Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function

The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.


Introduction
Consider the Maxwell-Bloch (MB) equations written in the form where x ∈ (0, L), L ≤ ∞, t ∈ R + = (0, +∞), λ ∈ R, E = E(t, x) and ρ = ρ(t, x, λ) are complex valued functions, N = N (t, x, λ) is a real function, subscripts mean the partial derivatives in t and x, and denotes complex conjugation.Hermitian conjugation will be denoted by * .It follows from the second and third equations of the system (1.1) that ∂ ∂t N 2 (t, x, λ) + |ρ(t, x, λ)| 2 = 0.This relation gives the well-known normalization condition N 2 (t, x, λ) + |ρ(t, x, λ)| 2 ≡ 1. (1.2) The system of Maxwell-Bloch (MB) equations is a system of integrable nonlinear PDEs which can be solved using the inverse scattering transform (IST) method.The MB equations were originally proposed by Lamb [16,17].Ablowitz, Kaup and Newell [1] have studied the coherent pulse propagation through a resonant two level optical medium and shown that the Maxwell-Bloch equations describing this phenomenon can be solved by the inverse scattering method.Zakharov introduced in the paper [24] the concepts of spontaneous and causal solutions of the MB equations that initiates the systematic application of the IST method to laser problems.This work was further evolved by Gabitov, Zakharov and Mikhailov [10,11,12] which have used the Marchenko integral equations and obtained a new version of IST method.They presented a description of general solutions of the MB equations and gave their classification.Generally, there are many pioneering works relating to the Maxwell-Bloch equations and only a small part of them are cited here.In this paper, we extend the IST method in the form of the matrix Riemann-Hilbert (RH) problem for solving the initial-boundary value problem (IBVP) for the MB equations with a periodic boundary condition.In [8,9], Fokas proposed the unified transform method for solving IBVPs for linear and for integrable nonlinear PDEs, the main idea of which is the simultaneous spectral analysis of both Lax operators (whose compatibility condition is provided by the satisfaction of corresponding nonlinear PDEs) to construct a unifying transform for solving certain IBVPs.We use the similar idea, namely, to construct the associated matrix RH problem, we use compatible solutions of the systems of Ablowitz-Kaup-Newell-Segur (AKNS) equations (in general, we consider the three systems), which are equivalent to the MB equations in the sense that their compatibility conditions are provided by the satisfaction of the corresponding MB equations.Notice that the compatibility condition of the AKNS system is the equivalent of the Lax representation, i.e., the compatibility condition of the Lax pair.Then we obtain the representation of a solution of the MB equations in terms of a solution of the associated RH problem and prove the theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem.
The MB equations are used to describe different physical phenomena, including self-induced transparency [1,17], superfluorescence [10,12] and others related to the problems on the propagation of an electromagnetic wave in a medium with distributed two-level atoms [2,17,20].Short reviews of the physical meaning of the MB equations and the application of the inverse scattering transform method can be found in [1,2,3,12,14,19], and in [13,23] for the reduced MB equations.We consider the problem on the propagation of an electromagnetic wave in a resonant medium with distributed two-level atoms.In (1.1), E(t, x) is the complex envelope of an electric field, N = N (t, x, λ) and ρ = ρ(t, x, λ) are the entries of the density matrix of a quantum two-level atom subsystem: N (t,x,λ) ρ(t,x,λ) ρ(t,x,λ) −N (t,x,λ) , the parameter λ is the deviation of the transition frequency of the given two-level atom from the mean frequency Ω, and L is the length of the attenuator.The weight function n(λ) (n(λ) ≥ 0, λ ∈ R) normalized by the condition ∞ −∞ n(λ) dλ = 1 describes the inhomogeneous broadening of the medium (the shape of the spectral line).Consider general initial and boundary conditions for the mixed problem for the MB equations: E(0, x) = E 0 (x), ρ(0, x, λ) = ρ 0 (x, λ), N (0, x, λ) = N 0 (x, λ), 0 < x < L ≤ ∞, (1.3) where E 0 (x), N 0 (x, λ), ρ 0 (x, λ) and E in (t) are infinitely differentiable for x ∈ (0, L), λ ∈ R and t ∈ R + .Taking into account (1.2), a function N 0 (x, λ) is defined by ρ 0 (x, λ), Here we choose the positive branch of the square root.This means that we consider the problem for an attenuator (self-induced transparency).
The following notation will be used in the paper: [A, B] is the commutator of the matrices A and B, i.e., [A, B] = AB − BA; I denotes the identity operator (matrix); the symbol "p.v." denotes the principal value of the Cauchy integral.Also, we will use the Pauli matrices It is known that the MB equations (1.5) are equivalent to the overdetermined system of linear differential equations (see, e.g., [2,12]) where V (t, x, λ) = iλσ 3 + H(t, x) − iG(t, x, λ), (1.11) F (t, x, s)n(s) s − λ ds, F (t, x, s) = N (t, x, s) ρ(t, x, s) ρ(t, x, s) −N (t, x, s) .
The system (1.8), (1.9) (where U , V of the form (1.10) and (1.11)) is commonly referred to as the system of the Ablowitz-Kaup-Newell-Segur (AKNS) equations for the MB equations.
In what follows, systems of matrix differential equations of the form (1.8), (1.9) and of the form (1.8), (1.12) will be called the AKNS systems.
In [6], the IBVP (1.5)-(1.7)has been studied in the case when n(λ) is the Dirac delta-function, i.e., without inhomogeneous broadening (an unbroadened medium), and asymptotic formulas for its solution in different sectors of the light cone has been obtained, as well as the fulfillment of the causality principle has been shown.The MB equations without inhomogeneous broadening has been also considered in [19], but different initial-boundary conditions were used there and the equations were written in a comoving frame of reference.In the present paper as well as in [6] the MB equations are considered in a laboratory frame.The work [19] provides a proper RH problem that generates the unique causal solution (i.e., the solution vanishes outside of the light cone) of the IBVPs for the MB equations.An application of the IST method to the IBVP (1.1), (1.3) and (1.4) with a smooth and fast decreasing input signal E in (t) was given in [15], using the simultaneous spectral analysis of equations (1.8) and (1.12) and the RH problem.The present paper shows that the IST method in the form of matrix Riemann-Hilbert problem can be used for the case of periodic boundary condition and give an integral representation for E(t, x), N (t, x, λ) and ρ(t, x, λ) through the solution of a singular integral equation.Also, it shows that the RH problem allows to study the long-time asymptotic behavior of a solution of the mixed problem for the MB equations, choosing as an example the problem when E(t, 0) = A 0 e iω 0 t , which is interesting itself.Later, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we will be able to obtain formulas for the asymptotics of a solution of the considered IBVP.
The paper has the following structure.In Section 1, the problem statement and its physical interpretation are given, as well as the AKNS systems for the MB equations and their compatibility conditions are considered.In Section 2, basic (compatible) solutions of the AKNS systems equivalent to the system of the MB equations are constructed and their properties are stated.These solutions will be used to construct the matrix Riemann-Hilbert problem associated with the IBVP (1.5)-(1.7)which is given in Section 3.Then, in Section 4, we obtain the theorems on the existence, uniqueness and smoothness of a solution of the associated RH problem and prove that this RH problem generates a solution of the IBVP (1.5)-(1.7).We give the representation of the solution of the considered IBVP in terms of the solution of the associated RH problem, and also the representation of the electric field envelope in terms of a solution of an integral equation and an equivalent associated RH problem.Also, we prove that the constructed RH problem provides the causality principle.Thus, the problem on the propagation of an electromagnetic wave generated by periodic input signal in a stable medium with distributed two-level atoms (attenuator) is solved.The presented results will be used later to obtain the asymptotics of a solution of the considered IBVP.In Appendix A, the analysis of the phase function is carried out, the results of which are used in proving the theorems.

Basic solutions of the systems of the AKNS equations
Let us suppose that a solution of the MB equations (1.5) exists, then the AKNS systems (1.8), (1.12) (as well as the AKNS system (1.8), (1.9)) are compatible and we can define their "basic" solutions with the properties enabling to obtain the matrix RH problem which generates a solution of the IBVP (1.5)-(1.7).The upper and lower bank equations (1.12) allow us to obtain solutions (which we call basic and denote Y ± , Z ± ; see Section 2.3) that have analytic continuations to the upper and lower complex half-planes C ± = {z ∈ C | ± Im z > 0} (Y ± have analytic continuations to C ± except for certain cuts).First, we will find a "background" solution of the AKNS system (1.8), (1.9), which will be used below.

The background solution
We seek the background solution of the AKNS system (1.8), (1.9) in the form (cf. [6]) where Φ 0 (t, x, λ) is a matrix function with the unit determinant, and constants α 0 , β 0 , a 2 × 2 matrix function M 0 (λ) and scalar functions α(λ), β(λ) are to be determined.Consider the logarithmic derivative of Φ 0 in t: 2) be a matrix with the unit determinant, that is, On the other hand, the relation must be satisfied (because then Φ 0 satisfies the equation (1.8), where E(t, x) = E bg (t, x)).Comparing the obtained relations for the logarithmic derivative (Φ 0 ) t Φ −1 0 , we obtain that We need to find a(λ), b(λ) and α 0 such that α(λ)a(λ)b(λ) is constant, since E bg (t, x) is independent of λ.We introduce the function where z ∈ C (z = λ + iν) and E is a point in the complex plane which will be specified below (see (2.7)), take the branch cut for κ(z) along the closed interval and fix its branch by the asymptotics Further, we introduce the function where z ∈ C and E is a point mentioned above, and fix the branch of w(z) by taking the branch cut along E, E and by asymptotics Thus, κ(z) and w(z) are analytic for z ∈ C \ E, E .Now, we set then (a(λ) + b(λ)) 2 = κ 2 (λ) and Hence, E bg (t, x) = 2 Im E λ+α 0 λ−Re E e 2i(α 0 t+β 0 x) and it is independent of λ if and only if α 0 = − Re E. Therefore, Comparing E bg (t, 0) with E in (t) = A 0 e iω 0 t , we obtain that 2 Im E = A 0 and 2 Re E = −2α 0 = −ω 0 , and hence Thus, the matrix function Φ 0 (t, x, λ) satisfies the equation (Φ 0 ) t = −(iλσ 3 + H 0 (t, x))Φ 0 , and the matrix H 0 (t) := H 0 (t, 0) is defined by the input signal E in (t) (1.7): . (2.8) Further, using the second MB equation ρ bg t + 2iλρ bg = N bg E bg , we find that if ρ bg = ic 0 E bg , then N bg = −2c 0 (λ − Re E).Taking into account the normalization condition we obtain c 0 = 1 2w(λ) and ρ bg = and hence (2.9) Thus, we obtain the periodic solution in the form of a plane wave for the MB equations (1.5): where Im E = A 0 /2, α 0 = − Re E = ω 0 /2, β 0 has the form (2.9), and is the input signal (1.7).The third MB equation is evidently fulfilled.The functions ρ bg (t, x, z) and N bg (t, x, z) are analytic for z ∈ C \ E, E , i.e., they are analytic in the same domain as w(z).Note that ρ bg (t, x, λ) and the partial derivative (N bg ) λ (t, x, λ) are discontinuous in λ at the point Re E. To obtain the infinitely differentiable solution with respect to λ on the whole line R, we have to redefine the function w(z): we draw a branch cut connecting the points E and E via infinity without intersection of the real line and fix the branch of w(z) by the condition w(0) = |E|.The solution (2.10) is equivalent to the background solution obtained in [3, Section II.B], although the work [3] considers a different IBVP for the MB equations, written in a comoving frame of reference, with similar nonzero boundary conditions as t → +∞, where t is a retarded time and does not coincide with the variable t that is used in the present paper (which considers the MB equations in a laboratory frame).In comparison with [3] that considers a more general input signal and studies soliton solutions, the present work carries out a more rigorous analysis: carefully formulates the IST for the IBVP, give the representations of a solution of the IBVP in terms of the associated RH problems and provides conditions for the existence and uniqueness of solutions of the IBVP and the associated RH problems, as well as proves certain important properties of solutions.Now, let us consider the logarithmic derivative of Φ 0 (t, x, λ) (2.1) with respect to x: Since we need to obtain the x-equation of the AKNS system, i.e., the equation (1.9), then ds .
Notice that, generally, instead of condition 2.1 one can require that the function n(λ) belong to the class H * on [−Λ, Λ] [21], which means the following: n(λ) satisfies the Hölder condition on any closed interval in (−Λ, Λ) and near the endpoints ∓Λ can be represented as n(λ) = n i (λ) |λ±Λ| α i , i = 1, 2, respectively, where 0 ≤ α i < 1 and n 1 (λ), n 2 (λ) are defined and satisfy the Hölder condition on sufficiently small parts of [−Λ, Λ] that are adjacent to the endpoints −Λ, Λ, respectively (the contour [−Λ, Λ] is oriented from −Λ to Λ).Then the function η(z) (2.12) is analytic on C \ [−Λ, Λ], continuous up to (−Λ, Λ) and has weak singularities at the points ±Λ the singularities of the type O (z ∓ Λ) −ν , 0 < ν < 1 .Near the endpoints ±Λ, the Cauchy integral in (2.12) admits the estimate , where α 1 , α 2 were defined above (cf.[21]).A function η(z) will be used when constructing the RH problem as well as the basic solutions of the AKNS systems (see below).In order for the RH problem to have no singularities, including weak singularities, at the points ±Λ, we will not use this requirement and keep condition 2.1 for n(λ), although in the case of a more general RH problem it can be used.
Since we consider the problem on the propagation of an electromagnetic wave in the stable medium (i.e., the problem for an attenuator), we can assume that N (t, x, λ) The next known lemma will be used below.
Throughout the paper, A[i] denotes the i-th column of a matrix A.
In the theorem below, for convenience, we assume that L = +∞, i.e., x ∈ R + .For the case when x ∈ (0, L), L < +∞, the theorem remains valid and the proof is similar.
The matrix function Φ(t, λ) (2.15) was obtained above.In addition, it follows from the form of Φ(t, λ) that it is infinitely differentiable in t.
The existence of the matrix functions W ± (t, x, λ), Ψ(t, x, λ), and w ± (x, λ) are proved below.In view of the above, the existence of the solutions (2.14) and (2.16) follows from the existence of these matrix functions.
Along with proving the existence of the solutions, we will prove their properties (b), (c) and (d).Property (a) will be proved at the end.
Since E(t, x) ∈ C 1 (R + ×R + ) is bounded near (0, 0), i.e., bounded on any compact set [0, , where T 0 > 0, L 0 > 0 and Λ 0 > Λ are arbitrary numbers.In addition, we obtain that E(t, x) ∈ L 1 [a, b], 0 ≤ a ≤ b < ∞, with respect to x for each t ∈ R + and with respect to t for each x ∈ R + It follows from the above that W 0 (t, x, λ) ≡ ∥I∥ = 1 and where and using the method of mathematical induction (MMI) it is easy to prove that = e Cx and hence the series , and in view of the arbitrariness of L 0 , T 0 and Λ 0 it converges locally uniformly for all t, x ∈ R + and λ ) is continuous on this set) and continuously differentiable in x since W (t, x, λ) is a solution of the integral equation (2.19) .Moreover, since η(λ), α(t, x, λ) and Υ(t, x, λ) and therefore From the properties of the functions E, N , ρ as a solution of the considered IBVP (1.5)-(1.7) it follows that E t (t, x), N t (t, x, λ) and ρ t (t, x, λ) are bounded for (t, x) close to (0, 0) and where In the same way as above, we prove that the series ∞ n=0 W n t converges absolutely and locally uniformly on
It also follows from the above that there exists the function (analytic continuation) W (t, x, z), analytic in z on C\[−Λ, Λ] and continuous up to the boundary [−Λ, Λ], which satisfies the integral equation (2.19) for z = λ ∈ R \ [−Λ, Λ], Λ < ∞, and has the boundary values W ± (t, x, λ) satisfying the integral equations (2.21) for λ ∈ [−Λ, Λ], Λ ≤ ∞ also, W (t, x, z) is continuously differentiable in t, x for each z .For Λ < ∞, we extend the functions W ± (t, x, λ) by defining them equal to W ± (t, x, λ) := W (t, x, λ) for λ ∈ R \ [−Λ, Λ] accordingly, in a similar way we define all the boundary values included in W ± (t, x, λ), i.e., α ± (t, x, λ) := α(t, x, λ), η ± (λ) := η(λ), etc., for Further, taking into account where we obtain that W − (t, x, z)e −iη(z)x is bounded in z ∈ C cl − , and as z → ∞, z ∈ C − , it has the asymptotics of the form and for fixed x as z → ∞, z ∈ C − , it has the asymptotics It can be proved similarly that W + (t, x, z)e iη(z)x is bounded in z ∈ C cl + , and for fixed x as z → ∞, z ∈ C + , it has the asymptotic behavior Note that if the symbol O(•) is present in a matrix expression, then it denotes a matrix of the appropriate size whose entries have the indicated order.The functions κ(z) (2.Taking into account that w(z for each fixed t we obtain the asymptotics: It follows from the above that the functions Y ± (t, x, λ) have the analytic continuations respectively, bounded in the neighborhood of the point Re E, have the singularities of the type (z − E) −1/4 and (z − E) −1/4 respectively, and in addition, for fixed t and x, Thus, the proof of (c) is completed.To prove that there exists a solution Ψ(t, x, λ) of the t-equation (1.8) (for each x) satisfying the initial condition Ψ(0, x, λ) ≡ I, we prove that there exists a solution Ψ(t, x, λ) = Ψ(t, x, λ)e iλtσ 3 of the equivalent integral equation We represent the solution of (2.22) as the Neumann series Ψ = ∞ n=0 Ψ n , where Ψ 0 (t, x, λ) ≡ I, x, λ)e iλ(t−τ )σ 3 dτ , and prove that the series converges.Since e ±iλ(t−τ ) ≤ 1, then Ψ n (t, x, λ) , and by the MMI we obtain Ψ n (t, x, λ) ,x) .Since (due to the properties of E(t, x)) M (t, x) is bounded on any compact set in R + × R + , then the series ∞ n=0 Ψ n converges absolutely and uniformly in λ on R and in (t, x) on any compact set in R + × R + .Since the functions Ψ n (t, x, λ) are continuous and the series converges locally uniformly on R + × R + × R, then the sum of the series Ψ(t, x, λ) is continuous on R + × R + × R, and since Ψ(t, x, λ) is a solution of the integral equation (2.22), then it is continuously differentiable in t.Obviously, Ψ n (t, x, λ) are continuously differentiable in x.It is easily verified that the series ∞ n=0 Ψ n x (t, x, λ) converges absolutely and locally uniformly on R + × R + × R, and hence Ψ(t, x, λ) is continuously differentiable in x.The function Ψ(t, x, λ) = Ψ(t, x, λ)e −iλtσ 3 has the analytic continuation Ψ(t, x, z) for z ∈ C. Taking into account that Ψ(t, x, λ) has the integral representation (2.22) and that e ±izt → 0 for z ∈ C ± (respectively) as z → ∞ (t ∈ R + ), we obtain that the functions Ψ(t, x, z)e izt and Ψ(t, x, z)e −izt are bounded in z ∈ C cl + and z ∈ C cl − , respectively, and have the asymptotics Similar results can be obtained by using the integral representation for Ψ(t, x, λ) by a transformation operator (cf.[7]).Since H(0, x) ≡ 0 and F (0, x, λ) ≡ −σ 3 , then for t = 0 the x ± -equations (1.12) take the form These equations have the solutions w ± (x, λ) = e iη ± (λ)xσ 3 satisfying the initial conditions lim x→∞ w ± (x, λ)e −iη ± (λ)xσ 3 = I (recall that we prove the theorem, assuming that L = +∞ and x ∈ (0, L)).Obviously, det w ± (x, λ) ≡ 1. Due to the properties of η(z), there exists the function (analytic continuation) w(x, z) = e iη(z)xσ 3 , analytic in z on C \ [−Λ, Λ] and continuous up to [−Λ, Λ], which satisfies the x ± -equations (1.12) for z = λ ∈ R \ [−Λ, Λ] and have the boundary values w ± (x, λ) satisfying (1.12) for λ ∈ [−Λ, Λ], and in addition, It follows from the above that the functions Z ± (t, x, λ) have the analytic continuations Thus, the existence of the solutions Z ± (t, x, λ) (2.16), property (b) for them, and property (d) have been proved.

Formulation of the Riemann-Hilbert problem
Since Y ± (2.14) and Z ± (2.16) are the solutions of the compatible AKNS systems the t-and x ± -equations (1.8), (1.12), they are linearly dependent, namely, they satisfy the "scattering relations" Since the transition matrices (scattering matrices) T ± (λ) are independent of t and x, they can be presented in the form as shown above, Hence, where a(λ), b(λ) and κ(λ) are defined by (2.5) and (2.3).The spectral functions a(λ), b(λ) have the analytic continuations a(z), b(z) to C \ [E, E] which have the following asymptotics at infinity: It was assumed that the interval E, E is oriented downward.For z ∈ E, E the functions a(z), b(z) and κ(z) satisfy the relations where the signs "−" and "+" in subscripts denote nontangential boundary values from the left and right of an oriented contour, and in what follows this notation is also used.It is easy to see that a(z) has no zeros and, hence, a discrete spectrum is empty.
The "reflection coefficient" where Since w(λ) ∈ C ∞ (R), then r(λ) is infinitely differentiable as a function of the real variable λ.

Due to the relations a
where Im z ∈ (Im E, − Im E), Re z = Re E. It satisfies the symmetry relation h Note that the scattering relations (3.1) can be rewritten in the form and also, it follows from (3.1) that where W (f, g) = det(f, g) is the Wronskian (Wronski determinant) of the solutions f , g.In addition, h(z) where η(z) is defined by (2.12), and introduce the contour Σ = R ∪ (E, E) oriented from the left to the right for the real line R and downward for the interval (E, E) (see Figure 1).The analysis of the phase function θ(t, x, z) (3.4) is performed in Appendix A.
Using the scattering relations (3.1), the relations obtained for a(z), b(z), r(z) and h(z), and Theorem 2.2, we obtain the following.The matrix Basic problem RH 0 .Given a contour Σ = R ∪ (E, E) (see Figure 1) and functions r(z) (3.2) (z ∈ R) and h(z) (3.3) (z ∈ (E, E)), find a 2 × 2 matrix function M (t, x, z) satisfying the following conditions: and has the continuous boundary values x, z) is bounded near the point of the contour self-intersection Re E and has the singularities of the type (z − E) −1/4 , z − E −1/4 at the endpoints E, E; • M (t, x, z) satisfies the boundary condition (which is also called the jump condition) where J(t, x, z) is defined as Remark 3.1.The jump matrix J(t, x, z) for z = λ ∈ R takes two different form: ) which solves the basic RH problem RH 0 , and this matrix function defines the complex electric field envelope E(t, x) and the density matrix by the formulas (more precisely, the function F (t, x, λ) is defined by the formula (4.2) for λ ∈ R \ {Re E} and defined appropriately at the point λ = Re E so that F (t, x, λ) is continuous in λ ∈ R).
Proof .The existence of the solution M (t, x, z) (3. x, z), z ∈ Σ \ {Re E}, and N (t, x, z) → I as z → ∞.Using the same arguments as for the proof that det M (t, x, z) ≡ 1, we obtain that N (t, x, z) ≡ I. Consequently, M (t, x, z) ≡ M (t, x, z), which proves the uniqueness of the solution.
Below, we will prove that the basic problem RH 0 has a unique solution M (t, x, z), infinitely differentiable in t, x, and that the matrix function M (t, x, z) generates solutions of the AKNS systems (1.8), (1.12) and, as a result, a solution of the MB equations, and this solution satisfied the initial and boundary conditions.Also, an integral representation of the solution of the MB equations through the solution of a singular integral equation will be given.Theorem 4.3.For each t ∈ R + and each x ∈ (0, L), L ≤ ∞, the basic problem RH 0 has a unique solution M (t, x, z), and this solution is continuous in (t, x) ∈ R + × (0, L).
Proof .Let t and x be arbitrary fixed elements from R + and from (0, L) respectively.The boundary condition M − (t, x, z) = M + (t, x, z)J(t, x, z) on Σ \ {Re E} can be rewritten in the form of the jump condition Since M (∞) = I, then a solution of the basic problem RH 0 , if it exists, has the form Passing to the limit in the equality (4.6) as z → ζ + , where ζ + = lim z→ζ,z∈+side z, ζ ∈ Σ and "+ side" denotes a positive side (a boundary value on the left) of the oriented contour Σ (we exclude the points of self-intersection of the contour), and carrying out elementary transformations, we obtain the following singular integral equation for M + (t, x, ζ): Denote W (t, x, ζ) := M + (t, x, ζ)−I and introduce the singular integral operator K and operator function Π defined by We will prove that for any fixed t, x there exists a unique solution W (t, x, ζ) of the singular integral equation (cf.[15]) which is considered in the space of operator (2×2 matrix) functions , where B L 2 (Σ) is the (Banach) space of bounded linear operators from L 2 (Σ) into L 2 (Σ), since Σ is a Carleson curve (see, e.g., [22] and references therein, or [18]).The function I − J(t, x, s) belongs to L 2 (Σ) with respect to s.Thus, Π(t, x, ζ) ∈ L 2 (Σ) as a function of the variable ζ.Note that I − J(t, x, s) ∈ L ∞ (Σ) in s.Since Re J(t, x, z) is positive definite for z ∈ R, det J(t, x, z) ≡ 1, the contour Σ is symmetric with respect to the real axis R and J −1 (t, x, z) = J * (t, x, z) for z ∈ Σ \ R thus, the contour Σ and the matrix J −1 (t, x, z) satisfy the Schwarz reflection principle , then it follows from [25, Theorem 9.3, p. 984] that there exists the operator (I − K) −1 ∈ B L 2 (Σ) .Consequently, the singular integral equation (4.7) has a unique solution W (t, x, ζ) for any fixed t and x, which belongs to L 2 (Σ) with respect to ζ.Therefore, there exists the solution M (t, x, z) of the basic problem RH 0 , which can be obtained by the formula Since the operator I −K ∈ B L 2 (Σ) and the operator function Π ∈ L 2 (Σ) depends continuously on the parameters (t, x) ∈ R + × (0, L), then the inverse operator (I − K) −1 and the solution W (t, x, ζ) of the equation (4.7) also depend continuously on (t, x).Taking into account the form (4.8) of the solution M (t, x, z), we obtain that M (t, x, z) is continuous in (t, x) ∈ R + ×(0, L).
The proof of the uniqueness of the solution M (t, x, z) is carried out in the same way as for Theorem 4.1.■ Theorem 4.4.For any (t, x) ∈ R + × (0, L), L ≤ ∞, a solution of the basic problem RH 0 is infinitely differentiable in t and x.
Proof .By Theorem 4.3, the basic problem RH 0 has a unique solution M (t, x, z) which can be obtained by the formula (4.8).
In order for the solution M (t, x, z), as well as the solution W (t, x, ζ) of the singular integral equation (4.7), to be differentiable in t and x, it is necessary that the integrals that arise when differentiating the solution M (t, x, z) and the equation (4.7) be convergent.The matrix J(t, x, z) and its derivatives in t and x are responsible for the decrease of the integrands.
On unbounded parts of the contour Σ, i.e., for z ∈ R with |z| ≫ 1 (|z| is sufficiently large), the jump matrix has the form (see Remark 3.1) Consider the case when t ≤ x.In this case M (t, x, z) = G (1) (t, x, z) −1 , where G (1) is defined in (4.5) (see the proof of Theorem 4.2), and since θ(t, x, z) is infinitely differentiable in the parameters t and x, then M (t, x, z) is also infinitely differentiable in the parameters.Now, consider the case when t > x.
Consider Λ < ∞.Choose any numbers λ 1 , λ 2 ∈ R with |λ i | sufficiently large and such that λ 1 < −Λ, λ 1 < Re E, λ 2 > Λ, and perform the following "δ-transformation": where δ(z) is a solution of the following problem of conjugation of boundary values (RH problem): Note that r 2 (λ) < 0 for λ ∈ R, since Im w(λ) = 0 on R and Im E > 0. A unique solution of the problem given above is the function For M (1) (t, x, z) the boundary condition has the form M with J (1) (t, x, z) = δ −σ 3 + (z)J(t, x, z)δ σ 3 − (z) which after applying factorizations takes the form We introduce the smooth curves L 1 , L 3 and L 1 , L 3 , symmetric to L 1 , L 3 with respect to the real axis, the additional jump contour Σ = L 1 ∪ L 3 ∪ L 1 ∪ L 3 (the points λ 1 , λ 2 do not belong to Σ), the domains D i and the domains D i , symmetric to D i with respect to the real axis, i = 1, 2, 3, which are displayed in Figure 2. Thus, the complex z-plane is decomposed into the parts where Since e 2iθ(t,x,z) → 0 and e −2iθ(t,x,z) → 0 as z → ∞ for Im z > 0 and Im z < 0 respectively (see the case t > x, Λ < ∞ in Appendix A and Figure 5), then G (2) (t, x, z) → I and hence M (2) (t, x, z) → I as z → ∞.For M (2) (t, x, z) the jump (boundary) condition has the form M − (t, x, z) defined on the new contour Σ ∪ Σ whose orientation remains the same on Σ = R ∪ (E, E), and all branches of Σ (i.e., L i , L i , i = 1, 3) are oriented from the left to the right.It is easy to verify that As in the proof of Theorem 4.3, we obtain that for any fixed t, x there exists a unique solution W (2) (t, x, ζ) of the singular integral equation (the equation is considered in the space of operator functions W (2) where and then we find M (2) (t, x, z) by the formula Since e ±2iθ(t,x,z) → 0 as z → ∞, z ∈ C ± , respectively (see the case t > x, Λ < ∞ in Appendix A), then for sufficiently large z ∈ Σ (i.e., on the parts of the jump contour Υ contained in a neighborhood of infinity), the matrix I − J (2) (t, x, z) decreases exponentially and its derivatives of any order in t and x decrease fast enough so that the integrals that arise when differentiating are convergent.Consequently, the solution W (2) (t, x, ζ) of the equation (4.10) and the function M (2) (t, x, z) (4.11) are infinitely differentiable in t and x.Since M (2) (t, x, z) and G (2) (t, x, z) are infinitely differentiable in t, x, then M (1) (t, x, z) and M (t, x, z) also have this property.
Consider Λ = ∞ (as before, t > x).We introduce the smooth curves symmetric with respect to R, where ξ = x 4(t−x) > 0, which are such that (see Figures 3 and 4): -all sufficiently large z belonging to L 2 (ξ) and L 2 (ξ) (i.e., the parts of L 2 and L 2 contained in a neighborhood of infinity) are contained in the domain bounded from above by the curve γ + ξ and from below by the curve γ − ξ , where γ ± ξ are defined by the equality Then complex z-plane is decomposed into the parts Define the contour Σ = L 2 ∪ L 2 , where L 2 and L 2 are oriented from the left to the right.The orientation on Σ remains the same as above.Further, we transform the matrix M (t, x, z) into where Then M (2) + (t, x, z)J (2) (t, x, z), z ∈ Σ ∪ Σ \ {Re E}, where has the following form: As above, we consider the singular integral equation (4.10),where Υ = Σ = L 2 ∪ L 2 , and find M (2) (t, x, z) G (2) (t, x, z) → I and M (2) (t, x, z) → I as z → ∞ by the formula (4.11).Since e −2iθ(t,x,z) → 0 and e 2iθ(t,x,z) → 0 as z → ∞ for z from the domains bounded by R, γ + ξ and R, γ − ξ respectively (see Figure 6 in Appendix A), then for sufficiently large z ∈ Σ (i.e., on the parts of the jump contour Υ contained in a neighborhood of infinity), the matrix I −J (2) (t, x, z) decreases exponentially and its derivatives of any order t and x decrease fast enough.Consequently, the solution W (2) (t, x, ζ) of the singular integral equation, the function M (2) (t, x, z) and, hence, the function M (t, x, z) are infinitely differentiable in t and x. ■ Theorem 4.5.Let Φ(t, x, z) := M (t, x, z)e −iθ(t,x,z)σ 3 , ( where M (t, x, z) is a solution of the basic problem RH 0 .Then Φ(t, x, z) is a unique solution of the AKNS system where t ∈ R + , x ∈ (0, L), L ≤ ∞, and z ∈ C \ Σ cl , Σ cl = Σ ∪ E, E .In the system (4.14) and (4.15), the function H(t, x) is defined by where the function F (t, x, λ) is defined by where M (t, x, λ + i0) = M + (t, x, λ) is defined appropriately at the point λ = Re E so that F (t, x, λ) is continuous in λ ∈ R, and these functions satisfy the Maxwell-Bloch equations in the matrix form where t ∈ R + , x ∈ (0, L), λ ∈ R, and the initial and boundary conditions that is, the functions E(t, x), ρ(t, x, λ) and N (t, x, λ) satisfy the MB equations (1.5) and the initial and boundary conditions (1.6) and (1.7).
As shown above, in the case τ = t − x > 0 and, accordingly, ξ > 0, the curve defined by the equality Re(iθ(t, x, z)) = 0, where z = λ + iν and Λ < ∞, consists of the real line R and the closed level line γ ξ defined by the equality Π(λ, ν) = 1/ξ (the level line (A.3)).For any fixed ξ > 0, the interval [−Λ, Λ] is located inside the oval γ ξ , and γ ξ intersects the real line at the stationary points λ ± = λ ± (ξ) (λ − < −Λ and λ + > Λ) which are simple ones because n(s) ds (s−λ) 3 is negative (i.e., strictly negative) for λ < −Λ and positive for λ > Λ.When Λ → +∞ (ξ > 0 is fixed) the stationary points λ ± tend to ±∞.For τ > 0 (ξ > 0) and Λ = +∞, the real line ν = 0 is an asymptote for the level line (A.3) consisting of the two curves γ ± ξ , symmetric with respect to the real line.It is easy to verify that for t > x (ξ > 0) the signature table of Re(iθ(t, x, z)) (A.1) has the form presented in Figures 5 and 6.The signature table shows that the exponents e −2iθ(t,x,z) and e 2iθ(t,x,z) vanish exponentially on the subintervals (E, Re E) and Re E, E respectively when z and the interval E, E lie inside of γ ξ (when Λ < ∞) or between the curves γ ± ξ (when Λ = ∞).They increase in τ unboundedly on the subintervals (E, Re E) and Re E, E respectively if z and the interval E, E lie outside of γ ξ (when Λ < ∞) or on those parts of the subintervals (E, Re E) and Re E, E respectively that lie above the curve γ + ξ and below the curve γ − ξ if z also lies in the corresponding regions (when Λ = ∞).
3) and w(z) (2.4) are analytic in C \ [E, E], and, hence, the functionM 0 (z) = a(z) b(z)b(z) a(z) defined by a(z) and b(z) of the form (2.5) is also analytic in C\ E, E .Note that since κ(z) → 1 and κ −1 (z) → 1 as z → ∞, then M 0 (z) → I as z → ∞.Thus, the function Φ(t, λ) (2.15) has the analytic continuation Φ(t, z) for z ∈ C\ E, E (the orientation on [E, E] is chosen from up to down) which is continuous up to the boundary and has the singularities of the type (z − E) −1/4 , z − E −1/4 at the points E and E. Since det M 0 (z) ≡ 1, then det Φ(t, z) ≡ 1.

)
is a solution of the following basic matrix Riemann-Hilbert problem (the problem of conjugation of boundary values on a contour Σ), where the reflection coefficient r(z) and the function h(z) are uniquely determined by the initial and boundary conditions (1.6) and (1.7):

Figure 3 .
Figure 3.The contours Σ = R ∪ E, E and Σ = L 2 ∪ L 2 , the curves γ ± ξ and the domains D i , D i , i = 2, 4, in the case when the interval E, E lies inside the domain bounded by the curves γ + ξ , γ − ξ .

1 ξ
Re(iθ) = 0 on γ ± ξ ; see the case t > x, Λ = ∞ in Appendix A and Figure6; -the interval [E, E] lies in the domain bounded by L 2 and L 2 ; -the domain D 2 is bounded by L 2 and R ∪ [E, Re E); -the domain D 2 is bounded by L 2 and R ∪ Re E, E .
(by virtue of the normalization condition for M (t, x, z)), then by the Liouville theorem det M (t, x, z) ≡ 1.Hence, there exists M −1 (t, x, z) and it is analytic on C \ Σ cl and has continuous boundary values on Σ \ {Re E}.Now suppose that there is another solution M (t, x, z) of the basic RH problem RH 0 .Consider the function N (t, x, z) 5)of the basic RH problem RH 0 was proved in Section 3. Let us prove the uniqueness of the solution.First we show that if M (t, x, z) is a solution of this RH problem, then det M (t, x, z) ≡ 1.It follows from the properties of M (t, x, z) that the function det M (t, x, z) is analytic for z ∈ C \ Σ cl , where Σ cl = Σ ∪ E, E , and has continuous boundary values on Σ \ {Re E}, and detM − (t, x, z) = det M + (t, x, z) det J(t, x, z) = det M + (t, x, z), z ∈ Σ \ {Re E}, since det J(t, x, z) ≡ 1.Hence, det M (t,x, z) is analytic for all z ∈ C except for the points E, E, Re E which are removable singularities by virtue of the properties of M (t, x, z).Thus, after appropriate modifications (definitions) at these points, det M (t, x, z) is analytic for z ∈ C, and since det M (t, x, z) = 1 + O z −1 as z → ∞ The initial condition(1.6)forthe MB equations implies F (0, x, λ) ≡ −σ 3 .Furthermore, for z = λ + i0 (λ ̸ = Re E) the matrix function Φ(t, x, z) (4.3) satisfies the t-equation (1.8), i.e., W t = U W , with the same matrix function U (t, x, λ), and det Φ(t, x, z) = det M (t, x, z) ≡ 1.Consequently, the matrix function (the Cauchy matrix) {Re E} and bounded near λ = Re E. Thus, we can appropriately define M + (t, x, Re E) (see the proof of Theorem 4.1) so thatF (t, x, λ) is continuous in λ ∈ R.The proof of the uniqueness of Φ(t, x, z) is similar to the proof of the uniqueness of M (t, x, z) in Theorem 4.1.■ Theorems 4.3, 4.4 and 4.5 give the following corollary.
Corollary 4.6.The function H(t, x) (4.16) and the electric field envelope E(t, x) can also be defined by