A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

The Baker-Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schr\"odinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts we propose such a matrix function that has unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful in the study of the long time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations.


Introduction
We consider the Maxwell-Bloch (MB) equations written in the form Here, E = E(t, x) is a complex-valued function of the time t and the coordinate x, and ρ = ρ(t, x, λ) and N = N (t, x, λ) are complex-valued and real functions of t, x, and the additional parameter λ. Subindices refer to partial derivatives in t and x, and * means a complex conjugation.
Equations (1.1)-(1.3) are used in many physical models which deal with a classical electromagnetic field that interacts resonantly with quantum two-level objects -two-level atoms, which have only two energy position: upper and lower level. In particular, there are models of the self-induced transparency [1,2], and two-level laser amplifier [52,53]. For these models E = E(t, x) is the complex valued envelope of an electromagnetic wave of fixed polarization, so that the field in the resonant medium is E(t, x) = E(t, x)e iΩ(x−t) + E * (t, x)e −iΩ(x−t) .
Then N 2 (t, x, λ) + |ρ(t, x, λ)| 2 ≡ 1 for all t, which reflects the conservation of probability: the total probability that an atom can be found in the upper or lower level equals 1. We also put Ω = 1 in (1.1). For a given (at the initial time) polarization, the population is determined to within a sign N (0, x, λ) = ± 1 − |ρ(0, x, λ)| 2 .
If N (0, x, λ) > 0, then an unstable medium is considered (the so-called two-level laser amplifier). If N (0, x, λ) < 0, then a stable medium is considered (the so-called attenuator). The Maxwell-Bloch equations became well-known in soliton theory after Lamb [48,49,50,51]. Ablowitz, Kaup and Newell have firstly applied the inverse scattering transform to the Maxwell-Bloch equations in [1]. In some sense general solutions to the MB equations and their classifying were done by Gabitov, Zakharov and Mikhailov in [27]. Some asymptotic results for the MB equations were obtained by Manakov in [52] and, in a collaboration with Novokshenov, in [53]. Elliptic periodic waves in the theory of self-induced transparency were constructed by Kamchatnov in [35]. We cite here only a small number of pioneering papers relating to the Maxwell-Bloch equations. Some reviews on an application of inverse scattering transform to the MB equations can be found in [1,2,27,40], and for the reduced Maxwell-Bloch equations in [28,63].
In some cases it is convenient [42] to use equations where Thus there are two Lax pairs (t-and x + -equations and t-and The main goal of the paper is to give a construction of the Baker-Akhiezer function Ψ(t, x, z) associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts we propose such a matrix function Ψ(t, x, z) that has unit determinant and takes an explicit form through theta functions and Cauchy integrals. The construction proceeds also from the requirement that Ψ(t, x, z) must satisfy the following system of linear equations which depend on z ∈ C \ Σ where Σ is a contour containing (as a part) the real axis R of the complex plane, and Symmetries of F , G, H and equations (1.8), (1.9) provide the following symmetry of Ψ As a result of our construction we obtain also a solution to the Maxwell-Bloch equations (1.1)- (1.3). This solution is an analog of finite-gap solutions of soliton equations. The Baker-Akhiezer function theory, as an analogue of the Floquet theory for ODE's with periodic coefficients, was successfully developed many years ago, in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation (see, e.g., [3,23,24,25,31,32,33,34,46,47,54,58]). Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch system or for the Karpman-Kaup equations [29,39], which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. One more goal is applications in asymptotic analysis. The presence of inhomogeneous broadening n(λ) leads to noticeable complications in the Deift-Zhou method of steepest descent [15,16,21,22]. We have some progress in studying of a mixed problem where we come to a necessity of using of the declared matrix BA function. We believe that results of the paper will be useful for further development of the results obtained, for example, in [27,40,42,52,53] and for an investigation of rogue waves (about them see, e.g., [4,5,28,55]) to the Maxwell-Bloch equations.
It is worth notice that it is very difficult to implement the algorithm [3] (which uses a Riemann surface) for constructing the Baker-Akhiezer function associated with the Maxwell-Bloch system. The matter in fact of presence of a given broadening function n(λ) is difficult to reconcile with the Riemann surface, which is the basic component of the method. To overcome this difficulty, it will be necessary to use Cauchy integrals with meromorphic/multi-valued kernels on the Riemann surface, which are very nontrivial for understanding to a wide range of specialists.

Definition of the Baker-Akhiezer function and main results
In order to formulate our main results we start from the following definition of matrix Baker-Akhiezer function associated with the Maxwell-Bloch equations. First of all we fix the weight function n(λ) (λ ∈ R) which is smooth and satisfies (1.4). Let Σ j := (E j , E * j ), j = 0, 1, 2, . . . , N be a set of vertical open intervals on the complex plane C which together with the real line R constitute an oriented contour Σ = R ∪ N j=0 Σ j . The orientation of R is chosen from left to right, and each Σ j is oriented from top to bottom (Fig. 1). Boundary values of functions from the left and right of Σ we denote by signs ± respectively: Definition 2.1. Let a contour Σ, a set of real constants (φ 0 , φ 1 , . . . , φ N ) and a weight function n(λ) be given. A 2 × 2 matrix Ψ(t, x, z) is called the Baker-Akhiezer function associated with the Maxwell-Bloch equations if for any x, t ∈ R: • boundary values Ψ ± (t, x, z) are continuous except for the endpoints E j and E * j , j = 0, 1, . . . , N where Ψ ± (t, x, z) have square integrable singularities; • boundary values Ψ ± (t, x, z) are bounded at the points of self-intersection Re E j , j = 0, 1, . . . , N ; • Ψ(t, x, z) satisfies the jump conditions • Ψ(t, x, z) satisfies the symmetry condition These properties defines the matrix BA function uniquely and allow to construct Ψ in an explicit form through theta functions and Cauchy integrals. To formulate main results let us define some necessary ingredients. Let be roots whose branches are fixed by cuts along [E j , E * j ], j = 0, . . . , N and conditions w(z) z N +1 , κ(z) 1 as z → ∞. Define scalar functions f (z) and g(z) through Cauchy integrals where C f j , C g j are uniquely defined by linear algebraic equations A unique solvability of (2.5) and (2.6) is well-known. A detailed proof can be found in [61,Problem 9.4.2, or [62].
Theorem 2.2. Let a contour Σ, a set of real constants (φ 0 , φ 1 , . . . , φ N ) and a weight (smooth) function n(λ) be given. Let all requirements of Definition 2.1 are fulfilled. Then Ψ is unique and takes the form where constants f 0 and g 0 are equal to functions f (z) and g(z) are given by (2.3)-(2.6), and M (t, x, z) is a solution of the following RH problem: • boundary values M ± (t, x, z) are continuous, except for the endpoints E j and E * j where M ± have square integrable singularities; • M (t, x, z) satisfies the jump conditions The next theorem presents an explicit formula for M (t, x, z).
where Θ is theta-function (6.7) defined by the Fourier series and A(z), A(D) are Abel mapping (6.4), (6.5), K is a vector of Riemann constants (6.6). The dependence of M (t, x, z) in t and x is determined by vector-function with components Theorem 2.4. Let Ψ is defined by Theorems 2.2 and 2.3. Then for any z ∈ C \ Σ matrix Ψ(t, x, z) is smooth in t and x and satisfies AKNS equations where H(t, x) is given by and Matrix F (t, x, λ) is Hermitian, has unit determinant and presented by formula where Im E j , f 0 and g 0 are defined by (2.7) and (2.8). The dependence of the solution in t and x is determined by the N dimensional (linear in t and x) vector-function The density matrix F (t, x, λ) equals to The paper is organized as follows. In Section 3, we prove the Theorem 2.2. In Section 4, we give a construction of the phases f and g by Cauchy integrals, and in Section 5, we propose another representations for them using hyperelliptic integrals. In Section 6, explicit construction of M (t, x, z) is presented (the proof of Theorem 2.3). In Section 7, we deduce AKNS equations for Ψ(t, x, z) (the proof of Theorem 2.4). Section 8 describes finite-gap solutions to the MB equations (the proof of the Theorem 2.5). Section 9 contains final remarks.
3 Proof of the Theorem 2.2 and RH problem for M = M (t, x, z) Uniqueness. The matrix Ψ(t, x, z) has unit determinant. Indeed, since Ψ is a matrix of the second order then, due to definition of Ψ, det Ψ is analytic in z ∈ C \ Σ, continuous up to the contour Σ, except for the endpoints E j , E * j where it has weak singularities, and bounded at all self-intersection points Re E j . In view of (2.1), det J(x, z) ≡ 1, hence i.e., det Ψ has no jump at the contour Σ. Therefore det Ψ is analytic everywhere, except for a set of self-intersection points and endpoints of Σ where it has removable singularities. At and it is continuous across Σ with exception of end points E j , E * j and points of self-intersection Re E j . These points are removable singularities. Hence Φ(t, x, z) has an analytic continuation for z ∈ C and it tends to identity matrix as z → ∞. By Liovilles's theorem Φ(t, Existence. To prove the existence of the Baker-Akhiezer function we use an explicit construction of Ψ using different RH problems. To transform the initial RH problem to a form allowing an explicit solution, let us seek Ψ(t, x, z) in the form where constants f 0 and g 0 , scalar functions f (z) and g(z) and matrix M (t, x, z) are to be determined. The symmetry of Ψ (1.10) produces symmetries of f (z) and g(z), i.e., they have to satisfy the conditions: f * (z * ) = f (z) and g * (z * ) = g(z), particularly f * 0 = f 0 and g * 0 = g 0 . Due to the definition of Ψ we obtain the RH problem (2.9), (2.10). Indeed, all above statements will be true if f (z) and g(z) possess properties: where f 0 and C f j are some real (as a result of the symmetry of Ψ) constants; • g(z) = g * (z * ) and where g 0 and C g j are some real constants. All constants f 0 , g 0 , C f j , C g j , j = 0, 1, 2, . . . , N , are determined in the next section where we prove formulas (2.3)-(2.8).
Asymptotics (3.2), (3.3) give that M (t, x, z) = I + O z −1 as z → ∞. The jumps of functions f (z) and g(z) provide the form of matrix (2.10). Indeed, for z ∈ Σ j , that gives (2.10). We stress that such a choice of the jump matrices on intervals Σ j (independent on z) provides solvability of the RH problem for M (t, x, z) in an explicit form in theta functions. The jump of the function g(z) on the real axis (z = λ) makes the matrix M (t, x, z) to be Re E j : The symmetries of M = M (z) follow from the symmetry of jump contour Σ with respect to the real axis and symmetric properties of scalar functions f (z) and g(z) and matrix Ψ = Ψ(z). Finally, it is important to emphasize a normalization condition which follows from the definition of Ψ.
The Riemann-Hilbert problems like (2.10) have already been encountered in different form in the so-called model problems (see, for example, publications [6,7,8,9,10,11,12,13,14,15,16,17,18,21,22,26,36,37,38,43,44,56,57,59,60,62]). All these papers were devoted to studying an asymptotic behavior of different problems arising in the soliton theory, in the theory of random matrix models, and also in the theory of integrable statistical mechanics. These model problems have auxiliary in nature, and for our constructions it is impossible to use the results of those articles directly. Therefore for the completeness of exposition we give in the next sections an explicit construction of scalar functions f (z), g(z) and matrix M (t, x, z) by using ideas of just cited articles and also of the paper [45].

Construction of the phases f and g by Cauchy integrals
In this section we give the construction of the phase functions f and g. We start from the case involving only one arc. In this case, the jump conditions for f and g are jumps of type (3.2) and (3.3) such that w(z) is analytic outside the arc and w(z) z as z → ∞, and introducẽ Then the jump conditions (4.1) reduce tõ Consequently, Particularly, f 0 is determined by Taking into account (3.1) it can be put C f 0 = 0 without loss of generality and hence f (z) = w(z) = (z − E 0 )(z − E * 0 ) and f 0 = − Re E 0 . Now consider the function g(z). In this case we haveg = gw −1 = −1 + O(1/z) as z → ∞, and thus Consequently, by the same reason as above with C g 0 = 0 and, particularly, Now consider the general case, where the contour consists of N + 1, N ≥ 1, arcs Σ j , j = 0, . . . , N . Define such that w(z) is analytic outside the arcs Σ j and w(z) z N +1 as z → ∞, and introducef andg as in (4.2). The jump conditions reduce tõ dξ. Hence Again, it is convenient to put C f 0 = 0. Then, in view off = O(1/z) as z → ∞, C f j have to satisfy the system of linear equations for m = 0, 1, 2, . . . , N − 1. Thus we have N equations (2.5) for N unknown constants C f j . It is well known (see, for example, [3,23,62,64]) that this system of linear algebraic equations has a unique solution {C f j } N j=1 . Then f (z) and f 0 takes the form (2.3) and (2.7). Now consider the function g(z). In view of (4.3), (4.4), and take into account that for N ≥ 1 g = O(1/z) as z → ∞ we havẽ where, by the same reasons as above we put C g 0 = 0. Consequently, g(z) takes the form (2.4). Now the requirement that g(z) given by (2.4) satisfies the asymptotic condition (3.3) leads to a system of N linear algebraic equation for C g j , j = 1, . . . , N . Indeed, if we use the asymptotic for large z expansion then, due to (3.3), it is evident that I 0 = I 1 = · · · = I N −2 = 0. Hence Re E j . This gives the system of linear algebraic equations (2.6). Similarly to (2.5), (2.6) has a unique solution. A detailed proof can be found in [61,Problem 9.4.2, or in [62]. The parameter g 0 is given by (2.8).
5 Representation of f (z) and g(z) through hyperelliptic integrals Here we give another representation of f (z) and g(z), using hyperelliptic integrals. We seek f (z) in the form 2 . In order to definef 0 ,f 1 , . . . ,f N −1 , we normalize f (z) by the conditions In other words, for z ∈ C \ N j=0 [E j , E * j ] the function f (z) can be considered as a hyperelliptic integral of the second kind with simple pole at infinity. Integral f (z) is uniquely fixed by the condition of zero a-periods [3]. They are A f j = 2 where B f j = b j df (z). The last equality becomes obvious if we use the definition of aand b-cycles of the hyperelliptic surface given by the function w(z) (see the next section and Fig. 2).

On the other hand, for
The function g(z) cannot be written as a hyperelliptic integral, but it is determined as a sum of the hyperelliptic integral −f (z) and Cauchy integrals where constants {C h j } N j=1 have to be determined. To prove this formula let us put h(z) := f (z) + g(z) w(z) .

Equations (3.2) and (3.3) provide the following properties of function h(z):
• where C h j = C g j + C f j are to be determined. Due to (5.1) C f j are already known: C f j = B f j , j = 1, . . . , N . Then h(z) can be written as a sum of Cauchy integrals and hence (5.2) follows. The asymptotic condition (3.3) leads to a system of N linear equation Similarly to (2.6) this system has a unique solution. In this case the parameter g 0 is equal to Substituting (2.7) in (5.3) and using equality C f j − C h j = −C g j we obtain which coincides with (2.8) and thus (5.2) is proved. Besides, we found relations (5.2) and (5.3) between the phase functions f (z) and g(z).
6 Explicit construction of the matrix M (t, x, z) In this section we present an explicit construction of M (t, x, z) which solves the RH problem (2.9), (2.10). The main ideas of such a construction are borrowed in [16,36,45]). First, define where cuts are chosen along [E j , E * j ], j = 0, . . . , N with orientation from top to bottom. The branch of root is fixed by the condition κ(∞) = 1. Then Notice also that Recall that RH problem for M (z) (2.9), (2.10)) is as follows: • boundary values M ± (t, x, z) are continuous except end-points E j and E * j where M ± have square integrable singularities; for j = 1, 2, . . . , N , and C f j , C g j , φ j are some given real constants (recall that First, consider the case N = 0. Then, by (6.2), M (t, x, z) ≡ M (z) can be constructed using κ(z) Expanding M (z) as z → ∞, In order to present an explicit solution of the RH problem in the general case (N ≥ 1), we introduce necessary facts from the theory of the Riemann manifolds by following closely to [3,36,45]. First, let X be the Riemann surface of genus N defined by the equation w 2 = P (z), where with cuts along Σ j = (E j , E * j ), j = 0, 1, 2, . . . , N . The Riemann surface X can be viewed as a double covering of the complex z-plane: two sheets of z-plane are glued along Σ j . The upper and lower sheets of X are denoted by X + and X − respectively; they are fixed by the relations where z = π(P) is the standard projection of P = (w, z) ∈ X on the Riemann sphere CP 1 . Thus each point on the z-plane has two preimages P ± = X ± , except for the branch points. Denote the preimage of z = ∞ on X ± by, respectively, ∞ ± . With the inclusion of two points (∞ + , ∞ − ), X becomes a compact Riemann surface of genus N . The square root P (z) turns into a meromorphic function on its own compact Riemann surface X , which have 2N + 2 zeros at E j and E * j , j = 0, 1, 2, . . . , N , and two poles at ∞ + and ∞ − , each of multiplicity N + 1. Further, we introduce the Abelian integrals where dω j (P) is a basis of holomorphic differentials on X The coefficients c jl are uniquely determined by the normalization conditions We have chosen a l -cycles as ovals on the upper sheet of X around the intervals E l ,Ê l , E l := E * l , l = 0, 1, 2, . . . , N , see Fig. 2. The normalized holomorphic differentials define the b-period matrix as where b l -cycle starts from (E 0 , E * 0 ), goes on the upper sheet to (E l , E * l ), and returns on the lower sheet to the starting point. This is a symmetric matrix with positive definite imaginary part.
Let e j = (0, . . . , 1, . . . , 0) be the unit vector in C N and Be j the j-th column of the matrix B. Denote by Λ ⊂ C N the lattice generated by the linear combinations, with integer coefficients, of the vectors e j and Be j for j = 1, 2, . . . , N . Then, by the definition, Jacobian variety of X is the complex torus Jac{X } = C N /Λ. The Abel mapping A : X → Jac{X } is defined as follows where the point P 0 is fixed by condition π(P 0 ) = E * 0 and Q is the integration variable. The Abel mapping is also defined for integral divisors D = P 1 + · · · + P m by summation A(D) = A(P 1 ) + · · · + A(P m ) (6.5) and In the hyperelliptic case, the Riemann constant vector K is defined by (cf. [64]) Associated with the matrix B there is the Riemann theta function defined for u ∈ C N by the Fourier series where (l, u) = l 1 u 1 + · · · + l N u N . It is an even function, i.e., Θ(−u) = Θ(u), and has the following periodicity properties Θ(u ± e j ) = Θ(u), Θ(u ± Be j ) = e ∓2πiu j −πiB jj Θ(u), where e j = (0, . . . , 0, 1, 0, . . . , 0) is the j-th basis vector in C N . This implies that the function where c, d ∈ C N are arbitrary constant vectors, has the periodicity properties h(u ± e j ) = h(u), h(u ± Be j ) = e ∓2πic j h(u).
It remains to choose the vectors d 1 and d 2 in such a way that M (z) is analytic at the zeros of the denominators in (6.11), i.e., the zeros of Θ(A(z) + d s ) and Θ(−A(z) + d s ) (s = 1, 2) are to be canceled by the zeros of κ(z) ± κ −1 (z).
Introduce the non-special divisor D = P 1 + · · · + P N such that D = D 1 + D 2 , where D 1 = P 1 + · · · + P N 1 ∈ X − , 0 ≤ N 1 ≤ N and a(z j ) = κ(z j ) + κ −1 (z j ) = 0, z j = π(P j ), j = 1, 2, . . . , N 1 , whereas D 2 = P N 1 +1 +· · ·+P N ∈ X + with b(z j ) = κ(z j )−κ −1 (z j ) = 0, j = N 1 +1, N 1 +2, . . . , N . Set d 1 = A(D) + K and d 2 = −A(D) − K. Then Θ(A(P) + A(D) + K) has N 1 zeroes P 1 , . . . , P N 1 on X + and N − N 1 zeroes P N 1 +1 , . . . , P N on X − [3], where the points P j and P j form a conjugated pair of points on X with π(P j ) = π(P j ) = z j ∈ C, j = 1, 2, . . . , N . Similarly, Θ(A(P) − A(D) − K) has N − N 1 zeroes P N 1 +1 , . . . , P N on X + and m zeroes P 1 , . . . , P N 1 on X − . Taking the restrictions of these Riemann theta functions on the upper sheet with the cut Γ = N j=0 (E j , E * j ), we have that are analytic in z ∈ C \ Γ with poles at z 1 , . . . , z N 1 , which are canceled in the products a(z)F 1 (z) and a(z)H 2 (z). Similarly, b(z)F 2 (z) and b(z)H 1 (z) are analytic in C \ Γ, since the poles z N 1 +1 , . . . , z N are canceled by the zeroes of b(z). Notice that the idea to cancel the poles of F j and H j by the zeros of κ ± κ −1 goes back to [16,19,20]. Thus matrix M (z) (6.13) satisfies all conditions to be a solution of the RH problem (2.9)-(2.10) if only do not vanish. Since the divisor D is non-special and ∞ ± / ∈ D, the denominator does not equal zero and takes a finite value. It is well known (cf. [3]) that the divisor of zeroes of Θ(A(z) + A(D) + K ± c) remain non-special if vector c is sufficiently small. Since all zeroes of F 1 (z) (H 2 (z)) belong to the mentioned divisor and this divisor does not contain infinity, then F 1 (∞) = 0 (as well as H 2 (∞) = 0). Moreover, taking into account the symmetries M (z) = σ 2 M * (z * )σ 2 , i.e., M 22 (z) = M * 11 (z * ) and M 21 (z) = −M * 12 (z * ), and unity determinant of the matrix M we obtain that gives the boundedness of all entries of matrix M (λ). In turn, it means that F 1 (∞) and H 2 (∞) can not vanish for any vector c. To prove the symmetry of M (z) let us consider matrixM (z) := σ 2 M * (z * )σ 2 . ThenM (z) and the original matrix M (z) solve the same RH problem and, due to the uniqueness of the solution of the RH problem, we haveM (z) ≡ M (z). Hence M (z) satisfies the symmetry conditions. We have constructed the matrix M (z) = M (t, x, z) that solves the required RH problem (2.9)-(2.10) and thus M provides analyticity of Ψ(t, x, z) in z ∈ C \ Σ, and also continuity up to the contour Σ (except for the endpoints E j and E * j , where Ψ has weak singularities). Formulas for entries M ij (t, x, z) of the matrix M (t, x, z) follows from (6.11) and (6.13). Expanding it at infinity, and taking into account that Θ(A(z) + B) is bounded and dA dz = O z −2 as z → ∞, we have Im E j and C(t, x) := − tC f +xC g +φ 2π .
Notice that C f and C g are determined when constructing f (z) and g(z) whereas E j = Re E j +i Im E j , j = 0, 1, 2, . . . , N , and real constants (φ 0 , φ 1 , φ 2 , . . . , φ N ) present itself free real parameters total number of which is equal to 3N + 3. Evidently, the constant φ 0 is defined modulo 2π, while (2πφ 1 , 2πφ 2 , . . . , 2πφ N ) can be regarded as a vector on the Jacobian Jac{X }. Formulas (6.15) will be used for a definition of finite-gap solutions to the MB equations.
In the theory of finite-gap integration [3], the divisor D is taken to be arbitrary, it defines poles of the Baker-Akhiezer vector function. In the absence of symmetry, such a vector function satisfies corresponding AKNS equations defined by two complex valued functions. These equations generate the focusing NLS equation for unique complex valued function if and only if they possess a symmetry which, in turn, take place if and only if the so called reality conditions are fulfilled. The left hand side of the equality is determined by 4N + 3 real parameters: branching points E j ∈ C (Im E j = 0, j = 0, 1, . . . , N ), projections z j = π(P j ) of the nonspecial divisor D = P 1 + · · · + P N , and q(0, 0) where q(x, t) is a finite-gap solution of NLSE.
The reality conditions reads as follows: the difference of the polynomials on the left-hand side must be a square of some polynomial N + 1-th degree with real coefficients. They contain N nonlinear relations (because independently from Re E j ) and hence the number of independent real parameters decreases to 3N + 3. This conditions were first obtained in [41] (see also [30]). The same conditions (6.16) characterize a set of finite-gap Dirac operators with anti-Hermitian potential matrices [41]. Therefore conditions (6.16) are also applicable to our case where the total number of free (real) parameters is also equals to 3N +3. In our case the divisor D is fixed by zeroes z j of the functions κ(z)±κ −1 (z). Thus, the reality conditions mean that there is a correspondence between (φ 0 , φ 1 , . . . , φ N ) and (f 0 , f 1 , . . . , f N −1 , |q(0, 0)| = |E(0, 0)|). However, this issue is beyond the scope of article. In our approach E 0 , E 1 , . . . , E N and φ 0 , φ 1 , . . . , φ N are independent. Due to the symmetry Ψ(t, x, z) = σ 2 Ψ * (t, x, z * )σ 2 the potential matrix H(t, x) is anti-Hermitian for any choice of the parameters (see the next section for details).
Proof . By the construction, Ψ is analytic with respect to c = C(t, x) which, in turn, is linear with respect to t and x. Hence Ψ(t, x, z) is smooth in t, x ∈ R. The matrix Ψ(t, x, z) is also analytic in z ∈ C \ Σ and has (due to (2.9) and (2.10)) the jump across Σ where the jump matrices are independent on t. The jump condition gives This relation, together with a continuity of Ψ ± outside of exceptional points (E j , E * j , Re E j ), implies that logarithmic derivative Ψ t (t, x, z)Ψ −1 (t, x, z) is analytic (entire) in z ∈ C. Indeed, since Ψ t (t, x, z)Ψ −1 (t, x, z) has no jump across Σ \ N j=0 Re E j then it can be extended to a continuous function because the exceptional points are removable singularities. We took into account the boundedness at the points of self intersection Re E j , weak singularities at the endpoints E j , E * j and the second order of Ψ. Further, since M (t, x, z) and M t (t, x, z) have the asymptotics: where [A, B] := AB − BA. Therefore, by Liouville's theorem, the logarithmic derivative is a polynomial where Using the symmetry σ 2 Ψ * (z * )σ 2 = Ψ(t, x, z) we find that U (z) = σ 2 U * (z * )σ 2 . This symmetry implies that H is anti-Hermitian, i.e., H = −H † . Hence q(t, x) = −p * (t, x) and we put q(t, x) := E(t, x)/2 where E(t, x) = −4im 12 (t, x)e 2i(tf 0 +xg 0 ) with m 12 (t, x) defined in (6.15). Thus Ψ(t, x, z) satisfies the first equation of (2.11) with matrix H given by (2.12).
Thus N (t, x, λ)) and ρ(t, x, λ) are smooth for all t, x ∈ R (λ = Re E j ) and bounded for all t, x, λ ∈ R.
In particulary, the simplest periodic solution to the MB equations takes the form of a plane wave E(t, x) = 2 Im E 0 e 2i(tf 0 +xg 0 )−iφ 0 , Some periodic and rational solutions of the reduced Maxwell-Bloch equations with n(λ) = δ(λ) were recently obtained in [63].