ(z)x and Y [2](t, x, z)e−izt+iµ(z)x, Y [2](t, x, z)eizt−iµ(z)x are analytic in C− and C+, respec- tively, continuous up to the boundary with exception of λ = 0 and have the following asymptotic behavior: Y [1](t, x, z)eizt−iµ(z)x = ( 1 0 ) + O(z−1), z ∈ Ω−, z →∞, (29) Y [1](t, x, z)e−izt+iµ(z)x = O(1), z ∈ D− \ {0}, z → 0, Y [2](t, x, z)e−izt+iµ(z)x = ( 0 1 ) + O(z−1), z ∈ Ω+, z →∞, (30) Y [2](t, x, z)eizt−iµ(z)x = O(1), z ∈ D+ \ {0}, z → 0. Formula (20) and Lemmas 3, 5 imply the following properties of Z(t, x, λ) = (Z[1](t, x, λ) Z[2](t, x, λ)): 1) Z(t, x, λ) (λ 6= 0) satisfies the t- and x-equations (7), (8); 2) Z(t, x, λ) = ΛZ(t, x, λ)Λ−1, λ ∈ R \ {0}; 3) detZ(t, x, λ) ≡ 1, λ ∈ R \ {0}; 4) the map (x, t) 7−→ Z(t, x, λ) (λ 6= 0) is smooth in t and x; 5) the maps z 7−→ Z[1](t, x, z) and z 7−→ Z[2](t, x, z) are analytic in Ω+ ∪D− and Ω−∪D+, respectively, and the asymptotic behavior of Z[1](t, x, z)eizt−ixµ(z), Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 133 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) Z[2](t, x, z)e−izt+ixµ(z) is as follows: Z[1](t, x, z)eizt−ixµ(z) = ( 1 0 ) + O(z−1), z ∈ Ω+, z →∞, (31) Z[1](t, x, z)eizt−ixµ(z) = O(1), z ∈ D− \ {0} z → 0, Z[2](t, x, z)e−izt+ixµ(z) = ( 0 1 ) + O(z−1), z ∈ Ω−, z →∞, (32) Z[2](t, x, z)e−izt+ixµ(z) = O(1), z ∈ D+ \ {0}, z → 0. Since the matrices Y (t, x, λ) and Z(t, x, λ) are the solutions of the t− and x−equations (7), (8), they are linear dependent. Consequently, there exists a transition matrix T (λ), independent of x and t, such that Y (t, x, λ) = Z(t, x, λ)T (λ). (33) The transition matrix is equal to T (λ) = Z−1(0, 0, λ)Y (0, 0, λ) = w−1(0, λ)Φ(0, λ), and hence T (λ) = ΛT (λ)Λ−1, λ ∈ R \ {0}, i.e., T (λ) has the form T (λ) = ( a(λ) b(λ) −b(λ) a(λ) ) . (34) The scattering relation (33) can be written in the form Y [1](t, x, λ) =a(λ)Z[1](t, x, λ)− b(λ)Z[2](t, x, λ), λ ∈ R \ {0}, (35) Y [2](t, x, λ) =a(λ)Z[2](t, x, λ) + b(λ)Z[1](t, x, λ), λ ∈ R \ {0}. (36) Relations (35), (36) give a(λ) = det(Z[1](t, x, λ), Y [2](t, x, λ)), a(λ) = det(Y [1](t, x, λ), Z[2](t, x, λ)), b(λ) = det(Y [2](t, x, λ), Z[2](t, x, λ)), b(λ) = −det(Z[1](t, x, λ), Y [1](t, x, λ)). It is easy to see that the matrix w(0, λ) = ( α(λ) −β(λ) β(λ) α(λ) ) (37) is the spectral function of the x-equation for t = 0, which is uniquely defined by the given initial functions E(0, x), ρ(0, x) and N (0, x), and the matrix Φ(0, λ) = ( A(λ) B(λ) −B(λ) A(λ) ) (38) 134 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems is the spectral function of the t-equation for x = 0, which is uniquely defined by the boundary condition E(t, 0). The functions α(λ), β(λ) and α(λ), β(λ) have analytic continuations in Ω+ ∪ ∪D− and Ω− ∪D+, respectively, the functions A(λ), B(λ) and A(λ), B(λ) have analytic continuations in C+ and C−, respectively. Let f?(z) := f(z) denote the Schwartz conjugate of a function f(z). Then an- alytic continuations are denoted as (α(z), β(z), α?(z), β?(z), A(z), B(z), A?(z), B?(z)) for z in the domains of their analyticity. They have the following asymp- totic behavior: α(z) =1 + O(z−1), β(z) =O(z−1), z →∞, z ∈ Ω+; α(z) =O(1), β(z) =O(1), z → 0, z ∈ D−; α?(z) =1 + O(z−1), β?(z) =O(z−1), z →∞, z ∈ Ω−; α?(z) =O(1), β?(z) =O(1), z → 0, z ∈ D+; A(z) =1 + O(z−1), B(z) =O(z−1), z →∞, z ∈ C+; A?(z) =1 + O(z−1), B?(z) =O(z−1), z →∞, z ∈ C−; A(z) =O(1), B(z) =O(1), z → 0, z ∈ C+; A?(z) =O(1), B?(z) =O(1), z → 0, z ∈ C−. The entries of the transition matrix T (λ) in the domains of their analyticity are equal to a(z) =α(z)A(z)− β(z)B(z), z ∈ Ω+; b(z) =α?(z)B(z) + β?(z)A(z), z ∈ D+; a?(z) =α?(z)A?(z)− β?(z)B?(z), z ∈ Ω−; b?(z) =α(z)B?(z) + β(z)A?(z), z ∈ D−. The spectral functions a(z) and b(z) are defined and smooth for z ∈ Σ\{0}. The determinant of T (z) ≡ 1 and, hence, a(z)a?(z) + b(z)b?(z) ≡ 1 for z ∈ Σ \ {0}. The spectral functions have the asymptotics: a(z) =1 + O(z−1), z →∞, z ∈ Ω+; (39) b(z) =O(1), z → 0, z ∈ D+; a?(z) =1 + O(z−1), z →∞, z ∈ Ω−; (40) b?(z) =O(1), z → 0, z ∈ D−. If the function a(z) has zeroes zj ∈ Ω+, j = 1, n, then a(zj) = det(Z[1](t, x, zj), Y [2](t, x, zj)) = 0. Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 135 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) Therefore, the vector columns are linear dependent: Y [2](t, x, zj) = γjZ[1](t, x, zj), γj = B(zj) α(zj) = A(zj) β(zj) , j = 1, n. (41) At the conjugate points zj ∈ Ω−, j = 1, n, a?(zj) = det(Y [1](t, x, zj), Z[2](t, x, zj)) = 0. Consequently, Y [1](t, x, zj) = γjZ[2](t, x, zj), γj = B?(zj) α?(zj) = A?(zj) β?(zj) , j = 1, n. (42) If the function b(z) has zeroes ζk ∈ D+, k = 1,m, then b(ζk) = det(Y [2](t, x, ζk), Z[2](t, x, ζk)) = 0. Therefore, Z[2](t, x, ζk) = ηkY [2](t, x, ζk), ηk = B(ζk) β?(ζk) = − A(ζk) α?(ζk) , k = 1,m. (43) At the conjugate points ζk ∈ D−, k = 1,m, b?(ζk) = −det(Z[1](t, x, ζk), Y [1](t, x, ζk)) = 0. Hence, Z[1](t, x, ζk) = −ηkY [1](t, x, ζk), ηk = B?(ζk) β(ζk) = −A ?(ζk) α(ζk) , k = 1,m. (44) Let us define the matrix M(t, x, z)=  ( Z[1](t, x, z)eizt−ixµ(z) Y [2](t, x, z) a(z) e−izt+ixµ(z) ) , z∈Ω+,( Y [1](t, x, z) a?(z) eizt−ixµ(z) Z[2](t, x, z)e−izt+ixµ(z) ) , z∈Ω−,( Y [2](t, x, z) b(z) eizt−ixµ(z) Z[2](t, x, z)e−izt+ixµ(z) ) , z∈D+,( Z[1](t, x, z)eizt−ixµ(z) −Y [1](t, x, z) b?(z) e−izt+ixµ(z) ) , z∈D−. (45) The matrix M is analytic for z ∈ C \ Σ if a(z) 6= 0 and b(z) 6= 0. It is meromorphic for z ∈ C \Σ and has poles at zj ∈ Ω+ and zj ∈ Ω−, where a(zj) = 136 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems 0 and a?(zj) = 0 j = 1, n. The matrix M also has poles at ζk ∈ D+ and ζk ∈ D−, where b(ζk) = 0 and b?(ζk) = 0, k = 1,m. It has the asymptotics M(t, x, z) = I + O(z−1) as z →∞. Let the contour Σ have the orientation shown in Fig. 1. Then the matrix (45) has jumps over the contour Σ, M(t, x, λ)− = M+(t, x, λ)J(t, x, λ), λ ∈ Σ \ {0}, where J(t, x, λ) =  1 + \|r(λ)\|2 −r(λ)e−2iθ(t,x,λ) −r(λ)e2iθ(t,x,λ) 1  , λ ∈ R, \|z\| > 1 2 , =  1 −r−1(λ)e−2iθ(t,x,λ) −r−1(λ)e2iθ(t,x,λ) 1 + \|r(λ)\|−2  , λ ∈ R, \|z\| < 1 2 , λ 6= 0; (46) =  0 −r(λ)e−2iθ(t,x,λ) r−1(λ)e2iθ(t,x,λ) 1  , λ ∈ Cup, =  0 r?(λ)e−2iθ(t,x,λ) −r?−1(λ)e2iθ(t,x,λ) 1  , λ ∈ Clow, (47) where r(λ) := b(λ)/a(λ) is defined on R ∪ Cup, r?(λ) := b?(λ)/a?(λ) is defined on R ∪ Clow, and θ(t, x, λ) = λt − xµ(λ), λ ∈ Σ \ {0}. It is easy to see that det J(t, x, λ) ≡ 1, λ ∈ Σ \ {0}. Suppose the number of zeroes is finite and these zeroes are simple poles of M(t, x, z), i.e., a(zj) = 0, a′(zj) = da(z) dz ∣∣∣ z=zj 6= 0, zj ∈ Ω+, j = 1, n, and b(ζk) = 0, b′(ζk) = db(z) dz ∣∣∣ z=ζk 6= 0, ζk ∈ D+, k = 1,m. Then res z=zj M [2](t, x, z) = γj a′(zj) e−2iθ(t,x,zj)M [1](t, x, zj), (48) res z=zj M [1](t, x, z) = γj a?′(zj) e2iθ(t,x,zj)M [2](t, x, zj), (49) res z=ζk M [2](t, x, z) = ηk b′(ζk) e−2iθ(t,x,ζk)M [1](t, x, ζk), (50) res z=ζk M [1](t, x, z) = ηk b?′(ζk) e2iθ(t,x,ζk)M [2](t, x, ζk), (51) where γj , γj are defined in (41), (42), and ηk, ηk are defined in (43), (44). Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 137 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) 4. Matrix Riemann–Hilbert Problems In this section we give a reconstruction of the solution to the MB equations in terms of the spectral functions a(λ), b(λ), which are defined through the spectral functions A(λ), B(λ), defined by input pulse E1(t), and α(λ), β(λ) defined by initial data E0(x), ρ0(x), N0(t). In the previous section we proved that the matrices (45) are the solutions of the following matrix RH problem: Find a 2× 2 matrix M(t, x, z) such that • M(t, x, z) is meromorphic in z ∈ C \ Σ (or analytic if a(z) 6= 0 and b(z) 6= 0) and continuous up to the set Σ \ {−1 2 , 0, 1 2}; RH1 • If a(zj) = a?(zj) = 0, j = 1, 2, . . . , n and all these zeroes are simple, then M(x, t, z) has poles at the points z = zj , z = zj , j = 1, 2, . . . , n, and the corresponding residues satisfy relations (48) and (49); RH2 • If b(ζj) = b?(ζj) = 0, j = 1, 2, . . . ,m and all these zeroes are simple, then M(x, t, z) has poles at the points z = ζj , z = ζj , j = 1, 2, . . . ,m, and the corresponding residues satisfy relations (50) and (51); RH3 • M−(t, x, λ) = M+(t, x, λ)J(t, x, λ), λ ∈ Σ \ {−1 2 , 0, 1 2}, where J(t, x, λ) is defined in (46) and (47); RH4 • M(t, x, z) is bounded in the neighborhoods of the points {−1 2 , 0, 1 2}; • M(t, x, z) = I +O(z−1), \|z\| → ∞. RH5 Theorem 2. Let the functions E(t, x), N (t, x) and ρ(t, x) be the solution of the mixed problem (5)–(6) for the Maxwell–Bloch equations. Let the corre- sponding spectral functions a(λ) and b(λ) have finite and simple zeroes in the domains of their analyticity. Then there exists the matrix M(t, x, z) which is the solution of the Riemann–Hilbert problem RH1–RH5, and the complex electric field envelope E(t, x) is defined by the relation E(t, x) =− lim z→∞ 4izM12(t, x, z). (52) P r o o f. The existence of the matrix M(t, x, z) follows from the above con- siderations. We only need to prove equation (52). The matrix M(t, x, z) defines the solution Φ(t, x, z) of the AKNS equations (7) and (8) by the formula Φ(t, x, z) = M(t, x, z)e−iθ(t,x,z)σ3 . Formula (52) follows from (7) and (RH5). Indeed, by substituting the last for- mula into equation (7), we find Mt + iz[σ3,M ] +HM = 0. (53) 138 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems Using (RH5), we put M(t, x, z) = I + m(t, x) z + o(z−1), where m(t, x) = lim z→∞ z(M(t, x, z)− I). This asymptotics and equation (53) give H(t, x) = −i[σ3,m(t, x)], and hence E(t, x) = −4im12 = − lim z→∞ 4izM12(t, x, z). Taking into account the well-known fact that a(z) can have multiple zeroes or infinitely many zeroes with limit points on the contour Σ or zeroes on Σ (the so-called spectral singularities), to avoid the complexities, we propose below another formulation of the matrix RH problem. We use the ideas proposed in [6, 7, 21] and introduce another set of solutions of the AKNS equations with suitable asymptotic behavior of the solutions in the vicinity of the origin (z = 0). 5. Basic Solutions with Well-Controlled Asymptotic Behavior at z = 0 We will use here the gauge AKNS equations defined by (11), (12) and (13), (14). Let X̂(t, x, λ) be a product of the matrices X̂(t, x, λ) = Ŵ (t, x, λ)Φ̂(t, λ), (54) where Ŵ (t, x, λ) satisfies the x-equation (12) for all t and Ŵ (t, 0, λ) = I, and Φ̂(t, λ) satisfies the t-equation (11) for x = 0 under the initial condition Φ̂(0, λ) = I. By the same way as in Lemmas 3.2 and 3.3, we prove the existence of representations Φ̂(t, λ) = e−itλσ3 + iλ t∫ −t K̂0(t, τ)e−iτλσ3dτ (55) and Ŵ (t, x, λ) = eixµ(λ)σ3 + x∫ −x L̂0(x, y, t)eiyµ(λ\|σ3dy+ iλ x∫ −x M̂0(x, y, t)eiyµ(λ\|σ3dy (56) Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 139 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) with some matrix functions K̂0(t, τ), L̂0(x, y, t), M̂0(x, y, t) which have the same symmetry properties as in Lemmas 2–5: L̂0(x, y, t) = ΛL̂0(x, y, t)Λ−1, etc. These representations guarantee the analyticity of X̂ in z ∈ C \ {0} and the following asymptotics as z → 0: X̂[1](t, x, z)eiθ(t,x,z) = ( 1 0 ) + O(z), Im z ≥ 0, z → 0; X̂[2](t, x, z)e−iθ(t,x,z) = ( 0 1 ) + O(z), Im z ≤ 0, z → 0. Due to Lemma 1, the matrix (54) X̂(t, x, λ) is a compatible solution of the gauge AKNS Eqs. (11), (12). Then, due to Sec. 2, the matrix X(t, x, λ) = g(t, x)X̂(t, x, λ) (57) with g defined by (15)–(17) is a solution of the AKNS system of Eqs. (7), (8). The matrix X is analytic in z ∈ C \ {0} and its columns have the asymptotics: X[1](t, x, z)eiθ(t,x,z) = g(t, x) ( 1 0 ) + O(z), Im z ≥ 0, z → 0; (58) X[2](t, x, z)e−iθ(t,x,z) = g(t, x) ( 0 1 ) + O(z), Im z ≤ 0, z → 0. (59) Finally, X is bounded in z for any compact set of the complex plane except the vicinity of the origin and for any fixed t and x. Let Ẑ(t, x, λ) be a product of the matrices Ẑ(t, x, λ) = Ψ̂(t, x, λ)ŵ(x, λ), (60) where Ψ̂(t, x, λ) satisfies the t-equation (11) for all x and Ψ̂(0, x, λ) = I, and ŵ(x, λ) satisfies the x-equation (12) with t = 0 under the initial condition ŵ(l, λ) = eilµ(λ)σ3 , where µ(λ) = λ + 1 4λ . If l = ∞, then the initial condition takes the form lim x→∞ ŵ(x, λ)e−ixµ(λ)σ3 = I. It is easy to see that due to Lemma 1, the matrix Ẑ(t, x, λ) is a compatible solution of the gauge AKNS Eqs. (11), (12). Again we prove the existence of representations Ψ̂(t, x, λ) = e−itλσ3 + iλ t∫ −t K̂Ψ(t, τ, x)e−iτλσ3dτ 140 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems and ŵ(x, λ) = eixµ(λ)σ3 + 2l−x∫ x L̂w(x, y)eiyµ(λ\|σ3dy + iλ 2l−x∫ x M̂w(x, y)eiyµ(λ\|σ3dy with some matrix functions K̂Ψ(t, τ, x), L̂w(x, y), M̂w(x, y) which have the men- tioned above symmetry properties. These representations guarantee the analytic- ity of the columns Ẑ[1](t, x, z) and Ẑ[2](t, x, z) of the matrix (60) in the domains Imµ(z) ≥ 0 and Imµ(z) ≤ 0, respectively, and give the following asymptotics as z → 0: Ẑ[1](t, x, z)eiθ(t,x,z) = ( 1 0 ) + O(z), Imµ(z) ≥ 0, z → 0; (61) Ẑ[2](t, x, z)e−iθ(t,x,z) = ( 0 1 ) + O(z), Imµ(z) ≤ 0, z → 0. (62) Due to Section 2, the matrix Z̃(t, x, λ) = g(t, x)Ẑ(t, x, λ) (63) with g defined by (15), (16), (17) is a solution of the AKNS equations (7), (8). The columns Z̃[1](t, x, z) and Z̃[2](t, x, z) of the matrix Z̃ are analytic in the domains Imµ(z) ≥ 0 and Imµ(z) ≤ 0, respectively, and these columns have the asymptotics: Z̃[1](t, x, z)eiθ(t,x,z) = g(t, x) ( 1 0 ) + O(z), Im z ≥ 0, z → 0; (64) Z̃[2](t, x, z)e−iθ(t,x,z) = g(t, x) ( 0 1 ) + O(z), Im z ≤ 0, z → 0. (65) Thus the construction of the compatible solutions (57) and (63) with well- controlled behavior (58), (59) and (64), (65) as z → 0 is finished. Introduce the corresponding spectral functions. They are generated by the linear dependence of the above solutions. Thus, for Imµ(λ) = 0, the matrices Z̃(t, x, λ) and X(t, x, λ) are linear dependent: Z̃(t, x, λ) = X(t, x, λ)T Z̃ (λ), T Z̃ (λ) = Z̃(0, 0, λ) = ŵ(0.λ) = ( α?0(λ) β0(λ) −β?0(λ) α0(λ) ) . (66) The spectral functions α0(λ), α?0(λ) = α0(λ) and β0(λ), β?0(λ) = β0(λ) are defined by the gauge x-equation (12) with t = 0 which, in turn, is determined by the function g(0, x) and the initial functions E0(x), ρ0(x) and N0(x). These spectral Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 141 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) functions are analytic in z ∈ D+ and, due to (61), (62), have the asymptotic behavior: α0(z) = 1 + O(z), β0(z) = O(z), z → 0, z ∈ D+ \ {0}. (67) The functions α?0(z) and β?0(z) are analytic in z ∈ D− and have the asymptotics: α?0(z) = 1 + O(z), β?0(z) = O(z), z → 0, z ∈ D− \ {0}. (68) Formulas (67), (68) yield that there exists ε > 0 and a set Oε = {z ∈ C : \|z\| < ε} such that α0(z) 6= 0, z ∈ D+ ∩Oε \ {0}; α?0(z) 6= 0, z ∈ D− ∩Oε \ {0}. The relation Z(t, x, λ) = X(t, x, λ)TZ(λ), Imµ(λ) = 0, where TZ(λ) = X−1(0, 0, λ)Z(0, 0, λ) = g−1(0, 0)w(0, λ), generates the spectral functions TZ(λ) = g−1(0, 0) ( α(λ) −β?(λ) β(λ) α?(λ) ) =: ( αg(λ) −β?g (λ) βg(λ) α?g(λ) ) , (69) where g−1(0, 0) is the unitary matrix, and the entries of the matrix w(0, λ) were defined in (37). Finally, the relation Y (t, x, λ) = X(t, x, λ)TY (λ), λ ∈ R \ {0}, defines the transition matrix TY (λ) = X−1(0, 0, λ)Y (0, 0, λ) = g−1(0, 0)Φ(0, λ), which has the form TY (λ) = g−1(0, 0) ( A(λ) B(λ) −B(λ) A(λ) ) =: ( Ag(λ) Bg(λ) −Bg(λ) Ag(λ) ) , Imλ = 0, (70) where the entries of the matrix Φ(0, λ) were defined in (38). The spectral functions defined by ŵ(0, λ) and w(0, λ) are related each other by the formula ŵ(0, λ) = g−1(0, 0)w(0, λ), Imµ(λ) = 0. (71) Indeed, the matrices w(x, λ) and g(0, x)ŵ(x, λ) are the solutions of (8). Therefore they are linear dependent, i.e., w(x, λ) = g(0, x)ŵ(x, λ)C(λ), where C(λ) = e−ilµ(λ)σ3g−1(0, l)eilµ(λ)σ3 . Without loss of generality, we put N0(l) = 142 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems 1. Then ρ0(l) = 0, g−1(0, l) = e−χ(0,l)σ3 where χ(0, l) = l∫ 0 h(0, x)dx + χ(0, 0). Since χ(0, 0) is a free parameter, we can put χ(0, l) = 0. Then ŵ(x, λ) = g−1(0, x)w(x, λ), and hence (71) is valid. Finally, from (66) and (69), we find( α?0(λ) β0(λ) −β?0(λ) α0(λ) ) = ( αg(λ) −β?g (λ) βg(λ) α?g(λ) ) . (72) All the introduced compatible solutions X, Y , Z, Z̃ of the AKNS equations (7), (8) and the spectral functions defined by T (λ) (34), T Z̃ (λ) (66), TZ(λ) (69), TY (λ) (70) allow us to formulate a new matrix RH problem which is regular, i.e, without poles. Moreover, this formulation does not require any additional assumptions on the absence of spectral singularities. 6. Regular Matrix Riemann–Hilbert Problem and a Solution of the Mixed Problem for the Maxwell–Bloch Equations Due to (39), a(z) = 1 + O(z−1) and a?(z) = 1 + O(z−1) as z →∞. Fig. 2. The domains OR, Oε and the oriented contour Γ. OR = {z ∈ C : \|z\| > R}, Oε = {z ∈ C : \|z\| < ε}, C+ R = {z ∈ C ∣∣ \|z\| = R, arg z ∈ (0, π)}, C−R = {z ∈ C ∣∣ \|z\| = R, arg z ∈ (π, 2π)}, C+ ε = {z ∈ C ∣∣ \|z\| = ε, arg z ∈ (0, π)}, C−ε = {z ∈ C ∣∣ \|z\| = ε, arg z ∈ (π, 2π)}, Γ := R ∪ Cε ∪ CR = R ∪ C+ ε ∪ C−ε ∪ C+ R ∪ C − R . Hence there exists R > 0 and a set OR = {z ∈ C : \|z\| > R} such that a(z) 6= 0, z ∈ Ω+ ∩OR; a?(z) 6= 0, z ∈ Ω− ∩OR. Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 143 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) Define the new contour Γ which contains the real axis R orientated from left to right and two circles of the radii ε and R oriented clockwise, i.e., Γ := R ∪ Cε ∪ CR (Fig. 2). Now, using the vectors Y [1], Y [2], Z[1], Z[2] from Sec. 3 and the vectors (58), (59), (64), (65), we can define a new matrix M reg(t, x, z)= ( Z[1](t, x, z)eiθ(t,x,z) Y [2](t, x, z) a(z) e−iθ(t,x,z) ) , z∈Ω+ ∩OR, (73) = ( X[1](t, x, z)eiθ(t,x,z) Z̃[2](t, x, z) α?g(z) e−iθ(t,x,z) ) , z∈D+ ∩Oε, (74) = ( X[1](t, x, z)eiθ(t,x,z) X[2](t, x, z)e−iθ(t,x,z) ) , ε < \|z\| < R, (75) = ( Z̃[1](t, x, z) αg(z) eiθ(t,x,z) X[2](t, x, z)e−iθ(t,x,z) ) , z∈D− ∩Oε, (76) = ( Y [1](t, x, z) a?(z) eiθ(t,x,z) Z[2](t, x, z)e−iθ(t,x,z) ) , z∈Ω− ∩OR, (77) which satisfies the following properties: • M reg(t, x, z) is analytic in z ∈ C \ Γ and continuous up to the set R \ {−R,−ε, ε,R}; RRH1 • M reg − (t, x, λ) = M reg + (t, x, λ)Jreg(t, x, λ), λ ∈ Γ \ {−R,−ε, 0, ε, R} RRH2 • M reg(t, x, z) is bounded in the vicinity of the points {−R,−ε, 0, ε, R}; RRH3 • M reg(t, x, z) = M0(t, x) +O(z), \|z\| → 0. RRH4 • M reg(t, x, z) = I +O(z−1), \|z\| → ∞. RRH5 Formulas (73)–(77) give the jump matrices on the circles: Jreg(t, x, λ) =  Ag(λ) a(λ) −Bg(λ) a(λ) e−2iθ(t,x,λ) −βg(λ)e2iθ(t,x,λ) αg(λ)  , λ ∈ C+ R ; (78) = ( 1 −r?g(λ)e−2iθ(t,x,λ) 0 1 ) , λ ∈ C+ ε ; (79) 144 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems Jreg(t, x, λ) = ( 1 0 rg(λ)e2iθ(t,x,λ) 1 ) , λ ∈ C−ε ; (80) =  α?g(λ) β?g (λ)e−2iθ(t,x,λ) B? g(λ) a?(λ) e2iθ(t,x,λ) A?g(λ) a?(λ)  , λ ∈ C−R (81) and on the real line: Jreg(t, x, λ) =  1 \|α?g(λ)\|2 r?g(λ)e−2iθ(t,x,λ) rg(λ)e2iθ(t,x,λ) 1  , λ ∈ (−ε, ε) \ {0}; (82) = ( 1 0 0 1 ) , ε < \|λ\| < R Imλ = 0; (83) =  1 \|a(λ)\|2 −r(λ)e−2iθ(t,x,λ) −r(λ)e2iθ(t,x,z) 1  , λ ∈ (−∞,−R) ∪ (R,∞), (84) where rg(λ) = βg(λ) αg(λ) , r?g(λ) = β?g (λ) α?g(λ) , r(λ) = b(λ) a(λ) , r(λ) = b(λ) a(λ) . The matrix M reg(t, x, z) is bounded in the vicinity of the points {−R,−ε, 0, ε, R} by the construction. Further, at the point z = 0, there ex- ist nontangential limits lim z→0±i0 M reg(t, x, z) = M0(t, x), where M0(t, x) = g(t, x). Hence, M reg(t, x, z) has a continuation up to a contin- uous function at the point z = 0 . Indeed, taking into account (58), (59), (61), (62), (67), (68) and (72), we have that lim z→0+i0 M reg(t, x, z) = lim z→0+i0 ( X[1](t, x, z)eiθ(t,x,z) Z̃[2](t, x, z) α0(z) e−iθ(t,x,z) ) = g(t, x) ( 1 0 0 1 ) ; and lim z→0−i0 M reg(t, x, z) = lim z→0−i0 ( Z̃[1](t, x, z) α0(z) eiθ(t,x,z) X[2](t, x, z)e−iθ(t,x,z) ) Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 145 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) = g(t, x) ( 1 0 0 1 ) . Asymptotics at the infinity is evident in view of (29), (30), (31), (32), (39), (40). Theorem 3. Let the functions E(t, x), N (t, x) and ρ(t, x) be a solution of the mixed problem (5)–(6) to the Maxwell–Bloch equations. Then there exists the matrix M(t, x, z) which solves the regular Riemann–Hilbert problem RRH1– RRH5. The matrix M(t, x, z) gives the field E(t, x) by the relation E(t, x) =− lim z→∞ 4izM12(t, x, z). (85) The entries N (t, x) and ρ(t, x) of the density matrix F (t, x) are defined as follows: F (t, x) = M0(t, x)σ3M −1 0 (t, x), M0(t, x) = lim z→0±i0 M(t, x, z). (86) P r o o f. The existence of the matrix M(t, x, z) follows from the above con- siderations. We only need to prove equations (85) and (86). The matrix M(t, x, z) defines the solution Φ(t, x, z) of the AKNS equations (7) and (8) by the formula Φ(t, x, z) = M(t, x, z)e−iθ(t,x,z)σ3 . Formulas (85) follow from (7) and (RRH5). Indeed, substituting the last formula into Eq. (7), we can find Mt + iz[σ3,M ] +HM = 0. (87) Using (RRH5), we put M(t, x, z) = I + m(t, x) z + o(z−1), where m(t, x) = lim z→∞ z(M(t, x, z)− I). This asymptotics and Eq. (87) give H(t, x) = −i[σ3,m(t, x)], and hence E(t, x) = −4im12 = − lim z→∞ 4izM12(t, x, z). Further, since lim z→0±i0 M reg(t, x, z) = M0(t, x), then the x-equation for M(t, x, z), Mx + i 4z Mσ3 = iz[σ3,M ] +HM + iF 4z M, gives F (t, x) = M0(t, x)σ3M −1 0 (t, x). Thus the mixed initial boundary value problem in the quarter xt-plane to the Maxwell–Bloch equations without spectral broadening is completely linearizable. 146 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems 7. The Study of the Matrix Riemann–Hilbert Problem It remains to consider the matrix Riemann–Hilbert problem independently of its origin. Namely, let the conjugation contour Γ = R ∪Cε ∪CR and all spectral functions be given. Let us consider the following problem: Find a 2×2 matrix M(t, x, z) such that the conditions RRH1–RRH5 are satisfied by the jump matrices given by (78)–(84). Let t and x be fixed. We look for the solution M(t, x, z) of the RH problem in the form M(t, x, z) = I + 1 2πi ∫ Γ P (t, x, s)[I − J(t, x, s)] s− z ds, z /∈ Γ. (88) The Cauchy integral (88) provides all properties of the RH problem (cf. [8]) if and only if the matrix Q(t, x, λ) := P (t, x, λ) − I satisfies the singular integral equation Q(t, x, z)−K[Q](t, x, z) = R(t, x, z), z ∈ Γ. (89) The singular integral operator K and the right-hand side R(t, x, z) are as follows: K[Q](t, x, z) := 1 2πi ∫ Γ Q(t, x, s)[I − J(t, x, s)] s− z+ ds, R(t, x, z) := 1 2πi ∫ Γ I − J(t, x, s) s− z+ ds. We consider this integral equation in the space L2(Γ) of the 2×2 matrix complex valued functions Q(z) := Q(t, x, z), z ∈ Γ. The norm of Q ∈ L2(Γ) is given by \|\|Q\|\|L2(Γ) = ∫ Γ tr(Q∗(z)Q(z))\|dz\| 1/2 =  2∑ j,l=1 ∫ Γ \|Qjl(z))\|2\|dz\| 1/2 . The operator K is defined by the jump matrix J(t, x, z) and the generalized function 1 s− z+ = lim z′→z,z′∈side+ 1 s− z′ . For the unique solvability of this integral equation the conjugation contour and the jump matrices on its non real part should be Schwartz symmetric. The contour Γ is symmetric with respect to the real axis, and the jump matrices satisfy the relations: J(t, x, λ)\|λ∈C+ ε = J∗−1(t, x, λ)\|λ∈C−ε , J(t, x, λ)\|λ∈C+ R = J∗−1(t, x, λ)\|λ∈C−R , Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 147 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) which can be easily verified under the clockwise orientation of the contours C±ε and C±R . We recall that the asterisk is the Hermitian conjugation. It means that the Schwartz symmetry principle is performed. Further, the matrix J(t, x, λ) + J∗(t, x, λ) is positive definite on the real axis (λ ∈ R). Then Theorem 9.3 from [31] (p. 984) guarantees the L2 invertibility of the operator Id−K (Id is the identical operator). The function R(t, x, z) belongs to L2(Γ) because I − J(t, x, z) ∈ L2(Γ) when z ∈ Γ, and the Cauchy operator C+[f ](z) := 1 2πi ∫ Γ f(s) s− z+ ds = f(z) 2 + p.v. 1 2πi ∫ Γ f(s) s− z ds is bounded in the space L2(Γ) [26]. Therefore, the singular integral equation (89) has a unique solution Q(t, x, z) ∈ L2(Γ) for any fixed x, t ∈ R, and formula (88) gives the solution of the above RH problem. The uniqueness can be proved in the same way as in [8] (p. 194–198). Thus the next theorem is valid. Theorem 4. Let a contour Γ and a jump matrix J(t, x, z) (78)–(84) sat- isfy the Schwartz symmetry principle. Let I − J(t, x, .) ∈ L2(Γ) ∩ L∞(Γ). Then for any fixed and real t, x the regular RH problem RRH1–RRH5 has the unique solution M(t, x, z) given by (88). Theorem 5. Let M(t, x, z) be the solution of the RH problem RRH1– RRH5 given by Theorem 4. If M(t, x, z) is absolutely continuous (smooth) in t and x, then Φ(t, x, z) (z ∈ C \ Γ) satisfies the AKNS equations Φt =− (izσ3 +H(t, x))Φ, (90) Φx = ( izσ3 +H(t, x) + i F (t, x) 4z ) Φ (91) almost everywhere (point-wise) with respect to t and x. The matrix H(t, x) is given by H(t, x) = −i[σ3,m(t, x)], m(t, x) = 1 π ∫ Γ (I +Q(t, x, λ))(J(t, x, λ)− I)dλ, (92) where Q(t, x, λ) is the unique solution of the singular integral equation (89). The matrix F (t, x) = M(t, x, 0)σ3M −1(t, x, 0) is Hermitian and it has the struc- ture F (t, x) = ( N (t, x) ρ(t, x) ρ∗(t, x) −N (t, x) ) . 148 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems P r o o f. The matrix Φ(t, x, z) := M(t, x, z)e(−izt+iη(z)x)σ3 is analytic in z ∈ C \ Γ and has a jump across the contour Γ, Φ−(t, x, λ) = Φ+(t, x, λ)J0(λ), λ ∈ Γ, where J0(λ) := e(iλt−iη(λ)x)σ3Jreg(t, x.λ)e(−iλt+iη(λ)x)σ3 is independent on t and x. This relation implies: dΦ−(t, x, λ) dt Φ−1 − (t, x, λ) = dΦ+(t, x, λ) dt Φ−1 + (t, x, λ), dΦ−(t, x, λ) dx Φ−1 − (t, x, λ) = dΦ+(t, x, λ) dx Φ−1 + (t, x, λ) for λ ∈ Γ. The relations obtained mean that the matrix logarithmic derivatives Φt(t, x, z)Φ −1(t, x, z) and Φx(t, x, z)Φ−1(t, x, z) are analytic in z ∈ C \ {0} with exception of self-intersection points of the contour Γ. The matrix M(t, x, z) and its derivative Mt(t, x, z) are analytic in z ∈ C \ Γ, and the Cauchy integral (88) gives the asymptotic formula M(t, x, z) = I + m±(t, x) z + O(z−2), z →∞, z ∈ C±. Hence, Φt(t, x, z)Φ −1(t, x, z) = −izσ3 + i[σ3,m+(t, x)] + O(z−1) = −izσ3 + i[σ3,m−(t, x)] + O(z−1), z →∞, where m−(t, x) = m+(t, x) = m(t, x) = i 2π ∫ Γ P (t, x, z)[I − J(t, x, z)]dz. Since M(t, x, z) is bounded up to the boundary, then z = 0, the end points and self-intersection points are removable singularities for Φt(t, x, z)Φ −1(t, x, z). Therefore, by Liouville’s theorem, this derivative is a polynomial U(z) := Φt(t, x, z)Φ −1(t, x, z) = −izσ3 −H(t, x), where H(t, x) := −i[σ3,m(t, x)] = ( 0 q(t, x) p(t, x) 0 ) . Using the Schwartz sym- metries of the jump matrix J(t, x, z), we show that U(z) = σ2U(z)σ2, where σ2 =( 0 −i i 0 ) . These reductions imply H(t, x) = −H∗(x, t), i.e., q(t, x) = −p(t, x), and we put q(t, x) := E(t, x)/2. Thus, Φ(t, x, z) satisfies equation (90), and a scalar function E(t, x) is defined by (92). The function E(t, x) is smooth in t and Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 149 M.S. Filipkovska, V.P. Kotlyarov, and E.A. Melamedova (Moskovchenko) x, because the matrix M(x, t, z) and hence m(t, x) are smooth in t and x by supposition. In the same way as above, we find that Φx(x, t, λ)Φ−1(x, t, λ) is a rational matrix function V (z) := Φx(x, t, λ)Φ−1(x, t, λ) = izσ3 +H(t, x) + iF (t, x) 4z because the two asymptotics are true: Φx(t, x, z)Φ−1(t, x, z) = izσ3 +H(t, x) + O(z−1), z →∞ and Φx(t, x, z)Φ−1(t, x, z) = iF (t, x) 4z + F0(t, x) + O(z), z → 0, where F (t, x) = M(t, x, 0)σ3M −1(t, x, 0) and F0(t, x) is some matrix. Moreover, the previous relations give that F0(t, x) ≡ H(t, x). Thus we can see that the matrix Φ(x, t, z) satisfies two differential equations (90) and (91). Their compat- ibility (Φxt(x, t, z) = Φtx(x, t, z)) gives the identity in z, Ux(z)− Vt(z) + [U(z), V (z)] = 0, U = −izσ3 −H, V = izσ3 +H + iF 4z , i.e., Ht(t, x) +Hx(t, x) + [ izσ3 +H(t, x), izσ3 +H(t, x) + iF (t, x) 4z ] = 0. This identity is equivalent to the system of matrix equations: Ht(t, x) +Hx(t, x) = 1 4 [σ3, F (t, x)], (93) Ft(t, x) =[F (t, x), H(t, x)]. (94) Using the Schwartz symmetries of the jump matrix J(t, x, z), we can find that F (t, x) is a Hermitian and traceless matrix, and we put F (t, x) = ( N (t, x) ρ(t, x) ρ∗(t, x) −N (t, x) ) . Matrix equations (93) and (94) are equivalent to the scalar equations (5). Thus, we have proved that the matrix Φ(t, x, z) satisfies Eqs. (90) and (91), which coincide with the AKNS Eqs. (7) and (8), and the scalar functions E(t, x), N (t, x), ρ(t, x) satisfy the MB equations (5) due to the compatibility of Eqs. (90) and (91). 150 Journal of Mathematical Physics, Analysis, Geometry, 2017, Vol. 13, No. 2 Maxwell–Bloch Equations and Matrix Riemann–Hilbert Problems The differentiability of the matrix M(t, x, z) (88) with respect to t and x can be proved if we assume that the boundary and initial conditions exponentially vanish at infinity or have a finite support. Then the introduced spectral functions will be analytic in some strip along the real axis of the complex plane. Further, using a factorization of the jump matrices into upper / lower triangular matrices, one can deform the contour into a complex plane in the neighborhoods of zero and infinity in such a way that on the deformed contour the new jump matrices will be exponentially close to the identity matrices when z → 0 and z →∞. Outside these points, the new contour coincides with Γ. This enables us to differentiate (many times) the singular integral Eq. (89) with respect to the parameters t and x under the integral sign, and thereby to prove the smoothness of the matrices Q(t, x, λ) = P (t, x, λ)−I andM(t, x, z) with respect to t and x. As a consequence, we obtain the smoothness of the matrix F (t, x) = M(t, x, 0)σ3M −1(t, x, 0) and, due to (92), the smoothness of the matrix H(t, x), i.e., the smoothness of the solution E(t, x), N (t, x), ρ(t, x) of the Maxwell–Bloch equations. It remains only to show that these functions give exactly the solution of the mixed problem. 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Maxwell-Bloch Equations without Spectral Broadening: Gauge Equivalence, Transformation Operators and Matrix Riemann-Hilbert Problems

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