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 the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the...
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| Cite this: | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
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| citation_txt | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
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| description | 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 the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the 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 a 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 the 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.
|
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 082, 27 pages
A Matrix Baker–Akhiezer Function
Associated with the Maxwell–Bloch Equations
and their Finite-Gap Solutions
Vladimir P. KOTLYAROV
B. Verkin Institute for Low Temperature Physics and Engineering,
47 Lenin Ave., 61103 Kharkiv, Ukraine
E-mail: kotlyarov@ilt.kharkov.ua
Received February 05, 2018, in final form August 02, 2018; Published online August 10, 2018
https://doi.org/10.3842/SIGMA.2018.082
Abstract. 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 equa-
tions such as Korteweg–de Vries equation, nonlinear Schrödinger 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.
Key words: Baker–Akhiezer function; Maxwell–Bloch equations; matrix Riemann–Hilbert
problems
2010 Mathematics Subject Classification: 34L25; 34M50; 35F31; 35Q15; 35Q51
1 Introduction
We consider the Maxwell–Bloch (MB) equations written in the form
Et + Ex = 〈ρ〉, 〈ρ〉 = Ω
∫ ∞
−∞
n(λ)ρ(t, x, λ)dλ, (1.1)
ρt + 2iλρ = NE , (1.2)
Nt = −1
2
(E∗ρ+ Eρ∗). (1.3)
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 conju-
gation.
Equations (1.1)–(1.3) are used in many physical models which deal with a classical elec-
tromagnetic field that interacts resonantly with quantum two-level objects – two-level atoms,
mailto:kotlyarov@ilt.kharkov.ua
https://doi.org/10.3842/SIGMA.2018.082
2 V.P. Kotlyarov
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)eiΩ(x−t) + E∗(t, x)e−iΩ(x−t).
N (t, x, λ) and ρ(t, x, λ) are entries of the density matrix F (t, x, λ) =
(
N (t,x,λ) ρ(t,x,λ)
ρ∗(t,x,λ) −N (t,x,λ)
)
. It
describes the atomic subsystem. The parameter λ is the deviation of transition frequency of
given two-level atom from its mean frequency Ω. The angular brackets in (1.1) mean averaging
with given weight function n(λ) > 0, such that∫ ∞
−∞
n(λ)dλ = 1. (1.4)
The weight function n(λ) characterizes inhomogeneous broadening. From (1.2) and (1.3) it
follows that
∂
∂t
(
N 2(t, x, λ) + |ρ(t, x, λ)|2
)
= 0.
We interest in solutions where initial data are subjected to the condition
N 2(0, x, λ) + |ρ(0, x, λ)|2 ≡ 1.
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].
A Lax pair for the Maxwell–Bloch system was first found in [1] by using results of [48, 49,
50, 51] (see also [2, 27]). It was shown that (1.1)–(1.3) are the compatibility condition of an
overdetermined linear system, known as the Ablowitz–Kaup–Newell–Segur (AKNS) equations
wt + iλσ3w = −H(t, x)w, (1.5)
wx − iλσ3 + iG(t, x, λ)w = H(t, x)w, (1.6)
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 3
where
σ3 =
(
1 0
0 −1
)
, H(t, x) =
1
2
(
0 E(t, x)
−E∗(t, x) 0
)
,
G(t, x, λ) = p.v.
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− λ
ds.
The symbol p.v. denotes the principal value integral. Differential equations (1.5) and (1.6)
are compatible if and only if E(t, x), ρ(t, x, λ) and N (t, x, λ) satisfy equations (1.1)–(1.3) (see, for
example, [2]). As shown in [51], ρ(t, x, λ) and N (t, x, λ) are related to the fundamental matrix
of (1.5). Indeed, let Φ(t, x, λ) be a solution of (1.5) such that detΦ(t, x, λ) ≡ 1 and Φ† be the
Hermitian-conjugated to Φ. Then F (t, x, λ) = Φ(t, x, λ)σ3Φ
†(t, x, λ) satisfies equation
Ft + [iλσ3 +H,F ] = 0.
It is a matrix form of the equations (1.2) and (1.3).
In some cases it is convenient [42] to use equations
wx − iλσ3w + iG±(t, x, λ)w = H(t, x)w, (1.7)
where
G±(t, x, λ) =
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− λ∓ i0
ds = p.v.
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− λ
ds± πi
4
F (t, x, λ)n(λ).
Thus there are two Lax pairs (t- and x+-equations and t- and x−-equations) for the MB equa-
tions. Equations (1.5) and (1.7) (as well as (1.5) and (1.6)) are compatible if and only if E(t, x),
ρ(t, x, λ) and N (t, x, λ) satisfy equations (1.1)–(1.3).
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 in-
tegrals. The construction proceeds also from the requirement that Ψ(t, x, z) must satisfy the
following system of linear equations
wt + izσ3w = −H(t, x)w, (1.8)
wx − izσ3w + iG(t, x, z)w = H(t, x)w, (1.9)
which depend on z ∈ C \ Σ where Σ is a contour containing (as a part) the real axis R of the
complex plane, and
G(t, x, z) =
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− z
ds, Im z 6= 0.
Symmetries of F , G, H and equations (1.8), (1.9) provide the following symmetry of Ψ
Ψ(t, x, z) = σ2Ψ∗(t, x, z∗)σ2, σ2 =
(
0 −i
i 0
)
. (1.10)
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
4 V.P. Kotlyarov
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 Rie-
mann 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 recon-
cile 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.
2 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 Σj := (Ej , 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
Σj . The orientation of R is chosen from left to right,
and each Σj is oriented from top to bottom (Fig. 1). Boundary values of functions from the left
and right of Σ we denote by signs ± respectively:
Ψ±(z) = lim
z′→z∈±side of Σ
Ψ(z′).
E0
E∗0
EN
E∗N
?
?
λ = Re z
Ej
E∗j
-- -
?
?
?
?
?
?
?
?
?
?
Figure 1. The oriented contour Σ = R ∪
N⋃
j=0
(Ej , E
∗
j ).
Definition 2.1. Let a contour Σ, a set of real constants (φ0, φ1, . . . , φN ) and a weight func-
tion 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:
• Ψ(t, x, z) is analytic in z ∈ C \ Σ, Σ := R ∪
N⋃
j=0
[Ej , E
∗
j ];
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 5
• boundary values Ψ±(t, x, z) are continuous except for the endpoints Ej 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 ReEj , j =
0, 1, . . . , N ;
• Ψ(t, x, z) satisfies the jump conditions
Ψ−(t, x, z) = Ψ+(t, x, z)J(x, z), z ∈ Σ,
where
J(x, z) =
(
e−
πxn(λ)
2 0
0 e
πxn(λ)
2
)
, z = λ ∈ R \
N⋃
j=0
ReEj , (2.1)
J(x, z) =
(
0 ie−iφj
ieiφj 0
)
, z ∈ Σj = (Ej , E
∗
j ), j = 0, 1, . . . , N ; (2.2)
• Ψ(t, x, z) satisfies the symmetry condition
Ψ(t, x, z) = σ2Ψ∗(t, x, z∗)σ2, σ2 =
(
0 −i
i 0
)
;
• Ψ(t, x, z) =
(
I +O
(
z−1
))
e−iz(t−x)σ3 as z →∞.
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
w(z) :=
√√√√ N∏
j=0
(z − Ej)(z − E∗j ), κ(z) := 4
√√√√ N∏
j=0
z − E∗j
z − Ej
, z ∈ C \
N⋃
j=0
[Ej , E
∗
j ]
be roots whose branches are fixed by cuts along [Ej , E
∗
j ], j = 0, . . . , N and conditions w(z) '
zN+1, κ(z) ' 1 as z →∞. Define scalar functions f(z) and g(z) through Cauchy integrals
f(z) =
w(z)
2πi
N∑
j=1
∫
Σj
Cfj
w+(ξ)(ξ − z)
dξ, (2.3)
g(z) =
w(z)
2πi
N∑
j=1
∫
Σj
Cgj
w+(ξ)(ξ − z)
dξ +
w(z)
4
∫
R
n(λ)
w(λ)(λ− z)
dλ, (2.4)
where Cfj , Cgj are uniquely defined by linear algebraic equations
N∑
j=1
Cfj
∫
Σj
ξkdξ
w+(ξ)
= 0, k = 0, . . . , N − 2,
N∑
j=1
Cfj
∫
Σj
ξN−1dξ
w+(ξ)
= −2πi, (2.5)
N∑
j=1
Cgj
∫
Σj
ξkdξ
w+(ξ)
= − iπ
2
∫
R
λkn(λ)
w(λ)
dλ, k = 0, . . . , N − 2,
6 V.P. Kotlyarov
N∑
j=1
Cgj
∫
Σj
ξN−1dξ
w+(ξ)
= 2πi− iπ
2
∫
R
λN−1n(λ)
w(λ)
dλ. (2.6)
A unique solvability of (2.5) and (2.6) is well-known. A detailed proof can be found in [61,
Problem 9.4.2, pp. 234–235] or [62].
Theorem 2.2. Let a contour Σ, a set of real constants (φ0, φ1, . . . , φN ) 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
Ψ(t, x, z) = e(itf0+ixg0)σ3M(t, x, z)e−(itf(z)+ixg(z))σ3 ,
where constants f0 and g0 are equal to
f0 = −
N∑
j=0
ReEj −
1
2πi
N∑
j=1
∫
Σj
Cfj ξ
N
w+(ξ)
dξ, (2.7)
g0 =
N∑
j=1
ReEj −
1
2πi
N∑
j=1
∫
Σj
Cgj ξ
N
w+(ξ)
dξ − 1
4
∫
R
λNn(λ)
w(λ)
dλ, (2.8)
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:
• M(t, x, z) is analytic in z ∈ C \
N⋃
j=0
[Ej , E
∗
j ];
• boundary values M±(t, x, z) are continuous, except for the endpoints Ej and E∗j where M±
have square integrable singularities;
• M(t, x, z) satisfies the jump conditions
M−(t, x, z) = M+(t, x, z)JM (t, x, z), z ∈ Σj = (Ej , E
∗
j ), j = 0, 1, . . . , N, (2.9)
JM (t, x, z) =
(
0 ie−i(tCfj +xCgj +φj)
iei(tCfj +xCgj +φj) 0
)
, z ∈ Σj = (Ej , E
∗
j ); (2.10)
• M(t, x, z) satisfies the symmetry condition M(t, x, z) = σ2M
∗(t, x, z∗)σ2;
• M(t, x, z) = I +O
(
z−1
)
as z →∞.
The next theorem presents an explicit formula for M(t, x, z).
Theorem 2.3. Under conditions of the Theorem 2.2 entries of matrix M(t, x, z) are
M11(t, x, z) =
κ(z) + κ−1(z)
2
Θ(A(∞) + A(D) + K)
Θ(A(z) + A(D) + K)
Θ(A(z) + A(D) + K + C(t, x))
Θ(A(∞) + A(D) + K + C(t, x))
,
M12(t, x, z) =
κ(z)− κ−1(z)
2
e−iφ0 Θ(A(∞) + A(D) + K)Θ(A(z)−A(D)−K−C(t, x))
Θ(A(z)−A(D)−K)Θ(A(∞) + A(D) + K + C(t, x))
,
M21(t, x, z) =
κ(z)− κ−1(z)
2
eiφ0 Θ(A(∞) + A(D) + K)
Θ(A(z)−A(D)−K)
Θ(A(z)−A(D)−K + C(t, x))
Θ(A(∞) + A(D) + K−C(t, x))
,
M22(t, x, z) =
κ(z) + κ−1(z)
2
Θ(A(∞) + A(D) + K)
Θ(A(z) + A(D) + K)
Θ(A(z) + A(D) + K−C(t, x))
Θ(A(∞) + A(D) + K−C(t, x))
,
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 7
where Θ is theta-function (6.7) defined by the Fourier series
Θ(u) =
∑
l∈ZN
exp {πi(Bl, l) + 2πi(l,u)}, (l,u) = l1u1 + · · ·+ lNuN ,
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
Cj(t, x) := −
tCfj + xCgj + φj
2π
, j = 1, 2, . . . , N.
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
Ψt = −(izσ3 +H(t, x))Ψ, Ψx = (izσ3 +H(t, x)− iG(t, x, z))Ψ, (2.11)
where H(t, x) is given by
H(t, x) = −iei(tf0+xg0)σ3 [σ3,m(t, x)]e−i(tf0+xg0)σ3 , (2.12)
m(t, x) = lim
z→∞
z(M(t, x, z)− I),
and
G(t, x, z) =
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− z
ds, z /∈ R.
Matrix F (t, x, λ) is Hermitian, has unit determinant and presented by formula
F (t, x, λ) = ei(tf0+xg0)σ3M(t, x, λ)σ3M
−1(t, x, λ)e−i(tf0+xg0)σ3 , (2.13)
λ 6= ReEj , j = 0, 1, 2, . . . , N.
Theorem 2.5. The associated with Ψ(t, x, z) finite-gap solution to the Maxwell–Bloch equations
(1.1)–(1.3) is given by
E(t, x) = EΘ
Θ(−A(∞) + A(D) + K + C(t, x))
Θ(A(∞) + A(D) + K + C(t, x))
e2i(tf0+xg0)−iφ0 , (2.14)
where
EΘ := 2
Θ(A(∞) + A(D) + K)
Θ(−A(∞) + A(D) + K)
N∑
j=0
ImEj ,
f0 and g0 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
C(t, x) := − tC
f + xCg + φ
2π
.
The density matrix F (t, x, λ) equals to(
N (t, x, λ) ρ(t, x, λ)
ρ∗(t, x, λ) −N (t, x, λ)
)
= ei(tf0+xg0)σ3M(t, x, λ)σ3M
−1(t, x, λ)e−i(tf0+xg0)σ3 . (2.15)
Moreover, the finite-gap solution E(t, x), N (t, x, λ)), ρ(t, x, λ) to the Maxwell–Bloch equations
(1.1)–(1.3) are smooth for t, x, λ ∈ R, except for the λj := ReEj, j = 0, 1, 2, . . . , N .
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.
8 V.P. Kotlyarov
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 Ej , E
∗
j where it has weak singularities, and bounded at all
self-intersection points ReEj . In view of (2.1), det J(x, z) ≡ 1, hence
det Ψ−(t, x, z) = det Ψ+(t, x, z), z ∈ Σ,
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
infinity det Ψ(t, x, z) = 1 + O
(
z−1
)
, hence det Ψ(t, x, z) ≡ 1 by Liouville theorem. In particular,
Ψ(t, x, z) is invertible for any z outside the exceptional set. Suppose that Ψ̃(t, x, z) is another
solution of the RH problem. Then Φ(z) := Ψ̃(t, x, z)Ψ−1(t, x, z) satisfies
Φ−(z) = Ψ̃−(t, x, z)Ψ−1
− (t, x, z) = Ψ̃+(t, x, z)J(x, z)J−1(x, z)Ψ−1
+ (t, x, z) = Φ+(z),
and it is continuous across Σ with exception of end points Ej , E
∗
j and points of self-intersection
ReEj . 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, x, z) =
Ψ̃(t, x, z)Ψ−1(t, x, z) ≡ I and therefore Ψ̃(t, x, z) ≡ Ψ(t, x, z), i.e., the matrix Ψ(t, x, z) is
unique. �
Existence. To prove the existence of the Baker–Akhiezer function we use an explicit construc-
tion 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
Ψ(t, x, z) = ei(tf0+g0x)σ3M(t, x, z)e−i(tf(z)+g(z)x)σ3 , (3.1)
where constants f0 and g0, 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 = f0 and g∗0 = g0.
Due to the definition of Ψ we obtain the RH problem (2.9), (2.10). Indeed, all above state-
ments will be true if f(z) and g(z) possess properties:
• f(z) is analytic in z ∈ C \
N⋃
j=0
[Ej , E
∗
j ];
• f(z) = f∗(z∗) and
f(z) = z + f0 +O(1/z), as z →∞; (3.2)
• f+(z) + f−(z) = Cfj , z ∈ Σj , j = 0, 1, . . . , N ,
where f0 and Cfj are some real (as a result of the symmetry of Ψ) constants;
• g(z) is analytic in z ∈ C \ (R ∪
N⋃
j=0
[Ej , E
∗
j ]);
• g(z) = g∗(z∗) and
g(z) = −z + g0 +O(1/z), as z →∞; (3.3)
• g+(z) + g−(z) = Cgj , z ∈ Σj , j = 0, 1, . . . , N ;
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 9
• g+(λ)− g−(λ) = πi
2 n(λ), λ ∈ R \
N⋃
j=0
ReEj ,
where g0 and Cgj are some real constants. All constants f0, g0, Cfj , Cgj , 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 func-
tions f(z) and g(z) provide the form of matrix (2.10). Indeed, for z ∈ Σj ,
JM (t, x, z) = e−i(tf+(z)+g+(z)x)σ3J(x, z)ei(tf−(z)+g−(z)x)σ3
=
(
0 ie−it(f+(z)+f−(z))−ix(g+(z)+g−(z))−iφj)
ieit(f+(z)+f−(z))+ix(g+(z)+g−(z))+iφj) 0
)
that gives (2.10). We stress that such a choice of the jump matrices on intervals Σj (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
continuous on R \
N⋃
j=0
ReEj :
JM (t, x, z) = e−i(tf+(z)+g+(z)x)σ3J(x, z)ei(tf−(z)+g−(z)x)σ3
= e−it(f+(z)−f−(z))σ3e−
πxn(λ)σ3
2 e−ix(g+(z)−g−(z))σ3
= e−
πxn(λ)σ3
2 e
πxn(λ)σ3
2 = I, z = λ ∈ R \
N⋃
j=0
ReEj
and thus M is analytic in z ∈ C \
N⋃
j=0
[Ej , 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
det Ψ(t, x, z) = detM(t, x, z) ≡ 1,
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].
4 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)
f+(z) + f−(z) = Cf0 , g+(z) + g−(z) = Cg0 (4.1)
across a single arc Σ0.
10 V.P. Kotlyarov
Define
w(z) :=
√
(z − E0)(z − E∗0)
such that w(z) is analytic outside the arc and w(z) ' z as z →∞, and introduce
f̃ :=
f
w
, g̃ :=
g
w
. (4.2)
Then the jump conditions (4.1) reduce to
f̃+(z)− f̃−(z) =
Cf0
w+
, g̃+(z)− g̃−(z) =
Cg0
w+
, z ∈ Σ0,
g̃+(λ)− g̃−(λ) =
iπn(λ)
2w(λ)
, λ ∈ R \ {ReE0}.
Due to the asymptotic conditions (3.2), f̃ = 1 + O(1/z) as z → ∞, and thus f̃ is (uniquely)
determined by Cf0 through Cauchy integral
f̃(z) = 1 +
1
2πi
∫
Σ0
Cf0
w+(ξ)(ξ − z)
dξ = 1 +
Cf0
2w(z)
.
Consequently,
f(z) = w(z)
(
1 +
1
2πi
∫
Σ0
Cf0
w+(ξ)(ξ − z)
dξ
)
= w(z) +
Cf0
2
.
Particularly, f0 is determined by
f0 = − 1
2πi
∫
Σ0
Cf0
w+(ξ)
dξ − 1
2
(E0 + E∗0) = −ReE0 +
Cf0
2
.
Taking into account (3.1) it can be put Cf0 = 0 without loss of generality and hence f(z) =
w(z) =
√
(z − E0)(z − E∗0) and f0 = −ReE0.
Now consider the function g(z). In this case we have g̃ = gw−1 = −1 + O(1/z) as z → ∞,
and thus
g̃(z) = −1 +
1
2πi
∫
Σ0
Cg0
w+(ξ)(ξ − z)
dξ +
1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ
= −1 +
1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ+
Cg0
2
.
Consequently, by the same reason as above with Cg0 = 0
g(z) = −w(z)
(
1− 1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ
)
,
and, particularly,
g0 = ReE0 −
1
4
∫
R
n(λ)
w(λ)
dλ.
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 11
Now consider the general case, where the contour consists of N + 1, N ≥ 1, arcs Σj , j =
0, . . . , N . Define
w(z) :=
√√√√ N∏
j=0
(z − Ej)(z − E∗j )
such that w(z) is analytic outside the arcs Σj and w(z) ' zN+1 as z → ∞, and introduce f̃
and g̃ as in (4.2). The jump conditions reduce to
f̃+(z)− f̃−(z) =
Cfj
w+
, g̃+(z)− g̃−(z) =
Cgj
w+
, z ∈ Σj , (4.3)
g̃+(z)− g̃−(z) =
iπn(λ)
2w(λ)
, λ ∈ R \
N⋃
j=0
{ReEj}. (4.4)
For N ≥ 1 we have
f(z) =
w(z)
2πi
N∑
j=0
∫
Σj
Cfj
w+(ξ)(ξ − z)
dξ
=
w(z)
2πi
N∑
j=0
(∫
Σj
Cf0
w+(ξ)(ξ − z)
dξ +
∫
Σj
Cfj − C
f
0
w+(ξ)(ξ − z)
dξ
)
.
Since w(z) is analytic in z ∈ C \
N⋃
j=0
Σj Cauchy theorem gives
1
2w(z)
=
1
2πi
N∑
j=0
∫
Σj
1
w+(ξ)(ξ − z)
dξ.
Hence
f(z) =
Cf0
2
+
w(z)
2πi
N∑
j=0
∫
Σj
Cfj − C
f
0
w+(ξ)(ξ − z)
dξ.
Again, it is convenient to put Cf0 = 0. Then, in view of f̃ = O(1/z) as z → ∞, Cfj have to
satisfy the system of linear equations
N∑
j=1
∫
Σj
ξmCfj
w+(ξ)
dξ = −2πiδm,N−1
for m = 0, 1, 2, . . . , N − 1. Thus we have N equations (2.5) for N unknown constants Cfj . It is
well known (see, for example, [3, 23, 62, 64]) that this system of linear algebraic equations has
a unique solution {Cfj }Nj=1. Then f(z) and f0 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
g̃ = O(1/z) as z →∞ we have
g̃(z) =
1
2πi
N∑
j=0
∫
Σj
Cgj
w+(ξ)(ξ − z)
dξ +
1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ,
12 V.P. Kotlyarov
where, by the same reasons as above we put Cg0 = 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 Cgj , j = 1, . . . , N . Indeed, if we use the asymptotic
for large z expansion
1
2πi
N∑
j=1
∫
Σj
Cgj
w+(ξ)(ξ − z)
dξ +
1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ =
∞∑
l=0
Il
zl+1
,
then, due to (3.3), it is evident that I0 = I1 = · · · = IN−2 = 0. Hence
g(z) = −
(
zN+1 + wNz
N + · · ·
)(IN−1
zN
+
IN
zN+1
+ · · ·
)
= −
(
zIN−1 + IN−1wN + IN + O
(
z−1
))
= −z + g0 + O
(
z−1
)
and thus IN−1 = 1, and g0 = −wN − IN where wN = −
N∑
j=0
ReEj . 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, pp. 234–235] or in [62]. The parameter g0 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
f(z) =
∫ z
E∗0
ϕ(λ)dλ with ϕ(λ) =
f̂(λ)
w(λ)
,
where w2(λ) =
N∏
j=0
(λ−Ej)(λ−E∗j ) ≡ λ2(N+1) + P2N+1λ
2N+1 + P2Nλ
2N + · · ·+ P0 = P (λ) and
f̂(λ) = λN+1 + f̂Nλ
N + f̂N−1λ
N−1 + · · · + f̂0. Asymptotics (3.2) of the function f(z) defines
f̂N =
P2N+1
2 . In order to define f̂0, f̂1, . . . , f̂N−1, we normalize f(z) by the conditions∫ E∗j
Ej
df = 0, j = 1, . . . , N.
In other words, for z ∈ C \
N⋃
j=0
[Ej , 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 Afj = 2
∫ E∗j
Ej
df = 0. Indeed, since ϕ+(λ) + ϕ−(λ) = 0
for z ∈ [Ej , E
∗
j ] and ϕ+(λ) + ϕ−(λ) = 2ϕ(λ) for z ∈ [ReEj ,ReEj+1], it is easy to check that
Cf0 =
∫ z
E∗0
(ϕ+(λ)+ϕ−(λ))dλ = 0 whereas Cfj for j > 0 are determined as the (nonzero) b-periods
of f(z)
Cfj = f+(z) + f−(z) =
∫ z
E∗0
(ϕ+(λ) + ϕ−(λ))dλ
= 2
j∑
l=1
∫ El
E∗l−1
ϕ(λ)dλ =: Bf
j 6= 0, j = 1, . . . , N, (5.1)
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 13
where Bf
j =
∫
bj
df(z). The last equality becomes obvious if we use the definition of a- and
b-cycles of the hyperelliptic surface given by the function w(z) (see the next section and Fig. 2).
On the other hand, for z ∈ R \
N⋃
j=0
{ReEj}
f+(z)− f−(z) =
∫ z
E∗0
(ϕ+(λ)− ϕ−(λ))dλ
= 2
j∑
l=1
∫ E∗l
El
ϕ+(λ)dλ =
j∑
l=1
Afl = 0, j = 1, . . . , N.
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
g(z) = −f(z) +
w(z)
2πi
N∑
j=0
∫
Σj
Chj
w+(ξ)(ξ − z)
dξ +
w(z)
4
∫
R
n(λ)
w(λ)(λ− z)
dλ, (5.2)
where constants {Chj }Nj=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):
• h(z) is analytic in z ∈ C \
(
R ∪
N⋃
j=0
[Ej , E
∗
j ]
)
;
• h(z) = O(1/z), as z →∞;
• h+(z)− h−(z) =
Chj
w(z) , z ∈ Σj , j = 0, 1, . . . , N ;
• h+(λ)− h−(λ) = πi
2w(λ)n(λ), λ ∈ R \
N⋃
j=0
ReEj ,
where Chj = Cgj + Cfj are to be determined. Due to (5.1) Cfj are already known: Cfj = Bf
j ,
j = 1, . . . , N . Then h(z) can be written as a sum of Cauchy integrals
h(z) =
1
2πi
N∑
j=0
∫
Σj
Chj
w+(ξ)(ξ − z)
dξ +
1
4
∫
R
n(λ)
w(λ)(λ− z)
dλ,
and hence (5.2) follows. The asymptotic condition (3.3) leads to a system of N linear equation
for Chj , j = 1, . . . , N provided that Ch0 = Cg0 = 0. The system are
N∑
j=1
Chj
∫
Σj
ξkdξ
w+(ξ)
= − iπ
2
∫
R
λkn(λ)
w(λ)
dλ, k = 0, . . . , N − 1.
Similarly to (2.6) this system has a unique solution. In this case the parameter g0 is equal to
g0 = −f0 −
1
2πi
N∑
j=1
∫
Σj
Chj ξ
N
w+(ξ)
dξ − 1
4
∫
R
λNn(λ)
w(λ)
dλ. (5.3)
Substituting (2.7) in (5.3) and using equality Cfj − Chj = −Cgj we obtain
g0 =
N∑
j=0
ReEj −
1
2πi
N∑
j=1
∫
Σj
Cgj ξ
N
w+(ξ)
dξ − 1
4
∫
R
λNn(λ)
w(λ)
dλ,
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).
14 V.P. Kotlyarov
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
κ(z) = 4
√√√√ N∏
j=0
z − E∗j
z − Ej
, z ∈ C \ Γ, Γ =
N⋃
j=0
[Ej , E
∗
j ],
where cuts are chosen along [Ej , E
∗
j ], j = 0, . . . , N with orientation from top to bottom. The
branch of root is fixed by the condition κ(∞) = 1. Then
κ−(z) = iκ+(z), z ∈ Γ. (6.1)
Notice also that
• κ(z) = (z − Ej)−1/4 + O(1) as z → Ej ,
• κ(z) = 1 +
N∑
j=0
Ej−E∗j
4z + O
(
z−2
)
, z →∞.
Recall that RH problem for M(z) (2.9), (2.10)) is as follows:
• M(t, x, z) is analytic in C \ Γ, Γ =
N⋃
j=0
[Ej , E
∗
j ];
• boundary values M±(t, x, z) are continuous except end-points Ej and E∗j where M± have
square integrable singularities;
• M−(t, x, z) = M+(t, x, z)JM (t, x, z), z ∈ Γ,
JM (t, x, z) =
(
0 ie−iφ0
ieiφ0 0
)
, z ∈ (E0, E
∗
0), (6.2)
=
(
0 ie−ixCfj −itCgj−iφj
ieixCfj +itCgj +iφj 0
)
, z ∈ (Ej , E
∗
j ) (6.3)
for j = 1, 2, . . . , N , and Cfj , Cgj , φj are some given real constants (recall that Cf0 = Cg0 = 0);
• M(t, x, z) = σ2M
∗(t, x, z∗)σ2;
• M(t, x, z) = I + O
(
z−1
)
, z →∞.
First, consider the case N = 0. Then, by (6.2), M(t, x, z) ≡ M(z) can be constructed
using κ(z)
M(z) =
κ(z) + κ−1(z)
2
κ(z)− κ−1(z)
2
e−iφ0
κ(z)− κ−1(z)
2
eiφ0
κ(z) + κ−1(z)
2
.
Expanding M(z) as z →∞,
M(z) = I +
m
z
+ O
(
z−2
)
,
we have
m =
E0 − E∗0
4
(
0 e−iφ0
eiφ0 0
)
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 15
and thus the simplest periodic solution of the Maxwell–Bloch equation associated with Ψ (2.1)
has the form of a plane wave (see (8.2)–(8.4) at the end of the paper).
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 w2 = P (z),
where
P (z) =
N∏
j=0
(z − Ej)(z − E∗j ),
with cuts along Σj = (Ej , 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 Σj . The upper
and lower sheets of X are denoted by X+ and X− respectively; they are fixed by the relations√
P (z) = ±zN+1
(
1 + O
(
z−1
))
, z = π(P)→∞, P ∈ X±,
where z = π(P) is the standard projection of P = (w, z) ∈ X on the Riemann sphere CP1.
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 Ej and E∗j , j = 0, 1, 2, . . . , N , and two poles at ∞+ and ∞−, each of multiplicity N + 1.
Further, we introduce the Abelian integrals
ωj(z) =
∫ z
E∗0
ψj(s)ds, j = 1, 2, . . . , N,
where dωj(P) is a basis of holomorphic differentials on X
ψj(z) =
N∑
i=1
cjiz
N−i√
P (z)
.
The coefficients cjl are uniquely determined by the normalization conditions∫
al
dωj(P) = 2
∫ E∗l
El
ψj+(z)dz = δjl, j, l = 1, 2, . . . , N.
We have chosen al-cycles as ovals on the upper sheet of X around the intervals
(
El, Êl
)
,
Êl := E∗l , l = 0, 1, 2, . . . , N , see Fig. 2.
Figure 2. a- and b-cycles.
16 V.P. Kotlyarov
The normalized holomorphic differentials define the b-period matrix as
Bjl =
∫
bl
dωj(P) = 2
l∑
k=1
∫ Ek
E∗k−1
ψj(z)dz,
where bl-cycle starts from (E0, E
∗
0), goes on the upper sheet to (El, E
∗
l ), and returns on the
lower sheet to the starting point. This is a symmetric matrix with positive definite imaginary
part.
Let ej = (0, . . . , 1, . . . , 0) be the unit vector in CN and Bej the j-th column of the matrix B.
Denote by Λ ⊂ CN the lattice generated by the linear combinations, with integer coefficients,
of the vectors ej and Bej for j = 1, 2, . . . , N . Then, by the definition, Jacobian variety of X is
the complex torus Jac{X} = CN/Λ. The Abel mapping A : X → Jac{X} is defined as follows
Aj(P) =
∫ P
P0
dωj(Q), j = 1, 2, . . . , N, (6.4)
where the point P0 is fixed by condition π(P0) = E∗0 and Q is the integration variable. The
Abel mapping is also defined for integral divisors D = P1 + · · ·+ Pm by summation
A(D) = A(P1) + · · ·+ A(Pm) (6.5)
and is extended to non-integral divisors D = D+ − D− (where D± are integral divisors) by
A(D) = A(D+) −A(D−). If the degree of the divisor D is zero, then A(D) is independent
of the chosen point P0. The Abel theorem states that if D = D+ − D− is the divisor of
a meromorphic function on the compact Riemann surface X and D+, D− are integral divisors of
zeros and poles, then A(D) = 0 in the Jacobian (mod Λ). Besides, for any non-special integral
divisor D = P1 + · · ·+PN of degree N , there exists a vector w(D) such that the Riemann theta
function Θ(A(P) + w(D)) defined on X with cuts along of the cycles aj and bj has precisely N
zeros at Pj , j = 1, . . . , N . The vector w(D) is defined by
w(D) = −A(D)−K.
In the hyperelliptic case, the Riemann constant vector K is defined by (cf. [64])
Kj =
1
2
N∑
l=1
Blj −
j
2
mod Λ. (6.6)
Associated with the matrix B there is the Riemann theta function defined for u ∈ CN by the
Fourier series
Θ(u1, . . . , un) =
∑
l∈ZN
exp {πi(Bl, l) + 2πi(l,u)}, (6.7)
where (l,u) = l1u1 + · · · + lNuN . It is an even function, i.e., Θ(−u) = Θ(u), and has the
following periodicity properties
Θ(u± ej) = Θ(u), Θ(u±Bej) = e∓2πiuj−πiBjjΘ(u),
where ej = (0, . . . , 0, 1, 0, . . . , 0) is the j-th basis vector in CN . This implies that the function
h(u) =
Θ(u + c + d)
Θ(u + d)
,
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 17
where c,d ∈ CN are arbitrary constant vectors, has the periodicity properties
h(u± ej) = h(u), h(u±Bej) = e∓2πicjh(u).
The Abelian integrals A(z) considered on the upper sheet of X (z ∈ C\Γ), have the properties
A−(z)−A+(z) = 0
(
mod ZN
)
, z ∈ R \ ∪Nj−0 ReEj , (6.8)
A−(z) + A+(z) = 0, z ∈ (E0, E
∗
0), (6.9)
A−(z) + A+(z) = Bej , z ∈ (Ej , E
∗
j ), j = 1, . . . , N. (6.10)
Indeed, since ψl(z+) + ψl(z−) = 0 on (Ej , E
∗
j ) and ψl(z) is continuous on C \ ∪Nj=0(Ej , E
∗
j ), it
easy to see that for l = 1, 2, . . . , N ,
Al(z+)−Al(z−) =
∫ z
E∗0
(ψl(s)− ψl(s))ds = 0, z± ∈ (−∞,ReE0) ∪ (ReE0,ReE1)
and
Al(z+)−Al(z−) = 2
j−1∑
k=1
∫ E∗k
Ek
ψl(s+)ds =
j−1∑
k=1
δkl = 0 (mod Z)
for z± ∈ (ReEj−1,ReEj), 1 < j ≤ N and for z± ∈ (ReEN ,+∞). On the other hand,
Al(z+) +Al(z−) =
∫ z
E∗0
(ψl(s+) + ψl(s−))ds = 0, z± ∈ (E0, E
∗
0)
and
Al(z+) +Al(z−) = 2
j∑
k=1
∫ Ek
E∗k−1
ψj(s)ds+
j−1∑
k=1
δkl = Bjl,
z± ∈ (Ej , E
∗
j ), j, l = 1, 2, . . . , N.
Now define (s = 1, 2)
Fs(z) =
Θ(A(z) + c + ds)
Θ(A(z) + ds)
, Hs(z) =
Θ(−A(z) + c + ds)
Θ(−A(z) + ds)
, z ∈ C \ Γ, (6.11)
where c,d1,d2 ∈ CN are arbitrary (so far) constant vectors. Then, by (6.8)–(6.10), we have
(s = 1, 2)
Fs−(z) = Fs+(z), Hs−(z) = Hs+(z), z ∈ R \
N⋃
j=0
ReEj
and
Fs−(z) = e−2πicjHs+(z), Hs−(z) = e2πicjFs+(z), z ∈ (Ej , E
∗
j )
for j = 0, 1, . . . , N , where c0 = 0.
Next, define the matrix-valued function
M̂(z) :=
(
F1(z) H1(z)
F2(z) H2(z)
)
.
18 V.P. Kotlyarov
Then we have
M̂−(z) = M̂+(z), z ∈ R \
N⋃
j=0
ReEj , (6.12a)
M̂−(z) = M̂+(z)
(
0 e2πicj
e−2πicj 0
)
, z ∈ (Ej , E
∗
j ), j = 0, 1, . . . , N, (6.12b)
with c0 = 0. Finally, taking into account (6.1) and (6.12), we define M by (provided F1(∞) 6= 0
and H2(∞) 6= 0)
M(z) :=
a(z)
F1(z)
F1(∞)
b(z)
H1(z)
F1(∞)
e−iφ0
b(z)
F2(z)
H2(∞)
eiφ0 a(z)
H2(z)
H2(∞)
, (6.13)
where a(z) := 1
2(κ(z) + κ−1(z)) and b(z) := 1
2
(
κ(z)− κ−1(z)
)
. Obviously, M(z) is analytic in
C \
N⋃
j=0
[Ej , E
∗
j ], M(z) = I + O
(
z−1
)
as z →∞, and, due to (6.12), it has the jumps
M−(z) = M+(z)
(
0 ie2πicje−iφ0
ie−2πicjeiφ0 0
)
, z ∈ (Ej , E
∗
j ), j = 0, 1, . . . , N,
if we take into account that a−(z) = ib+(z) and b−(z) = ia+(z) when z ∈ (Ej , E
∗
j ), j =
0, 1, . . . , N . These jumps are consistent with the jump conditions (6.2), (6.3) for the RH problem,
if we set cj := − tCfj +xCgj +φj
2π , j = 1, . . . , N (recall that c0 = 0) and hence the matrix M(z)
depends from t, x ∈ R additionally, i.e., M(z) = M(t, x, z).
It remains to choose the vectors d1 and d2 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) + ds) and Θ(−A(z) + ds) (s = 1, 2) are
to be canceled by the zeros of κ(z)± κ−1(z).
The zeros of κ(z)±κ−1(z) are those of κ2(z)± 1, and hence of κ4(z)− 1. By the definition
of κ(z), equation κ4(z)− 1 = 0 reads r(z) = 1, where
r(z) :=
N∏
j=0
z − E∗j
z − Ej
.
Since
N∑
j=0
(Ej − E∗j ) = 2i ImEj 6= 0, equation r(z) = 1 reduces to
0 =
N∏
j=0
(z − E∗j )−
N∏
j=0
(z − Ej) = 2i
N∑
j=0
ImEj
N−1∏
l=0
(z − zl)
with some finite zl, l = 0, . . . , N − 1.
Introduce the non-special divisor D = P1 + · · · + PN such that D = D1 + D2, where D1 =
P1 + · · ·+PN1 ∈ X−, 0 ≤ N1 ≤ N and a(zj) = κ(zj)+κ−1(zj) = 0, zj = π(Pj), j = 1, 2, . . . , N1,
whereas D2 = PN1+1+· · ·+PN ∈ X+ with b(zj) = κ(zj)−κ−1(zj) = 0, j = N1+1, N1+2, . . . , N .
Set d1 = A(D) + K and d2 = −A(D) − K. Then Θ(A(P) + A(D) + K) has N1 zeroes
P ′1, . . . ,P ′N1
on X+ and N − N1 zeroes P ′N1+1, . . . ,P ′N on X− [3], where the points P ′j and Pj
form a conjugated pair of points on X with π(P ′j) = π(Pj) = zj ∈ C, j = 1, 2, . . . , N . Similarly,
Θ(A(P) − A(D) − K) has N − N1 zeroes PN1+1, . . . ,PN on X+ and m zeroes P1, . . . ,PN1
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 19
on X−. Taking the restrictions of these Riemann theta functions on the upper sheet with the
cut Γ =
N⋃
j=0
(Ej , E
∗
j ), we have that
F1(z) =
Θ(A(z) + c + d1)
Θ(A(z) + d1)
, H2(z) =
Θ(−A(z) + c + d2)
Θ(−A(z) + d2)
are analytic in z ∈ C \Γ with poles at z1, . . . , zN1 , which are canceled in the products a(z)F1(z)
and a(z)H2(z). Similarly, b(z)F2(z) and b(z)H1(z) are analytic in C \ Γ, since the poles
zN1+1, . . . , zN are canceled by the zeroes of b(z). Notice that the idea to cancel the poles
of Fj and Hj 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
F1(∞) =
Θ(A(∞+) + A(D) + K + c)
Θ(A(∞+) + A(D) + K)
, H2(∞) =
Θ(A(∞+) + A(D) + K− c)
Θ(A(∞+) + A(D) + K)
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 F1(z) (H2(z)) belong to the mentioned divisor and this divisor does not contain infinity,
then F1(∞) 6= 0 (as well as H2(∞) 6= 0). Moreover, taking into account the symmetries
M(z) = σ2M
∗(z∗)σ2, i.e., M22(z) = M∗11(z∗) and M21(z) = −M∗12(z∗), and unity determinant
of the matrix M we obtain(
M11(z)M22(z)−M12(z)M21(z)
)
|z=λ = |M11(λ)|2 + |M12(λ)|2 ≡ 1, λ 6= ReEj (6.14)
that gives the boundedness of all entries of matrix M(λ). In turn, it means that F1(∞)
and H2(∞) can not vanish for any vector c. To prove the symmetry of M(z) let us consider
matrix M̃(z) := σ2M
∗(z∗)σ2. Then M̃(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 have M̃(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 Ej and E∗j , where Ψ has weak singularities).
Formulas for entries Mij(t, x, z) of the matrix M(t, x, z) follows from (6.11) and (6.13). Ex-
panding it at infinity,
M(t, x, z) = I +
m(t, x)
z
+ O
(
z−2
)
, z →∞,
and taking into account that Θ(A(z) + B) is bounded and dA
dz = O
(
z−2
)
as z →∞, we have
m12(t, x) = E0e
−iφ0 Θ(−A(∞) + A(D) + K + C(t, x))Θ(A(∞) + A(D) + K)
Θ(−A(∞) + A(D) + K)Θ(A(∞) + A(D) + K + C(t, x))
, (6.15)
m21(t, x) = E0e
iφ0 Θ(A(∞)−A(D)−K + C(t, x))Θ(A(∞) + A(D) + K)
Θ(A(∞)−A(D)−K)Θ(A(∞) + A(D) + K−C(t, x))
,
where E0 = 1
4
N∑
j=0
(Ej − E∗j ) = i
2
N∑
j=0
ImEj and C(t, x) := − tCf+xCg+φ
2π . Notice that Cf and Cg
are determined when constructing f(z) and g(z) whereas Ej = ReEj+i ImEj , j = 0, 1, 2, . . . , N ,
and real constants (φ0, φ1, φ2, . . . , φN ) present itself free real parameters total number of which is
20 V.P. Kotlyarov
equal to 3N + 3. Evidently, the constant φ0 is defined modulo 2π, while (2πφ1, 2πφ2, . . . , 2πφN )
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 func-
tion 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
N∏
j=0
(z − Ej)(z − E∗j )− |q(0, 0)|2
N∏
j=1
(z − zj)(z − z∗j )
=
(
zN+1 + fNz
N + fN−1z
N−1 + · · ·+ f0
)2
(6.16)
are fulfilled. The left hand side of the equality is determined by 4N + 3 real parameters:
branching points Ej ∈ C (ImEj 6= 0, j = 0, 1, . . . , N), projections zj = π(Pj) of the non-
special divisor D = P1 + · · · + PN , 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 zj fN = −
N∑
j=0
ReEj) 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 zj of the functions κ(z)±κ−1(z). Thus, the reality conditions mean that there is
a correspondence between (φ0, φ1, . . . , φN ) and (f0, f1, . . . , fN−1, |q(0, 0)| = |E(0, 0)|). However,
this issue is beyond the scope of article. In our approach E0, E1, . . . , EN and φ0, φ1, . . . , φN are
independent. Due to the symmetry Ψ(t, x, z) = σ2Ψ∗(t, x, z∗)σ2 the potential matrix H(t, x) is
anti-Hermitian for any choice of the parameters (see the next section for details).
7 AKNS equations for Ψ(t, x, z)
Here we prove Theorem 2.4.
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
Ψ−(t, x, z) = Ψ+(t, x, z)J(x, z),
where the jump matrices
J(x, z) =
(
e−
πxn(λ)
2 0
0 e
πxn(λ)
2
)
, λ ∈ R \
N⋃
j=0
ReEj ,(
0 ie−iφj
ieiφj 0
)
, z ∈ Σj = (Ej , E
∗
j ), j = 0, 1, . . . , N,
are independent on t. The jump condition gives
∂Ψ−(t, x, z)
∂t
Ψ−1
− (t, x, z) =
∂Ψ+(t, x, z)
∂t
Ψ−1
+ (t, x, z), z ∈ Σ.
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 21
This relation, together with a continuity of Ψ± outside of exceptional points (Ej , E
∗
j ,ReEj), im-
plies that logarithmic derivative Ψt(t, x, z)Ψ
−1(t, x, z) is analytic (entire) in z ∈ C. Indeed, since
Ψt(t, x, z)Ψ
−1(t, x, z) has no jump across Σ \
N⋃
j=0
ReEj 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 ReEj , weak singularities at the endpoints Ej , E
∗
j
and the second order of Ψ. Further, since M(t, x, z) and Mt(t, x, z) have the asymptotics:
M(t, x, z) = I +
m(t, x)
z
+ O
(
z−2
)
,
dM(t, x, z)
dt
=
dm(t, x)/dt
z
+O
(
z−2
)
, z ∈ C±,
as z →∞, it follows that
Ψt(t, x, z)Ψ
−1(t, x, z)
= −izσ3 + iei(tf0+xg0)σ3 [σ3,m(t, x)]e−i(tf0+xg0)σ3 + O
(
z−1
)
, z →∞,
where [A,B] := AB − BA. Therefore, by Liouville’s theorem, the logarithmic derivative is
a polynomial
U(z) := Ψt(t, x, z)Ψ
−1(t, x, z) = −izσ3 −H(t, x),
where
H(t, x) := −iei(tf0+xg0)σ3 [σ3,m(t, x)]e−i(tf0+xg0)σ3 =
(
0 q(t, x)
p(t, x) 0
)
.
Using the symmetry σ2Ψ∗(z∗)σ2 = Ψ(t, x, z) we find that U(z) = σ2U
∗(z∗)σ2. 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) = −4im12(t, x)e2i(tf0+xg0) with m12(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).
In contrast with previous case logarithmic derivative Ψx(t, x, z)Ψ−1(t, x, z) is analytic in
z ∈ C± only. Indeed, since jump matrix J(z) (9.1) is independent on t and x for z ∈ Σ \ R,
then this logarithmic derivative is continuous across the contour Σ\R, while it is not continuous
across the real line because the corresponding jump matrix (2.1) J(x, λ) = e−
πn(λ)xσ3
2 depends
on x. The endpoints of the contour Σ are removable singularities by the same reasons as above.
Further, the asymptotic behavior at infinity gives
Ψx(t, x, z)Ψ−1(t, x, z) = izσ3 +H(t, x) +O
(
z−1
)
, z ∈ C±, z →∞.
The jump condition Ψ− = Ψ+e
−πxn(λ)σ3
2 (λ is real) yields
Ψx(t, x, λ+ i0)Ψ−1(t, x, λ+ i0)−Ψx(t, x, λ− i0)Ψ−1(t, x, λ− i0) =
πn(λ)
2
F (t, x, λ),
where
F (t, x, λ) := Ψ(t, x, λ+ i0)σ3Ψ−1(t, x, λ+ i0) = Ψ(t, x, λ− i0)σ3Ψ−1(t, x, λ− i0)
= ei(tf0+xg0)σ3M(t, x, λ)σ3M
−1(t, x, λ)e−i(tf0+xg0)σ3 ,
λ 6= ReEj , j = 0, 1, 2, . . . , N.
Therefore Ψx(t, x, z)Ψ−1(t, x, z)− izσ3 −H(t, x) is represented through Cauchy integral
Ψx(t, x, z)Ψ−1(t, x, z)− izσ3 −H(t, x) =
1
4i
∫ ∞
−∞
F (t, x, s)n(s)
s− z
ds, z /∈ R.
22 V.P. Kotlyarov
Due to the symmetries of M(t, x, λ) we find that F (t, x, λ) is Hermitian. Since tr(Ψx(t, x, λ±
i0)Ψ−1(t, x, λ± i0)) = (det Ψ(t, x, λ± i0))′x ≡ 0 and trσ3 = trH(t, x) = 0 then trF (t, x, λ) = 0
and, hence, F (t, x, λ) has the structure
F (t, x, λ) :=
(
N (t, x, λ) ρ(t, x, λ)
ρ∗(t, x, λ) −N (t, x, λ)
)
.
Thus Ψ(t, x, z) satisfies two differential equations
Ψt = U(t, x, z)Ψ, U(t, x, z) = −izσ3 −H(t, x),
Ψx = V (t, x, z)Ψ, V (t, x, z) = izσ3 +H(t, x)− iG(t, x, z),
where
G(t, x, z) =
1
4
∫ ∞
−∞
F (t, x, s)n(s)
s− z
ds, z /∈ R.
For real z = λ ∈ R we have two differential in x equations
Ψx = V ±(t, x, λ)Ψ, V ±(t, x, λ) = iλσ3 +H(t, x)− iG±(t, x, λ),
where G±(t, x, λ) := G(t, x, λ± i0).
The compatibility condition (Ψxt(t, x, λ± i0) = Ψtx(t, x, λ± i0)) gives the identity in λ
Ux(t, x, λ)− V ±t (t, x, λ) +
[
U(t, x, λ), V ±(t, x, λ)
]
= 0.
This identity is equivalent to
Ht(t, x) +Hx(t, x)− 1
4
∫ ∞
−∞
[σ3, F (t, x, s)]n(s)ds
=
i
4
∫ ∞
−∞
Ft(t, x, s) + [isσ3 +H(t, x), F (t, x, s)]
s− λ∓ i0
n(s)ds
and it is possible if and only if the left and right hand sides are equal zero, i.e.,
Ht(t, x) +Hx(t, x)− 1
4
∫ ∞
−∞
[σ3, F (t, x, s)]n(s)ds = 0,
Ft(t, x, λ) + [iλσ3 +H(t, x), F (t, x, λ)] = 0.
These matrix equations are equivalent to the MB equations (1.1)–(1.3). Thus we proved that the
matrices Ψ(t, x, λ ± i0) satisfy equations (2.11) (with coefficients (2.12), (2.13)) which coincide
with AKNS system (1.8) and (1.9). Hence scalar functions E(t, x), N (t, x, λ) and ρ(t, x, λ) satisfy
the Maxwell–Bloch equations (1.1)–(1.3). �
8 Finite-gap solutions to the MB equations
Here we prove the Theorem 2.5.
Proof. Taking into account (6.15) we have
E(t, x) = EΘe−iφ0 Θ(−A(∞) + A(D) + K + C(t, x))
Θ(A(∞) + A(D) + K + C(t, x))
e2i(tf0+xg0), (8.1)
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 23
where
EΘ := 2
Θ(A(∞) + A(D) + K)
Θ(−A(∞) + A(D) + K)
N∑
j=0
ImEj
is a constant. Hence (8.1) gives (2.14). Equation (2.15) follows from (2.13). Formulas for entries
Mij(t, x, z) of the matrix M(t, x, z) follows from (6.11) and (6.13). Finally, the analytical depen-
dence of all ingredients of the construction with respect to c = C(t, x) provides the smoothness
of the solution of the MB equations. The important property (6.14) follows from (2.12) and
a chain of equalities
N 2(t, x, λ)) + |ρ(t, x, λ)|2 = −detF (t, x, λ) = −detM(t, x, λ)σ3M
−1(t, x, λ) = 1.
Thus N (t, x, λ)) and ρ(t, x, λ) are smooth for all t, x ∈ R (λ 6= ReEj) 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 ImE0e2i(tf0+xg0)−iφ0 , (8.2)
ρ(t, x, λ) = − i ImE0
w(λ)
e2i(tf0+xg0)−iφ0 , (8.3)
N (t, x, λ) =
λ− ReE0
w(λ)
, (8.4)
where w(λ) =
√
(λ− E0)(λ− E∗0), λ ∈ R and
tf0 + xg0 = (x− t) ReE0 −
x
4
∫ ∞
−∞
n(λ)dλ
w(λ)
.
Some periodic and rational solutions of the reduced Maxwell–Bloch equations with n(λ) = δ(λ)
were recently obtained in [63].
9 Final remarks
Many asymptotic problems deal with contours of another structure and, consequently, another
RH problems. Namely, let Σ = R ∪
N⋃
j=0
Σj ∪
N⋃
j=1
Γj where Σj = (Ej , Êj), j = 0, 1, . . . , N , and
Γj = (Êj−1, Ej), j = 1, . . . , N (Fig. 3). Σ has to be symmetric with respect to the real line,
therefore we suppose that E∗j = ÊN−j and Ê∗j = EN−j . Denote through Re Ẽ a unique point of
self-intersection of contour Σ. Here Ẽ = EN/2 for even N and Ẽ = E[N/2]+1 for odd N where
[N/2] is the integer part of N/2.
Definition 9.1. Let a contour Σ, a set of real constants (φ0, φ1, . . . , φN ) 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:
• Ψ(t, x, z) is analytic in z ∈ C \ Σ where Σ is a closure of Σ;
• boundary values Ψ±(t, x, z) are continuous with exception of endpoints Ej and E∗j , j =
0, 1, . . . , N where they have square integrable singularities;
24 V.P. Kotlyarov
--
�
�
�����
@@I
@
@
@
@
@@
@
@
@@ �
�
�
�
��
E0
Ê6 = E∗0
Ẽ
Ẽ∗
Σ2Γ2
Ê1
E2 Ê2
λ = Re z
Figure 3. Oriented contour Σ.
• Ψ(t, x, z) satisfies the jump conditions:
Ψ−(t, x, z) = Ψ+(t, x, z)J(x, z), z ∈ Σ,
where
J(x, z) =
(
e−
πxn(λ)
2 0
0 e
πxn(λ)
2
)
, z = λ ∈ R \ Re Ẽ,
J(x, z) =
(
0 ie−iφ0
ieiφ0 0
)
, z ∈ Σj =
(
Ej , Êj
)
, j = 0, 1, . . . , N, (9.1)
J(x, z) =
(
e−iφj 0
0 eiφ̂j
)
, z ∈ Γ̂j =
(
Êj−1, Ej
)
, j = 1, . . . , N,
• Ψ(t, x, z) satisfies the symmetry condition Ψ(t, x, z) = σ2Ψ∗(t, x, z∗)σ2, where σ2 =
(
0 −i
i 0
)
;
• Ψ(t, x, z) =
(
I +O
(
z−1
))
e−iz(t−x)σ3 as z →∞.
By the same way as above it is possible to obtain results similar those are formulated in
Theorems 2.2–2.5. In this case another Riemann surface with the same branch points arises but
with a different basis of cycles, that corresponds to a different choice of jump matrices (Details
can be found in [45] for Ψ associated with the nonlinear Schrödinger equation).
The paper presents the matrix Baker–Akhiezer function associated with the Maxwell–Bloch
equations. We used the matrix Riemann–Hilbert problem posed on the complex plane with
a finite set of cuts. Such a Baker–Akhiezer function having the unit determinant, satisfies
the AKNS equations for the Maxwell–Bloch system and generates the finite-gap quasi-periodic
solution to the MB equations.
The matrix Baker–Akhiezer function will be useful for applying to Cauchy problems with
periodic (quasi-periodic) finite-gap initial data as well as for the initial-boundary value problems
with such type of initial and boundary functions. The suggested RH problem will be also useful
for studying the long time/large space asymptotic behavior of solutions of different initial-
boundary value problems to the MB equations by the way as, for example, in [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]. The focusing nonlinear
Schrödinger equation and its finite-gap solutions are widely used for modeling of the so-called
rogue waves. Some recent results in this field can be found in [4, 5, 55]. In this regard, we
hope that the results of paper will be useful for an investigation of the rogue waves of the
Maxwell–Bloch equations.
A Matrix Baker–Akhiezer Function Associated with the Maxwell–Bloch Equations 25
Acknowledgments
The author thanks to the referees for careful reading of the manuscript and valuable recommen-
dations.
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1 Introduction
2 Definition of the Baker–Akhiezer function and main results
3 Proof of the Theorem 2.2 and RH problem for M=M(t,x,z)
4 Construction of the phases f and g by Cauchy integrals
5 Representation of f(z) and g(z) through hyperelliptic integrals
6 Explicit construction of the matrix M(t,x,z)
7 AKNS equations for (t,x,z)
8 Finite-gap solutions to the MB equations
9 Final remarks
References
|
| id | nasplib_isofts_kiev_ua-123456789-209768 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:18:45Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kotlyarov, V.P. 2025-11-26T11:26:16Z 2018 A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34L25; 34M50; 35F31; 35Q15; 35Q51 arXiv: 1802.01622 https://nasplib.isofts.kiev.ua/handle/123456789/209768 https://doi.org/10.3842/SIGMA.2018.082 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 the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the 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 a 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 the 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. The author thanks the referees for careful reading of the manuscript and valuable recommendations. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions Article published earlier |
| spellingShingle | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions Kotlyarov, V.P. |
| title | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_full | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_fullStr | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_full_unstemmed | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_short | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_sort | matrix baker-akhiezer function associated with the maxwell-bloch equations and their finite-gap solutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209768 |
| work_keys_str_mv | AT kotlyarovvp amatrixbakerakhiezerfunctionassociatedwiththemaxwellblochequationsandtheirfinitegapsolutions AT kotlyarovvp matrixbakerakhiezerfunctionassociatedwiththemaxwellblochequationsandtheirfinitegapsolutions |