How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave
We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically a finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this so...
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| description | We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically a finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 045, 32 pages
How Discrete Spectrum and Resonances Influence
the Asymptotics of the Toda Shock Wave
Iryna EGOROVA a and Johanna MICHOR b
a) B. Verkin Institute for Low Temperature Physics and Engineering,
47, Nauky Ave., 61103 Kharkiv, Ukraine
E-mail: iraegorova@gmail.com
b) Faculty of Mathematics, University of Vienna,
Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
E-mail: Johanna.Michor@univie.ac.at
URL: http://www.mat.univie.ac.at/~jmichor/
Received January 21, 2021, in final form April 26, 2021; Published online May 01, 2021
https://doi.org/10.3842/SIGMA.2021.045
Abstract. We rigorously derive the long-time asymptotics of the Toda shock wave in
a middle region where the solution is asymptotically finite gap. In particular, we describe
the influence of the discrete spectrum in the spectral gap on the shift of the phase in the
theta-function representation for this solution. We also study the effect of possible resonances
at the endpoints of the gap on this phase. This paper is a continuation of research started
in [arXiv:2001.05184].
Key words: Toda equation; Riemann–Hilbert problem; steplike; shock
2020 Mathematics Subject Classification: 37K40; 35Q53; 37K45; 35Q15
1 Introduction
The Toda shock wave describes the motion of an infinite chain of particles with nonlinear nearest
neighbor interactions when the chain is excited with shock type initial conditions. We are
interested in the effect the eigenvalues in the spectral gap of the associated Lax operator have
on the asymptotic behavior of the shock wave. The Toda shock wave is generated by the solution
of the following initial value problem for the Toda lattice [23, 24]
d
dt
b̃(n, t) = 2
(
ã(n, t)2 − ã(n− 1, t)2
)
,
d
dt
ã(n, t) = ã(n, t)
(
b̃(n+ 1, t)− b̃(n, t)
)
, (n, t) ∈ Z× R+, (1.1)
with a steplike initial profile
{
ã(n, 0), b̃(n, 0)
}
such that
ã(n, 0)→ a±, b̃(n, 0)→ b±, as n→ ±∞, (1.2)
where a± > 0 and b± ∈ R satisfy the condition
b− + 2a− < b+ − 2a+. (1.3)
This condition fixes the position of the background spectra relative to each other; their mutual
location produces essentially different types of asymptotic solutions [20]. The notion of the Toda
shock wave [4, 5] was traditionally associated with symmetric initial data
ã(n− 1, 0) = ã(−n, 0), b̃(n, 0) = −b̃(−n, 0), (1.4)
mailto:iraegorova@gmail.com
mailto:Johanna.Michor@univie.ac.at
http://www.mat.univie.ac.at/~jmichor/
https://doi.org/10.3842/SIGMA.2021.045
2 I. Egorova and J. Michor
and the background constants a− = a+ = 1
2 , b+ = −b− > 1. The asymptotic of the solution
of (1.1) for the particular case
ã(n, 0) =
1
2
, b̃(n, 0) = b sgnn, n ∈ Z, where sgn 0 = 0, (1.5)
was studied in the pioneering work [25] by Venakides, Deift, and Oba in 1991. By use of the Lax–
Levermore approach they established that in a middle region of the half plane (n, t) ∈ Z× R+,
the asymptotic of the shock wave (1.1), (1.5) is described by a 2-periodic solution of the Toda
lattice. They also showed that the asymptotic undergoes a phase shift caused by the presence
of a single eigenvalue at λ = 0. We refer to this middle region of periodic asymptotics as VDO
region,1 compare Figure 1.
In this paper, we offer a derivation and rigorous justification of the asymptotic for (1.1)–(1.3)
in the VDO region using the vector Riemann–Hilbert problem (RHP) approach. We allow more
general initial data (1.2) with arbitrary positive a± and b± satisfying (1.3). In particular, the
novel features are:
� an arbitrary discrete spectrum,
� possible resonances at the edges of the continuous spectrum,
� no symmetry assumption (1.4),
� a partial revision of results in [25] including estimates on the error terms,
� a finite gap (two band) asymptotic due to spectra of different length.
The vector RHP approach in the context of the Toda problem was proposed in [6] and
further developed in [2, 12, 17, 18, 19]. We use standard conjugations/deformations such as the
g-function technique [8] which proved its efficiency in steplike cases. A suitable g-function for
the VDO region replaces the standard phase function and makes it possible to apply the lense
mechanism. It also provides a characterization of the boundaries of the sectors (see Figure 2)
where the asymptotics are given by a finite gap solution of (1.1) with unaltered phase. We des-
cribe the g-function for the VDO region as an Abel integral on the Riemann surface associated
with the continuous two band spectrum of the underlying Jacobi operator of (1.1) in Section 3.
Before we state our main theorem, let us first note that without loss of generality it is
sufficient to study the case of background spectra [b− 2a, b+ 2a] ∪ [−1, 1]. Indeed, assume that
the vector-function
(
ã(t), b̃(t)
)
=
{
ã(n, t), b̃(n, t)
}
n∈Z is the solution of (1.1)–(1.3). Then the
function (a(t), b(t)) given by
a(n, t) =
1
2a+
ã
(
n,
t
2a+
)
, b(n, t) =
1
2a+
b̃
(
n,
t
2a+
)
− b+, n ∈ Z,
satisfies the initial value problem
d
dt
b(n, t) = 2
(
a(n, t)2 − a(n− 1, t)2
)
,
d
dt
a(n, t) = a(n, t)(b(n+ 1, t)− b(n, t)), (n, t) ∈ Z× R+, (1.6)
a(n, 0)→ 1
2
, b(n, 0)→ 0, n→ +∞; a(n, 0)→ a, b(n, 0)→ b, n→ −∞,
with
b+ 2a < −1, (1.7)
where we denoted b := b− − b+, a := a−
2a+
.
1Precise boundaries for the VDO region in our general case are given by (3.5)–(3.7).
How Discrete Spectrum and Resonances Influence the Asymptotics 3
Hence it suffices to study the shock wave (1.6)–(1.7). We assume that the initial data tend
to the background constants exponentially fast with some small rate ρ > 0,
∞∑
n=1
eρn
(∣∣∣∣a(n, 0)− 1
2
∣∣∣∣+ |b(n, 0)|+ |a(−n, 0)− a|+ |b(−n, 0)− b|
)
<∞. (1.8)
Figure 1 demonstrates the behavior of the Toda shock wave corresponding to the initial data
a(n, 0) = 1
2 , n ∈ Z; b(n, 0) = −4, n < 0; b(0, 0) = −1.7, b(n, 0) = 0, n > 0, at a large but
fixed time t = 200. Such initial data have one eigenvalue in the gap and the background spectra
are of equal length. Hence the asymptotic of the shock wave in the VDO region is periodic
with period 2 and exhibits one phase shift. In the left and right modulation regions (MR) the
asymptotic is a modulated single-phase quasi-periodic Toda solution as discussed in [11].
����� ��
-��� -��� � ��� ���
���
���
���
���
�(�� ���)
0.5
1.0
1.5
2.0
a(n, 200)
−400 −200 0 200 400
��� ����
-��� -��� � ��� ���
-���
-���
-���
-���
-���
-���
-���
-���
���
�(�� ���)
−4.0
−3.5
−3.0
−2.5
−2.0
−1.5
−1.0
−0.5
0.0
b(n, 200)
−400 −200 0 200 400
Figure 1. Numerically computed Toda shock wave with one eigenvalue.
The initial data (1.8) can have a finite discrete spectrum. We enumerate the eigenvalues λj
in the gap (b+2a,−1) increasingly starting from the leftmost; let ℵ be the number of eigenvalues
in the gap. Given an arbitrary2 small ε > 0, the VDO region consists of ℵ+ 1 disjoint regions{
(n, t) :
n
t
∈ Ijε = [ξj + ε, ξj−1 − ε]
}
as depicted in Figure 2, where ξj are the points ξ at which the level line Re g(λ, ξ) = 0 of
the g-function (cf. Section 3) crosses R at λj . We denote by ξ0 and ξℵ+1 the points where
Re g(b + 2a, ξ0) = 0 and Re g(−1, ξℵ+1) = 0, and the rays n
t = ξℵ+1 and n
t = ξ0 determine
2The maximal value of ε which is admissible for our purpose is specified in Section 4.
4 I. Egorova and J. Michor
n
t
n
t ∈ I
j
ε
n
t ∈ I
1
ε
n
t = ξ0
n
t = ξ0 − ε
n
t = ξ1 + ε
n
t = ξj−1
n
t = ξj
n
t = ξℵ+1
Figure 2. The VDO region.
the outer boundaries of the VDO region. With each interval Ijε we associate a shift phase ∆j
(not depending on ε) expressed in terms of the initial scattering data for the solution of (1.6)
and (1.8) (see (4.16) below). For each ∆j one finds via Jacobi’s inversion problem (3.12) the
initial Dirichlet divisor and the unique finite gap solution
{
â(n, t,∆j), b̂(n, t,∆j)
}
from the
isospectral set associated with the two-band spectrum S := [b− 2a, b+ 2a] ∪ [−1, 1]. Our main
result is
Theorem 1.1. Let {a(n, t), b(n, t)} be the solution of the initial value problem (1.6)–(1.7), (1.8)
and let n → ∞, t → ∞ with n
t ∈ I
j
ε , where ε > 0 is an arbitrary, sufficiently small number.
Let
{
â(n, t,∆j), b̂(n, t,∆j)
}
be the finite gap solution associated with the spectrum S and with
the phase ∆j given by (3.16), (4.16), (4.6), (4.12), (4.11), (2.23). Then there exists C(ε) > 0
such that
b(n, t) = b̂(n, t,∆j) +O
(
e−C(ε)t
)
, a(n, t) = â(n, t,∆j) +O
(
e−C(ε)t
)
. (1.9)
The shift of the phases at the point λj =
zj+z
−1
j
2 ∈ (b+2a,−1) of the discrete spectrum is given by
∆j −∆j+1 = 2
∫ q
q1
log
∣∣∣ zjs−1
s−zj
∣∣∣ ∣∣P̃−1(s)
∣∣ds∫ q1
−1 P̃−1(s) ds
, P̃(s) =
√
(s− q)(s− q1)
(
s− q−1
1
)(
s− q−1
)
,
where q and q1 are defined by (2.6).
Remark 1.2. (i) For initial data (1.8) the scattering data consist of the modulo of the right
transmission coefficient |T (λ)| given on [b − 2a, b + 2a], the right reflection coefficient R(λ)
on [−1, 1] and the discrete spectrum on R \ ([b− 2a, b+ 2a] ∪ [−1, 1]). As expected, we see that
R(λ) and the discrete spectrum to the right of λj do not influence the asymptotic in the sector
n
t ∈ I
j
ε .
(ii) The error terms in (1.9) are of order O
(
e−C(ε)t
)
and thus significantly better than the
estimate O
(
t−1
)
one would expect by analogy with the error estimates in the modulation re-
gions [11]. The error terms in (1.9) were obtained by a careful analysis of the relations between
the analytic continuation of the scattering functions. These relations allowed us to prove that
there are no parametrix points [7] in the RHP for the VDO region.
(iii) We use vector RHP statements instead of matrix statements (as do [1, 10, 14, 15, 22] in
the case of the KdV equation with steplike initial data), because the matrix statements for the
shock wave are ill-posed for certain arbitrary large values of n and t in the class of invertible
How Discrete Spectrum and Resonances Influence the Asymptotics 5
matrices with L2-integrable singularities on the jump contour, for both the initial and model
RHPs. This fact for Toda shock can be established similarly as for the KdV case [13]. One would
have to admit then additional poles for solutions outside the discrete spectrum in the matrix
statements. This makes proving uniqueness of the solutions far more difficult. The statements
of the RHPs in vector form together with additional symmetries to be posed on contours, jump
matrices and on the solutions itself imply uniqueness almost straightforward. However, for the
final small-norm arguments we need to construct an invertible matrix model RHP solution.
It has poles and it might not be unique, but the corresponding error vector function has no
poles. Such a solution is given in Lemma 5.4.
(iv) Unlike to KdV, for the Toda equation with non-overlapping background spectra, the
statements of the RHPs associated with left and right initial data look identical. The proper
choice of the initial statement for the RHP can essentially simplify the further analysis in a given
region of space-time variables Z×R+ (cf. [12]). For the VDO region both choices are appropriate.
2 Notations and statement of the initial holomorphic RHP
To maintain generality of the presentation while keeping notations short, we formulate all pre-
liminary facts on the inverse scattering transform for the steplike initial profile in terms of the
spectral variables z± associated with the initial data (1.2), (1.3). First of all, let us list some
well known properties of the scattering data for the steplike Jacobi operator H̃(t) involved in the
Lax representation d
dtH̃(t) =
[
H̃(t), Ã(t)
]
for the initial value problem (1.1), (1.2). This problem
has a unique solution (cf. [23]). Assume that the coefficients of the initial Jacobi operator H̃(0)
tend to the limiting (or background) constants a±, b± with a first summable moment of pertur-
bation, that is, n(ã(n, 0)−a±) ∈ `1(Z±) and n(b̃(n, 0)− b±) ∈ `1(Z±). Then the unique solution{
ã(t), b̃(t)
}
of (1.1) satisfies
n
(
ã(n, t)− a±
)
∈ `1(Z±), n
(
b̃(n, t)− b±
)
∈ `1(Z±). (2.1)
With this condition fulfilled, introduce some notations and notions.
� The background Jacobi operators
H±y(n) := a±y(n− 1) + b±y(n) + a±y(n+ 1), n ∈ Z,
have spectra σ± = [b± − 2a±, b± + 2a±] which do not overlap, and by (1.3) satisfy
supσ− < inf σ+.
� The Joukovski maps z± = z±(λ) of the spectral parameter λ are given by
λ = b± + a±
(
z± + z−1
±
)
, z± : clos(C \ σ±) 7→ |z±| ≤ 1.
The map z+ 7→ z− is one-to-one between the domains D+ and D−, where
D± = {z± : |z±| < 1, z± /∈ z±(σ∓)}.
The functions
{
z±n±
}
n∈Z are called the free exponents. They solve the background spectral
equations H±y(n) = λy(n).
� The operator H̃(t) has an absolutely continuous spectrum on the set σ+ ∪ σ− and a finite
discrete spectrum σd, which we divide into three parts,
σleft
d = {λj ∈ σd : λj < b− − 2a−}, σright
d = {λj ∈ σd : λj > b+ + 2a+},
σgap
d = {λj ∈ σd : b− + 2a− < λj < b+ − 2a+}.
The points z±j = z±(λj) ∈ D± ∩ (−1, 1), λj ∈ σd, are also called points of the discrete
spectrum.
6 I. Egorova and J. Michor
� The Jost solutions of the spectral equation
ã(n− 1, t)ψ±(λ, n− 1, t) +
(
b̃(n, t)− λ
)
ψ±(λ, n, t) + ã(n, t)ψ±(λ, n+ 1, t) = 0
are normalised as
lim
n→±∞
(z±)∓nψ±(λ, n, t) = 1, λ ∈ clos(C \ σ±).
We can consider them as functions of z± in the closures of D±. Their Wronskian
W̃ (λ) := a(n− 1, 0)
(
ψ−(λ, n− 1, 0)ψ+(λ, n, 0)− ψ+(λ, n− 1, 0)ψ−(λ, n, 0)
)
(2.2)
is an important spectral characteristic of the steplike scattering problem. It can be treated
as an analytic function of z± in closD±. The points z±j are its simple zeros, and ψ±(λj , n, t)
are the (dependent) eigenfunctions.
� The normalising constants are introduced by(∑
n∈Z
ψ2
±(λj , n, t)
)−2
= γ±j (t) = γ±j (0)ez
±
j −(z±j )−1
.
� The scattering relations
T±(λ, t)ψ∓(λ, n, t) = R±(λ, t)ψ±(λ, n, t) + ψ±(λ, n, t) (2.3)
hold on the sets |z±| = 1.
� The time evolution of the scattering data is given by
R±(λ, t) = R±(λ, 0)e±z±∓z
−1
± for |z±| = 1,
|T±(λ, t)|2 = |T±(λ, 0)|2e±z∓∓z
−1
∓ for |z∓| = 1. (2.4)
Relations (2.3) and (2.4) hold under condition (2.1) which guarantees existence and good ana-
lytical properties of the Jost solutions. The function |T±(λ, 0)|2 cannot be continued analytically
outside the domain |z∓| = 1. However, if the initial data tend to the limiting constants expo-
nentially fast with a rate ρ > 0 (cf. (1.8)), then the right hand side of the ±-scattering relation
continues analytically in the domain 1 − ρ < |z±| ≤ 1, and the respective equality (2.3) is pre-
served. In particular, the reflection coefficient R±(λ, 0) continues in the domain 1−ρ < |z±| ≤ 1,
and the function χ(λ) defined in (2.23) below can be continued analytically in both domains.
The vector RHP connected with the scattering problem for H̃(t) can be stated in two ways,
based either on the right or left scattering data. The correct choice of the scattering data which
significantly simplifies the further analysis depends on the region of the (n, t) half-plane for
which the asymptotic of (1.1)–(1.3) should be derived. In our situation, the VDO region on
the Z×R+ half plane could be analysed via left or right RHP and both cases are equivalent in
structure and complexity of steps. To state a proper vector RHP we proceed as follows.
Let M be the two-sheeted Riemann surface associated with the function
w(λ) =
((
(λ− b−)2 − 4(a−)2
)(
(λ− b+)2 − 4(a+)2
))1/2
,
with glued cuts along σ+ and σ−. Denote a point on M by p = (λ,±). On the upper sheet of M
introduce two 1 × 2 vector-functions M±(p, n, t) := M±(p) (here variables n and t are treated
as parameters) by
M±(p) =
(
T±(λ, t)ψ∓(λ, n, t)(z±(λ))n, ψ±(λ, n, t))(z±(λ))−n
)
, p = (λ,+).
How Discrete Spectrum and Resonances Influence the Asymptotics 7
The first component of each function is a meromorphic function of p on the upper sheet on M
with simple poles at points of σd and known residues. At infinity M±(p) have finite values and
the product of components is equal to 1 [12]. Let us extend each function M± to the lower
sheet by
M±(p∗) = M±(p)σ1, (2.5)
where σ1 = ( 0 1
1 0 ) is the first Pauli matrix and p∗ = (λ,−) is the involution point for p = (λ,+).
With this extension, both functions have jumps along the boundaries of the sheets on M, and
these jumps can be easily evaluated. The jump problems together with normalisation conditions
M±1 (∞+)M±1 (∞−) = 1, residue conditions at points of σd and σ∗d, and symmetry condition (2.5),
form the content of the left and right RHPs associated with (1.1)–(1.3).
In this paper, we use the traditional RHP statement based on the right scattering data
in terms of the variable z+. As discussed in the introduction, we restrict ourselves to the case
a+ =
1
2
, b+ = 0, a− = a > 0, b− = b, b+ 2a < −1,
and denote by S = [b − 2a, b + 2a] ∪ [−1, 1] the continuous spectrum of the Jacobi operator
involved in (1.6). To ease notations, we omit from here on the subscript “+” in the notations
and set
z(λ) := z+(λ), λ =
z + z−1
2
, |z| ≤ 1,
q = z(b− 2a), q1 = z(b+ 2a), zj := z+(λj), γj := γ+
j (0). (2.6)
Remark 2.1. We use the formal notation zj ∈ σd, σgap
d if λj =
zj+z
−1
j
2 ∈ σd, σgap
d , respectively.
Let us enumerate the points zj starting from σgap
d , that is,
−1 < zℵ < · · · < z1 < q1, ℵ = Cardσgap
d .
All remaining points of the discrete spectrum will lie outside of σgap
d .
Further notations are
R(z) := R+(λ, 0) for |z| = 1,
ψ(z, n, t) = ψ+(λ, n, t) for |z| ≤ 1,
T (z, t) := T+(λ, t), ψleft(z, n, t) = ψ−(λ(z), n, t) for z ∈ D, (2.7)
where
D = {z : |z| < 1, z /∈ [q1, q]}. (2.8)
The domain D is in one-to-one correspondence with the upper sheet of M (we treat sheets as
open sets) with
z = λ−
√
λ2 − 1↔ p = (λ,+), z ∈ D.
The domain
D∗ =
{
z : z−1 ∈ D
}
(2.9)
corresponds to the lower sheet by
z−1 = λ+
√
λ2 − 1↔ p∗ = (λ,−).
8 I. Egorova and J. Michor
Therefore, the meromorphic RHP for M+(p) on M can be reformulated as an equivalent mero-
morphic RHP for m(z) = M+(p(z)) on the z-plane, with jumps along the unit circle T =
{z : |z| = 1} and intervals [q1, q] and
[
q−1, q−1
1
]
. In this paper, we propose a slightly different
(holomorphic) statement of the initial RHP, which is equivalent to the RHP for M+(p) on M,
and therefore has a unique solution (cf. [12]). This statement is specific for the domain VDO,
where we derive the asymptotics, and allows us to skip several of the standard transformations,
such as the reformulation of the meromorphic problem as a holomorphic problem and one of two
steps corresponding to opening of lenses.
Let us choose a large natural number N � 1 and set
δ ≤ N−1 min
zi 6=zj∈σd
{
|zi − q|, |zi − zj |, |zi − q1|, ||zi| − 1|
}
. (2.10)
We can always assume that δ < ρ, where ρ is the decay rate from (1.8). Then the right Jost
solutions ψ(z, n, t) and ψ(z−1, n, t) are holomorphic functions in an δ-vicinity of the unit circle T,
and the standard scattering relation
T (z, t)ψleft(z, n, t) = ψ
(
z−1, n, t
)
+R(z, t)ψ(z, n, t)
is continued analytically in the open ring
Ω̃δ = {z : 1− δ < |z| < 1}.
With our choice of δ, there are no points of the discrete spectrum in Ω̃δ, moreover,
inf
zj∈σd
dist(zj , Cδ) > (N − 1)δ,
where we denoted
Cδ = {z : |z| = 1− δ}. (2.11)
In particular, the continuation of the initial reflection coefficient R(z) is an analytic function
in Ω̃δ. Set
Dδ,j = {z : |z − zj | < δ}, Tδ,j = {z : |z − zj | = δ}, D∗δ,j =
{
z : z−1 ∈ Dδ,j
}
,
Dδ = D \
(
Ω̃δ ∪
⋃
σd
Dδ,j
)
, D∗δ =
{
z : z−1 ∈ Dδ
}
. (2.12)
In D \ Σδ, where
Σδ = I ∪ Cδ ∪
⋃
σd
Tδ,j , (2.13)
with I := [q1, q], introduce the vector-function m(z) = (m1(z, n, t),m2(z, n, t)) by3
m(z) =
(
T (z, t)ψleft(z, n, t)z
n, ψ(z, n, t)z−n
)
, z ∈ Dδ,(
ψ(z−1, n, t)zn, ψ(z, n, t)z−n
)
, z ∈ Ω̃δ,(
T (z, t)ψleft(z, n, t)z
n, ψ(z, n, t)z−n
)
Aj(z), z ∈ Dδ,j .
(2.14)
Here
Aj(z) =
(
1 0
(z − zj)−1γjz
2n+1
j et(zj−z
−1
j ) 1
)
. (2.15)
3This is a function of z, and n and t are treated as large parameters.
How Discrete Spectrum and Resonances Influence the Asymptotics 9
Lemma 2.2 ([12]). We have
m1(0, n, t) =
∞∏
k=n
2a(k, t), lim
z→0
1
2z
(m1(z, n, t)m2(z, n, t)− 1) = b(n, t). (2.16)
Extend m(z) to D∗ \ Σ∗δ with Σ∗δ =
{
z : z−1 ∈ Σδ
}
by
m(z−1) = m(z)σ1. (2.17)
Formula (2.17) implies that the vector function (2.14), considered as a piecewise-analytic function
in C, has jumps along the circle Cδ, along the interval I and the small circles Tδ,j , as well as
along their images C∗δ , I∗ and T∗δ,j under the map z → z−1. However, m(z) does not have a jump
along the unit circle |z| = 1, i.e., it is holomorphic in the ring 1− δ < |z| < (1− δ)−1. The fact
that m(z) does not have any singularities at zj ∈ σd is established in [6] and [19].
The symmetry condition (2.17) plays a crucial role in establishing uniqueness of the solution
for RHPs, and we cannot violate it. For this reason, the initial RHP and all its further transfor-
mations (deformations and conjugations) should satisfy the following symmetry constraints (i)
and (ii). Let Σ be the jump contour of a generic RHP.
(i) The jump contour Σ should be symmetric with respect to the map z 7→ z−1, i.e., with every
point z it also contains z−1.
(ii) Symmetric parts of Σ are oriented in such a way that the jump matrix ṽ(z) of the problem
m̃+(z) = m̃−(z)ṽ(z) and the solution itself satisfy the symmetries
ṽ(z) = σ1ṽ
(
z−1
)
σ1, z ∈ Σ; m̃(z) = m̃
(
z−1
)
σ1, z ∈ C \ Σ.
Constraint (ii) implies that the orientation of symmetric parts of Σ is as follows: if a point z
moves along a part of the contour K ⊂ Σ ∩ {z : |z| < 1} in the positive direction, then the
point z−1 moves simultaneously in the positive direction of the symmetric part K∗, where K ∩
K∗ = ∅. Except for the lense mechanisms where triangle matrices are used, all conjugations of
the solution vector consist of multiplication by diagonal matrices of the form [d(z)]−σ3 , that is,
in transformations m̃(z) 7→ m̃(z)[d(z)]−σ3 , where d : C \Σ→ C is a sectionally analytic function
and σ3 = ( 1 0
0 −1 ). On all such conjugations we pose the symmetry constraint (iii):
(iii) The contour Σ of a non-analyticity for d(z) should be symmetric with respect to z 7→ z−1.
Moreover, the function d(z) should satisfy either the property
d
(
z−1
)
= d(z)−1, z ∈ C \ Σ,
or the property
d
(
z−1
)
= d(z) 6= 0, z ∈ C \ Σ, d(0) = 1.
Recall that m(z) in (2.14) has bounded positive limits of both components at 0 and∞, moreover,
by Lemma 2.2,
m1(0)m2(0) = 1, m1(0) > 0. (2.18)
This is a normalization condition. The properties of the conjugation matrices [d(z)]−σ3 listed
above allow to preserve the normalisation condition for all transformations.
The phase function of our problem is given by
Φ(z) = Φ
(
z,
n
t
)
=
1
2
(
z − z−1
)
+
n
t
log z, z ∈ clos(C \ R−).
10 I. Egorova and J. Michor
Set ξ = n
t and consider the cross points of the 0-level lines for the function Re Φ(z, ξ), that is,
the lines described by
1
2
(
z − z−1
)
+ ξ log |z| = 0,
for different values of ξ ∈ R. One of the level lines for all ξ ∈ R is evidently the unit circle |z| = 1.
If ξ ≥ 1 the other two lines are located in the domains |z| < 1 and |z| > 1 and are symmetric
with respect to the map z 7→ z−1. An elementary analysis shows that the point −1 < z0(ξ) < 0,
where the respective level line crosses the real axis moves monotonously from −1 to 0 when ξ
runs the interval [1,+∞). Thus, the points z0(ξ) and z−1
0 (ξ) meet at point −1 for ξ = 1. The
same analysis shows that for ξ ∈ [−∞, 0) the cross point z0(ξ) moves monotonously along the
interval [0, 1] and z0(1) = z−1
0 (1) = 1. When ξ ∈ [−1, 1], the points z0(ξ) and z−1
0 (ξ) = z0(ξ) lie
on the unit circle.
To state the RHP for which m(z) in (2.14) is the unique solution,4 we introduce orientations
on the jump contour Σδ ∪Σ∗δ according to the symmetry requirements above. The contour Cδ is
oriented counterclockwise, C∗δ is oriented clockwise. On the two symmetric parts
I := [q, q1], I∗ :=
[
q−1, q−1
1
]
, (2.19)
the orientation5 is taken from right to left on I and from left to right on I∗. Moreover, all Tδ,j
and T∗δ,j are supposed to be oriented counterclockwise. Then m+(z) (resp. m−(z)) will denote
the limit from the positive (resp. negative) side of the contour. We assume that these limits
exist and m(z) extends to a continuous function on the sides of Σδ ∪ Σ∗δ except possibly at the
end points of I and I∗,
J :=
{
q, q1, q
−1, q−1
1
}
, (2.20)
where the square root (not L2-integrable!) singularities are admissible.
Recall that m1(z) = m2
(
z−1
)
= O(z − q̃)−1/2 as z → q̃ ∈ {q, q1} iff W (q̃) = 0, where
W (z) = W̃ (λ(z)) (cf. (2.2)) is the Wronskian of the Jost solutions, z ∈ closD. If the Wronskian
vanishes at q̃, we call q̃ a resonant point. The general situation is non-resonant, that is, W (q̃) 6= 0.
Note that as a function of z, the Wronskian takes complex conjugated values on the sides of the
contours (2.19).
T∗δ,j
σ1v
(
z−1
)
σ1
Tδ,j
Aj(z)
Cδ
C∗δ
I∗
σ1v
(
z−1
)
σ1
I
(
1 0
χ(z)e2tΦ(z) 1
)
(
1 0
R(z)e2tΦ(z) 1
) σ1v
(
z−1
)
σ1
Figure 3. Jump matrix v(z) in Theorem 2.3.
4We formulate a RHP which is equivalent to the initial RHPs considered in [11] or [12] in the domain under
consideration. Uniqueness is proven in [12].
5In what follows, for a < b the notation [b, a] means that the contour is oriented from b to a.
How Discrete Spectrum and Resonances Influence the Asymptotics 11
Theorem 2.3 ([11, 12]). Let δ > 0 be given as in (2.10). For all (n, t) ∈ Z × R+ and z ∈ C,
the function (2.14), (2.17) is the unique solution of the following RH problem: to find a vector-
function m(z) = (m1(z),m2(z)) holomorphic in C \ (Σδ ∪ Σ∗δ) which
� is continuous up to the boundary, except of possibly points in J (2.20),
� satisfies conditions (2.17), (2.18) and the jump condition
m+(z) = m−(z)v(z), z ∈ Σδ ∪ Σ∗δ , (2.21)
v(z) =
(
1 0
χ(z)e2tΦ(z) 1
)
, z ∈ I,(
1 0
R(z)e2tΦ(z) 1
)
, z ∈ Cδ,
Aj(z), z ∈ Tδ,j , zj ∈ σd,
σ1v(z−1)σ1, z ∈ Σ∗δ ,
(2.22)
where Φ(z) = Φ(z, nt ) is the phase function, matrices Aj(z) are defined by (2.15) and R(z)
is the holomorphic continuation of the initial right reflection coefficient (2.7). The func-
tion χ(z) is defined by
χ(z) :=
(
z − z−1
)[
ζ−1 − ζ
]
(z − i0)
|W (z)|2
, z ∈ I, (2.23)
with ζ(z), z ∈ D, connected with z by the Joukovski map λ = b + a(ζ + ζ−1). Here
W (z) = W̃ (λ(z)) is the Wronskian of the Jost solutions defined in (2.2),
� in vicinities of the points in J , m(z) has the following behavior:
– if q̃ ∈ {q, q1} is nonresonant, then m(z) = (O(1), O(1)) as z → q̃, z → q̃−1,
– if q̃ ∈ {q, q1} is resonant, i.e., if χ(z) = O
(
1√
z−q̃
)
as z → q̃, then
m(z) =
(
O
(
1√
z − q̃
)
, O(1)
)
, z → q̃,
m(z) =
(
O(1), O
(
1√
z − q̃−1
))
, z → q̃−1.
Remark 2.4. According to (1.8), W (z) admits an analytic continuation in a small vicinity of
the interval I (see [11, equation (2.11)]). Respectively, in this vicinity there exists an analytic
continuation X(z) of χ(z) such that X±(z) = ±i|χ(z)| for z ∈ I. The function X(z) does not
have other jumps in this vicinity.
Note that in the VDO region, the off-diagonal matrix elements of the jump matrix v(z) grow
exponentially with respect to t for z ∈ I∪I∗. The same is true for v(z) on those contours Tδ,j∪T∗δ,j
which correspond to the origins 0 > zj > z0(ξ). The remaining parts of the jump matrices are
asymptotically close to the identity matrix as t→∞. In the next sections we perform a series of
conjugation/deformation steps which transform the initial RHP of Theorem 2.3 to the equivalent
problem with a jump matrix which is asymptotically close as t → ∞ to a piecewise constant
matrix with respect to z. This limiting matrix also depends on ξ as a piecewise constant matrix,
and the respective (so called model) RHP has a unique solution which can be found explicitly
in terms of the Riemann theta-function. Let us emphasize that for the VDO region we propose
transformations which lead to the absence of any additional parametrix problems. The first
12 I. Egorova and J. Michor
transformation of the initial problem solution is associated with the so called g-function method
first introduced for the KdV equation in [8]. In our case, this g-function is a normalized Abel
integral associated with the two-sheeted Riemann surface glued via the cuts along the continuous
spectrum S = [b−2a, b+2a]∪ [−1, 1]. In fact, the g-function is a linear combination (dependent
on ξ) of the normalized Abel differentials of the second and third type which are involved
in the exponential part of the Baker–Akhiezer function corresponding to the finite gap solution
of the Toda lattice associated with the two-band spectrum S. It is crucial for our endeavor to
understand in detail the properties of the g-function as a function of the spectral parameter λ.
The next section is devoted to this subject.
3 g-function as an Abel integral and its connection
with the Baker–Akhiezer function
Let M be the two-sheeted Riemann surface associated with S, i.e., with the function
R1/2(λ) = −
√(
λ2 − 1
)(
(λ− b)2 − 4a2
)
,
with sheets ΠU and ΠL glued along the cuts over the intervals [b − 2a, b + 2a] and [−1, 1].
The indices U and L label the upper and lower sheets of the surface. We denote by p = (λ,±)
the points of M, with (∞,±) :=∞±; and p∗ = (λ,∓) for p = (λ,±) denotes the sheet exchange
map. Choose a canonical basis of a and b cycles on M as follows: the cycle b surrounds the
interval [b− 2a, b+ 2a] counterclockwise on ΠU and the cycle a passes from b+ 2a to −1 on the
upper sheet and back on the lower sheet. The part of this cycle on ΠU we denote by JU and
treat it as a contour on M. Its projection lies on the interval [b+ 2a,−1] which we call the gap.
The lower part of a has the same projection on the gap, and is considered as the contour JL
which passes from −1 to b+ 2a on ΠL.
Let Ω0 be the Abel differential of the second kind on M with second order poles at ∞+
and∞− and let ω∞+,∞− be the Abel differential of the third kind with logarithmic poles at∞+
and ∞−, both normalized as∫
a
Ω0 =
∫
a
ω∞+,∞− = 0.
As it is known,
Ω0 =
(λ− ν1)(λ− ν2)
R1/2(λ)
dλ, ω∞+,∞− =
λ− ν3
R1/2(λ)
dλ, (3.1)
where νi ∈ R for i = 1, 2, 3. Moreover, ν3 ∈ (b+ 2a,−1), and at least one of the points ν1 or ν2
also lies in the gap (b+ 2a,−1). Consider the Abel integral given by
g(p, ξ) :=
∫ p
1
Ω0 + ξ
∫ p
1
ω∞+,∞− ,
where ξ ∈ R is a parameter. On ΠU we denote it by g(λ, ξ), that is,
g(λ, ξ) =
∫ λ
1
(λ− ν1)(λ− ν2)− ξ(λ− ν3)
R1/2(λ)
dλ =
∫ λ
1
(λ− µ1(ξ))(λ− µ2(ξ))
R1/2(λ)
dλ. (3.2)
Since ∫ −1
b+2a
(λ− µ1(ξ))(λ− µ2(ξ))
R1/2(λ)
dλ = 0, (3.3)
How Discrete Spectrum and Resonances Influence the Asymptotics 13
then µi(ξ) ∈ R, these points do not coincide, and at least one of them belongs to the gap.
By definition of Ω0 we have
(λ− ν1)(λ− ν2)
R1/2(λ)
= −1 +O
(
λ−2
)
,
that is,
(λ− µ1(ξ))(λ− µ2(ξ))
R1/2(λ)
+ 1 =
−µ1(ξ)− µ2(ξ) + b
λ
+O
(
λ−2
)
=
ξ
λ
+O
(
λ−2
)
,
which implies
µ1(ξ) = b− ξ − µ2(ξ). (3.4)
By (3.3),∫ −1
b+2a
(λ− µi(ξ))(λ− b+ ξ + µi(ξ)))
R1/2(λ)
dλ = 0,
that is, µi(ξ) are the zeros of the quadratic equation
µ2
∫
a
dλ
R1/2(λ)
+ µ(ξ − b)
∫
a
dλ
R1/2(λ)
−
∫
a
λ2 + λ(ξ − b)
R1/2(λ)
dλ = 0.
With the notations
Γ1 =
∫
a
λ2dλ
R1/2(λ)∫
a
dλ
R1/2(λ)
, Γ2 =
∫
a
λdλ
R1/2(λ)∫
a
dλ
R1/2(λ)
, (3.5)
we infer
µ1,2(ξ) =
1
2
(
b− ξ ±
√
(b− ξ)2 + 4(Γ1 + (ξ − b)Γ2)
)
. (3.6)
Lemma 3.1. The functions µi(ξ), i = 1, 2, are monotonically decreasing with respect to ξ ∈ R.
For ξ ∈ (ξℵ+1, ξ0),6 where
ξℵ+1 = b+
1− Γ1
1 + Γ2
, ξ0 = b+
Γ1 − (b+ 2a)2
b+ 2a− Γ2
, (3.7)
we have µi(ξ) ∈ (b+ 2a,−1), i = 1, 2.
Proof. Differentiating (3.6) with respect to ξ implies
2
d
dξ
µi(ξ) = −1± ξ − b+ 2Γ2√
(b− ξ)2 + 4(Γ1 + (ξ − b)Γ2)
.
We observe that
Γ1 > 1, Γ2 < −1, |Γ1| > |Γ2|.
Inequality
|ξ − b+ 2Γ2| <
√
(b− ξ)2 + 4(Γ1 + (ξ − b)Γ2)
6In [12] and [11], these values were denoted by ξ′cr,1 and ξ′cr.
14 I. Egorova and J. Michor
holds if Γ2
2 < Γ1. The last one follows from the Cauchy inequality∣∣∣∣ ∫
a
λ dλ
R1/2(λ)
∣∣∣∣ <
√∫
a
λ2dλ
R1/2(λ)
√∫
a
dλ
R1/2(λ)
.
Thus, µi(ξ) are monotonically decreasing with respect to ξ. Assume that µ1(ξ) < µ2(ξ). A trivial
analysis shows that the value ξ0 corresponds to the location µ1(ξ0) = b+ 2a, that is,
2(b+ 2a) = b− ξ0 −
√
(b− ξ0)2 + 4(Γ1 + (ξ0 − b)Γ2).
This implies the second equation in (3.7). The location µ2(ξℵ+1) = −1 provides the first formula
in (3.7). From (3.4) it follows that
ξ0 = −2a− µ1(ξ0), ξℵ+1 = b+ 1− µ2(ξℵ+1),
that is,
ξ0 − ξℵ+1 = |b+ 2a| − 1 + µ2(ξℵ+1)− µ1(ξ0) > 0,
since µ1(ξ0), µ2(ξℵ+1) ∈ (b+ 2a,−1). �
Let ε > 0 be an arbitrary small number and let ξ0 and ξℵ+1 be defined by (3.7) and (3.5).
For any ξ ∈ [ξℵ+1 + ε, ξ0 − ε], both points µ1(ξ) and µ2(ξ) are inner points of the gap. The set
of level lines Re g = 0 consists of the two intervals [b − 2a, b + 2a] and [−1, 1] and an infinite
contour which intersects the real axis at µ0(ξ) such that
µ1(ξ) < µ0(ξ) < µ2(ξ). (3.8)
Lemma 3.2. The real-valued function µ0(ξ) implicitly given by Re g(µ0(ξ), ξ) = 0 is monotonic
with d
dξµ0(ξ) < 0 for ξ ∈ (ξℵ+1, ξ0). Moreover,
lim
ξ→ξ0
µ0(ξ) = lim
ξ→ξ0
µ1(ξ) = b+ 2a, lim
ξ→ξℵ+1
µ0(ξ) = lim
ξ→ξℵ+1
µ2(ξ) = −1. (3.9)
Proof. By (3.6) we have µ1(ξ)µ2(ξ) = (b − ξ)Γ2 − Γ1. This implies with (3.4) that µ0(ξ) is
given implicitly by∫ −1
µ0(ξ)
(λ− µ1(ξ))(λ− µ2(ξ))
R1/2(λ)
dλ =
∫ −1
µ0(ξ)
λ2 + (ξ − b)λ+ ((b− ξ)Γ2 − Γ1)
R1/2(λ)
dλ = 0.
Differentiating with respect to ξ implies
d
dξ
µ0 =
∫ −1
µ0
λ− Γ2
R1/2(λ)
dλ
R1/2(µ0)
(µ0 − µ1)(µ0 − µ2)
.
Since the second multiplier is negative, it is sufficient to prove that the integral is positive. But∫ −1
µ0
λ− Γ2
R1/2(λ)
dλ =
∫ −1
µ0
λ
R1/2(λ)
dλ−
∫ −1
µ0
dλ
R1/2(λ)
∫ −1
b+2a
λ
R1/2(λ)
dλ
(∫ −1
b+2a
dλ
R1/2(λ)
)−1
,
that is, we have to prove that∫ −1
µ0
λ dλ
R1/2(λ)∫ −1
µ0
dλ
R1/2(λ)
>
∫ −1
b+2a
λ dλ
R1/2(λ)∫ −1
b+2a
dλ
R1/2(λ)
.
This inequality is true by the mean value theorem. Equalities (3.9) were proven in [12]. �
How Discrete Spectrum and Resonances Influence the Asymptotics 15
Let us recall the Baker–Akhiezer function for a finite gap solution
{
â(n, t), b̂(n, t)
}
of the
Toda lattice equation associated with the spectrum S = [b− 2a, b+ 2a] ∪ [−1, 1] and the initial
Dirichlet divisor p0 = (λ(0, 0), σ(0, 0)), σ(0, 0) ∈ {+,−}. In our case the divisor consists of one
point on the Riemann surface M with projection on the closed gap of the spectrum and it will
later depend on the slow variable ξ.
Let ζ be the holomorphic Abel differential on M normalized as
∫
a ζ = 1 and let
∫
b ζ =: τ ∈
iR+ be its b-period. Introduce the Abel map A(p) :=
∫ p
b−2a ζ. It is an odd function on M,
A(p∗) = −A(p). Moreover, it has a jump along the a-cycle, which we interpret as a union
JU ∪ JL. Then A+(p)−A−(p) = −τ as p ∈ JU ∪ JL. We set A(b+ 2a) := A+(b+ 2a) = −τ/2
and A(p0) = A+(p0), where p0 ∈ JU ∪ JL is the initial Dirichlet divisor.
Let Ξ = τ
2 + 1
2 be the Riemann constant and
Λ =
∫
b
ω∞+,∞− ∈ iR, U =
∫
b
Ω0 ∈ iR, (3.10)
be the b-periods of the Abel differentials (3.1). Following [23], we introduce the notations
Z(p, n, t) := A(p)−A(p0)− n Λ
2πi
− t U
2πi
− Ξ,
Z(n, t) := Z(∞+, n, t).
Evidently, θ(Z(p, 0, 0)) = 0 iff p = p0, where
θ(v) := θ(v | τ) =
∑
m∈Z
exp
(
πim2τ + 2πimv
)
is the Jacobi theta function. Recall that the time-dependent Baker–Akhiezer function for the
finite gap Toda lattice solution with spectral data as above has the form [23]
Ψ(p, n, t) = C(n, t)
θ(Z(p, n, t))
θ(Z(p, 0, 0))
exp
(
n
∫ p
1
ω∞+,∞− + t
∫ p
1
Ω0
)
= C(n, t)
θ(Z(p, n, t))
θ(Z(p, 0, 0))
exp(tg(p, ξ)), ξ =
n
t
.
Here C(n, t) is a positive constant (with respect to p) which provides the equalities
lim
p→∞±
Ψ(p∗, n, t)Ψ(p, n, t) = 1, Ψ(p, 0, 0) = 1,
and
C(n+ 1, t)
C(n, t)
=
√
θ(Z(n− 1, t))
θ(Z(n+ 1, t))
> 0.
As is known, for each n and t fixed, the Baker–Akhiezer function is a meromorphic function of p
on M with a simple pole at p0. Respectively, the vector function (Ψ(p∗, n, t),Ψ(p, n, t)) does not
have jumps on M, and the vector function
m̂(p) :=
(
Ψ(p∗, n, t) exp
(
−tg
(
p∗,
n
t
))
,Ψ(p, n, t) exp
(
−tg
(
p,
n
t
)))
,
has an evident jump along a,
m̂+(p) = m̂−(p)e−(nΛ+tU)σ3 , p ∈ JU ∪ JL.
16 I. Egorova and J. Michor
Here we took into account that for p ∈ JU ∪ JL,[ ∫ p
1
ω∞+,∞−
]
+
−
[ ∫ p
1
ω∞+,∞−
]
−
= −Λ,
[ ∫ p
1
Ω0
]
+
−
[ ∫ p
1
Ω0
]
−
= −U.
Note that since g(p∗, ξ) = −g(p, ξ),
m̂1(∞+)m̂2(∞+) = 1, m̂(p) = m̂(p∗)σ1.
The function
f(p) := f∞
θ(Z(p, 0, 0))
θ
(
A(p)− 1
2
) , f∞ :=
√
θ
(
A(∞+)− 1
2
)
θ
(
A(∞−)− 1
2
)
θ(Z(∞+, 0, 0))θ(Z(∞−, 0, 0))
,
has a simple pole at the branch point b+ 2a and a simple zero at p0. Moreover,
f(∞+)f(∞−) = 1, f+(p∗) = f−(p∗)e−i∆, f+(p) = f−(p)ei∆, p ∈ JU ,
where
∆ = −2π
(
A(p0) +
τ
2
)
∈ R. (3.11)
Note that (3.11) can be rewritten as the Jacobi inversion problem∫ p0
b+2a
ζ = −∆
2π
(mod 1) (3.12)
and allows us to compute uniquely the divisor point p0 for any given real valued ∆. Summing
up the considerations above, we proved the following
Lemma 3.3. The vector function
m̃(p) =
(
m̃1(p, n, t), m̃2(p, n, t)
)
=
(
Ψ(p∗, n, t)f(p∗) exp
(
−tg
(
p∗,
n
t
))
,Ψ(p, n, t)f(p) exp
(
−tg
(
p,
n
t
)))
= C̃(n, t)
(
θ(Z(p∗, n, t))
θ
(
A(p∗)− 1
2
) , θ(Z(p, n, t))
θ
(
A(p)− 1
2
)), where C̃(n, t) := C(n, t)f∞,
solves the following RHP on M: to find a holomorphic vector-function m̃(p) on M \
(
JU ∪ JL
)
,
which satisfies
� the jump condition
m̃+(p) = m̃−(p)e−(nΛ+tU+i∆)σ3 , (3.13)
� the symmetry condition
m̃(p∗) = m̃(p)σ1 for p ∈M \
(
JU ∪ JL
)
, (3.14)
� the normalization condition m̃1(∞+)m̃2(∞+) = 1.
� Both components of m̃(p) have simple poles at the branch point p = b + 2a and no other
singularities.
How Discrete Spectrum and Resonances Influence the Asymptotics 17
Note that the constant C̃(n, t) in Lemma 3.3 satisfies
C̃(n+ 1, t)
C̃(n, t)
=
√
θ(Z(n− 1, t))
θ(Z(n+ 1, t))
> 0.
Together with Theorem 9.48 of [23] it implies
Corollary 3.4. For the vector function m̃(p) = m̃(p, n, t) the following holds
m̃1(∞+, n, t)
m̃1(∞+, n+ 1, t)
=
√
θ(Z(n− 1, t))θ(Z(n+ 1, t))
θ(Z(n, t))
=
â(n, t)
ã
,
where ã = CapS is the logarithmic capacity of the spectrum S.
Introduce the product
h̃(p) := m̃1(p)m̃2(p) = C̃2(n, t)
θ(Z(p∗, n, t))
θ
(
A(p∗)− 1
2
) θ(Z(p, n, t))
θ
(
A(p)− 1
2
) ,
and let h(λ) := h̃(p) for p = (λ,+). The function h̃(p) has a double pole on M at the branch
point b+ 2a, that is, h(λ) has a simple pole at b+ 2a. Moreover, θ(Z(p, n, t)) has the only zero
at p(n, t) = (λ(n, t),±) ∈ JU ∪ JL, which is the unique solution of the Jacobi inversion problem∫ p(n,t)
p0
ζ = n
Λ
2πi
+ t
U
2πi
, (3.15)
θ(Z(p∗, n, t)) has a simple zero at the involution point p∗(n, t). Thus h(λ(n, t)) = 0, and it is
a simple zero of h. We observe that from the jump and symmetry conditions it follows that h̃(p)
does not have jumps on M, moreover, h̃(p) = h̃(p∗), p ∈M. This means that h(λ) does not have
jumps along the spectrum S and on the gap [b + 2a,−1]. The normalisation condition implies
limλ→∞ h(λ) = 1. Hence h(λ) is a meromorphic function on C, i.e., h(λ) = λ−λ(n,t)
λ−b−2a .
Corollary 3.5. Let λ(n, t) ∈ [b + 2a,−1] be the projection on C of the Dirichlet eigenvalue
p(n, t) given by (3.15). Then
lim
p→∞+
p (m̃1(p)m̃2(p)− 1) = b+ 2a− λ(n, t).
We recall that the trace formula in our case looks like
b̂(n, t) =
1
2
(1 + b− 2a+ b+ 2a− 1− 2λ(n, t)) = b− λ(n, t).
Therefore,
lim
p→∞+
p (m̃1(p)m̃2(p)− 1) = b̂(n, t) + 2a.
Since the problem (3.12) has a unique solution p0 for any real ∆, we can treat ∆ as the
initial data to choose the representative
{
â(n, t), b̂(n, t)
}
for the isospectral set of finite gap
potentials with spectrum S. To emphasise this dependence we denote the representative
as
{
â(n, t,∆), b̂(n, t,∆)
}
. In turn, the solution of the RHP with jump (3.13) we denote as
m̃(p, n, t,∆). We proved the following
18 I. Egorova and J. Michor
Theorem 3.6. Let m̃(p, n, t,∆) be the unique solution of the RHP in Lemma 3.3. Then
lim
p→∞+
ã
m̃1(p, n, t,∆)
m̃1(p, n+ 1, t,∆)
= â(n, t,∆),
lim
p→∞+
p (m̃1(p, n, t,∆)m̃2(p, n, t,∆)− 1)− 2a = b̂(n, t,∆),
where
{
â(n, t,∆), b̂(n, t,∆)
}
is the finite gap solution of (1.6) with two band spectrum S =
[b− 2a, b+ 2a] ∪ [−1, 1] and initial Dirichlet divisor p0 given by the Jacobi inversion (3.12).
Remark 3.7. For convenience of the reader, we recall from [23] that the finite gap solution
{â(n, t,∆), b̂(n, t,∆)} corresponding to the initial phase ∆ and spectrum S is given by
â(n, t,∆) = ã
√
θ
( (n−1)Λ
2πi + tU
2πi −
∆
2π +A
)
θ
( (n+1)Λ
2πi + tU
2πi −
∆
2π +A
)
θ
(
nΛ
2πi + tU
2πi −
∆
2π +A
) ,
b̂(n, t,∆) = b̃+
1
Y
∂
∂w
log
θ
( (n−1)Λ
2πi + tU
2πi −
∆
2π +A+ w
)
θ
(
nΛ
2πi + tU
2πi −
∆
2π +A+ w
) , (3.16)
where
A :=
1
2
+
∫ b−2a
∞+
ζ, Y =
∫
a
dλ
R1/2(λ)
, b̃ = b− 1
Y
∫
a
λ dλ
R1/2(λ)
,
ζ is the normalized holomorphic Abel differential, ã = CapS, and U , Λ are defined by (3.10).
4 Reduction of the initial RHP to the model RHP
From now we work again in the variable z. Let us identify the upper sheet ΠU of the Riemann
surface M with the domain (2.8) and the lower sheet with (2.9). The image of JU under the
map p 7→ z we denote by J , preserving the orientation, i.e., J = [q1,−1] is oriented from right
to left. As for the image J∗ of JL, we change its orientation in accordance with our symmetry
requirements, i.e., J∗ =
[
q−1
1 ,−1
]
is oriented from left to right. The other contours used here are
already defined by (2.19), (2.11), (2.12), (2.13). In this section, we perform two transformations
(steps) which transform the solution of the initial RHP (Theorem 2.3) to the solution of the RHP
with the jump matrix which is close as t→∞ to a piecewise constant jump matrix everywhere
on the jump contour, without exceptional points (parametrices).
Step 1. Set
yk = yk(ξ) = z(µk(ξ)) ∈ J for k = 0, 1, 2,
where µk(ξ) are defined in Section 3. From (3.8) we have y2(ξ) < y0(ξ) < y1(ξ). With these
definitions at hand, the g(p)-function (3.2) is given in terms of z by
g(z, ξ) =
1
2
∫ z
1
(s− y1)
(
s− y−1
1
)
(s− y2)
(
s− y−1
2
)√
(s− q1)
(
s− q−1
1
)
(s− q)
(
s− q−1
) ds
s2
, z ∈ C \ (−∞, 1).
Thus, the level lines Re g(z, ξ) = 0 which are different from the unit circle T and intervals I
and I∗, cross the real axis at the points y0(ξ) and y−1
0 (ξ). Similar to [11, Lemma 3.1] and [12,
Lemma 5.3] we establish the following
How Discrete Spectrum and Resonances Influence the Asymptotics 19
Lemma 4.1. The function g(z) satisfies the following properties
(a) g(z) is single valued on C \
[
q−1, q
]
and g
(
z−1
)
= −g(z) for z ∈ C \
[
q−1, q
]
,
(b) Re g(z) = 0 for z ∈ I ∪ I∗ ∪ {z : |z| = 1},
(c) g(q) = g
(
q−1
)
= 0,
(d) g−(z) = −g+(z) for z ∈ I ∪ I∗,
(e) Φ(z)− g(z) = K(ξ) +O(z) as z → 0, where K(ξ) ∈ R and
d
dξ
K(ξ) = − log (2ã) . (4.1)
Here ã is the logarithmic capacity of the set S, cf. [12, Lemma 5.4],
(f) g+(z)− g−(z) = −U − ξΛ for z ∈ J , and
g+(z)− g−(z) = U + ξΛ for J∗, where U and Λ are defined by (3.10).
The signature table for Re g(z, ξ) is given in Figure 4.
T∗
δ,j Tδ,j
Re g < 0 Re g < 0Re g > 0 T
qq1 0−1q−1
1q−1 y−1
0
y0
Figure 4. Signature table of Re g(z, ξ) for ξ ∈ Iε.
Recall that we enumerated the eigenvalues of the problem (1.6), (1.8) starting from the gap
(b + 2a,−1) in ascending order, i.e., λ1 is the minimal eigenvalue and λℵ is the maximal one.
The respective zj are enumerated in descending order. Choose a small ε1 > 0 such that
y1(ξ0 − ε1) > z1 + 2δ, y2(ξℵ+1 + ε1) < zℵ − 2δ.
Smaller values of ε or δ do not affect these conditions. From Lemma 3.2 it follows that there
are unique values ξi ∈ (ξℵ + ε1, ξ0 − ε1) such that
y0(ξj) = zj ∀j = 1, . . . ,ℵ.
Evidently, ξℵ+1 < · · · < ξj+1 < ξj < · · · < ξ0. We denote
Iε := [ξℵ+1 + ε, ξ0 − ε] \
ℵ⋃
j=1
(ξj − ε, ξj + ε) =
ℵ+1⋃
j=1
Ijε , Ijε := [ξj + ε, ξj−1 − ε]. (4.2)
From the considerations above it is straightforward to get the following
Lemma 4.2. For any arbitrary small positive ε < ε1 one can choose δ > 0 such that for all
ξ ∈ Iε the following inequalities are valid
inf
zj∈σd
inf
z∈Tδ,j
|Re g(z, ξ)| − |Φ(z, ξ)− Φ(zj , ξ)| > C(ε) > 0. (4.3)
20 I. Egorova and J. Michor
Note that the infimum in (4.3) is taken along the circles around all points of the discrete
spectrum σd. The VDO region{
(n, t) ∈ Z× R+ :
n
t
∈ Iε
}
consists of ℵ+ 1 nonintersecting sectors separated by arbitrary small sectors as depicted in Fi-
gure 2. If σgap
d = ∅, then ℵ = 0 and the VDO region is the simply connected region{
(n, t) ∈ Z× R+ :
n
t
∈ [ξℵ+1 + ε, ξ0 − ε]
}
.
For each ξ ∈ Iε we divide the eigenvalues based on their relative location with respect to the
point µ0(ξ) (respectively, y0(ξ)) and introduce the Blaschke product
Π(z) = Π(z, ξ) =
∏
y0(ξ)<zk<0
|zk|
z − z−1
k
z − zk
. (4.4)
Note that
Π
(
z−1
)
= Π−1(z), Π(0) > 0. (4.5)
In fact,
Π(z, ξ) = Πj(z) =
∏
zj<zk<0
|zk|
z − z−1
k
z − zk
for ξ ∈ Ijε . (4.6)
Set also (cf. (2.12))
E(z) = E(z, ξ) =
1
z−zj
zjγje
2tΦ(zj)
−zjγje2tΦ(zj)
z−zj 1
, z ∈ Dδ,j , zj ∈ (y0, 0),
σ1E
(
z−1
)
σ1, z ∈ D∗δ,j , zj ∈ (y0, 0),
I, z ∈ C \
⋃
zj∈(y0,0) Dδ,j ∪ D∗δ,j .
(4.7)
The matrix E is not an identity matrix only in small vicinities of those zj which lie in the domain
where Re g(z, ξ) > 0.
From properties (a), (e) of Lemma 4.1, property (4.5) and (4.7) it follows that if a vector m
satisfies (2.17) and (2.18), so does the vector
m(1)(z) = m(z)E(z)
[
Π(z)et(Φ(z)−g(z))]−σ3 , z ∈ C \ (Σδ ∪ Σ∗δ ∪ J ∪ J∗), (4.8)
where σ3 is the third Pauli matrix. A straightforward computation using Theorem 2.3 and Lem-
ma 4.1 shows that if m(z) satisfies (2.21), (2.22), then m(1)(z) given by (4.8) solves the jump
problem
m
(1)
+ (z) = m
(1)
− (z)v(1)(z), z ∈ Σδ ∪ Σ∗δ ∪ J ∪ J∗,
How Discrete Spectrum and Resonances Influence the Asymptotics 21
where
v(1)(z) =
(
et(g+(z)−g−(z)) 0
Π−2(z)χ(z) e−t(g+(z)−g−(z))
)
, z ∈ I,(
et(g+(z)−g−(z)) 0
0 e−t(g+(z)−g−(z))
)
, z ∈ J,(
1 0
Π−2(z)R(z)e2tg(z) 1
)
, z ∈ Cδ,
Aj(z), z ∈ Tδ,j , zj /∈ (y0, 0),
Bj(z), z ∈ Tδ,j , zj ∈ (y0, 0),
σ1(v(1)(z−1))σ1, z ∈ Σ∗δ ∪ J∗.
Here
Aj(z) = Aj(z, ξ) =
(
1 0
γjzj
Π2(z)(z−zj)e2tHj(z) 1
)
,
Bj(z) = Bj(z, ξ) =
(
1 0
Π2(z)(z−zj)
γjzj
e−2tHj(z) 1
)
,
and
Hj(z) = Hj(z, ξ) = Φ(zj)− Φ(z) + g(z, ξ).
With our choice of the VDO region we evidently have
Lemma 4.3. Uniformly with respect to ξ ∈ Iε
sup
zj∈σd
sup
z∈Tδ,j
(‖Aj(z)− I‖+ ‖Bj(z)− I‖) ≤ C1(ε)e−C(ε)t, (4.9)
sup
z∈Cδ∪C∗δ
∥∥v(1)(z)− I
∥∥ ≤ C2(ε)e−C(ε)t. (4.10)
Here ‖ · ‖ is a norm of 2× 2 matrices.
Remark 4.4. The function m(1)(z) inherits the singularities of m(z) described in Theorem 2.3.
Recall that these singularities essentially depend on the presence or absence of resonances
at points (2.20). In the next step we apply the lense mechanism around I and I∗, which will
at the same time weaken these singularities. We will use ` = −1, 0, 1 to indicate singularities as
follows. In C \ [q−1, q] introduce a function Q(z) such that
Q4(z) =
(z−q)(z−q1)
(zq1−1)(zq−1) , q nonresonant, q1 nonresonant (` = −1),
(zq1−1)(zq−1)
(z−q)(z−q1) , q resonant, q1 resonant (` = 1),
(zq1−1)(z−q)
(zq−1)(z−q1) , q nonresonant, q1 resonant (` = 0),
(zq−1)(z−q1)
(zq1−1)(z−q) , q resonant, q1 nonresonant (` = 0),
(4.11)
with the branch of the forth root defined by the condition Q(1) = 1. Evidently, Q(z−1) = Q−1(z)
and Q(0) > 0. It has jumps on I ∪ I∗ for ` = 0 and on I ∪ I∗ ∪ J ∪ J∗ for ` = ±1. The function
Ω(z, s) =
1
2s
s+ z
s− z
22 I. Egorova and J. Michor
can be considered as the Cauchy kernel for symmetric contours, because Ω(z, s) = 1
z−s(1 + o(1))
as z → s, and
Ω
(
z, s−1
)
d
(
s−1
)
= Ω
(
z−1, s
)
ds.
Using this kernel allows us to preserve the symmetry condition. Set
P(z) =
√
(z − q)(z − q1)
(
z − q−1
1
)(
z − q−1
)
z−2, z ∈ C \ (I ∪ I∗). (4.12)
This function satisfies the symmetries
P(z−1) = P(z) for z ∈ C \ (I ∪ I∗) and P−(z) = −P+(z) for z ∈ I ∪ I∗.
Define
S(z) =
1
2πi
∫ q−1
q
Ω(z, s)f(s) ds, f(s) :=
log(Π−2(s)Q−2(z)|χ(s)|)
P+(s) , s ∈ I,
i∆̃
P(s) , s ∈ J,
f
(
s−1
)
, s ∈
[
q−1,−1
]
,
where
∆̃ = ∆̃(ξ) = −i
∫
I
log
(
Q−2(s)Π−2(s)|χ(s)|
)
P+(s)
ds
s
(∫
J
ds
sP(s)
)−1
.
It is straightforward to verify that S(z) solves the scalar RH problem
S+(z) = S−(z) + f(z), z ∈
[
q, q−1
]
,
S(z−1) = −S(z), z ∈ C \
[
q, q−1
]
,
S(z) = O(z), z → 0.
The above considerations imply that the function
F(z) = eP(z)S(z), z ∈ C \
[
q−1, q
]
, (4.13)
is the unique solution of the following RHP with jump along I ∪ I∗ ∪ J ∪ J∗,
(i) F+(z)F−(z) = Π−2(z)|χ(z)|Q−2(z) for z ∈ I,
(ii) F+(z) = F−(z)ei∆̃ for z ∈ J ,
(iii) F(z−1) = F−1(z) for z ∈ C \
[
q−1, q
]
,
(iv) F(0) > 0, F(1) = 1.
Since χ(s)Q−2(s) 6= 0 as s ∈ I ∪ I∗ and it is a continuous function on I ∪ I∗, then F(z) also has
nonzero finite limiting values as z → q̃ ∈ J (cf. [21]). It is straightforward to obtain
Lemma 4.5. The function F (z) := F(z)Q(z), defined for z ∈ C \
[
q−1, q
]
by (4.11)–(4.13),
solves the following RHP
(i) F+(z)F−(z) = Π−2(z)|χ(z)| for z ∈ I,
(ii) F+(z) = F−(z)ei∆ for z ∈ J ,
(iii) F
(
z−1
)
= F−1(z) for z ∈ C \
[
q−1, q
]
,
(iv) F (0) > 0, F (1) = 1,
How Discrete Spectrum and Resonances Influence the Asymptotics 23
where
∆ = ∆(ξ) = −i
∫
I
log
(
Q−2(s)Π−2(s, ξ)|χ(s)|
)
P+(s)
ds
s
(∫
J
ds
sP(s)
)−1
+ `π. (4.14)
In a vicinity of q̃ ∈ {q, q1} we have F (z) = C(z− q̃)1/4(1+o(1)) if q̃ is a nonresonant point, and
F (z) = C(z − q̃)−1/4(1 + o(1)) if q̃ is a resonant point. The jumps of F along the contours I∗
and J∗ as well as its behavior at q−1, q−1
1 are uniquely defined by the symmetry (iii).
Note that the only dependence of ∆(ξ) on ξ is due to the Blaschke product (4.4) depending
on ξ. It means that ∆(ξ) has constant values on every interval Ijε . By (4.6) we obtain that
∆(ξ) = ∆j for ξ ∈ Ijε , j = 1, . . . ,ℵ+ 1, (4.15)
where
∆j = −i
∫
I
log
(
Q−2(s)Π−2
j (s)|χ(s)|
)
P+(s)
ds
s
(∫
J
ds
sP(s)
)−1
+ `π. (4.16)
Given these preparations, we can implement the next deformation step.
Step 2. Introduce two symmetric contours Lδ and L∗δ =
{
z : z−1 ∈ Lδ
}
surrounding I and I∗
counterclockwise at a small distance such that
min
zj∈σd
dist
(
zj ,Ωδ
)
� δ and dist
(
y0(ξ),Ωδ
)
� δ ∀ξ ∈ Iε, (4.17)
where Ωδ and Ω∗δ are the enclosed regions so that Lδ = ∂Ωδ and L∗δ = ∂Ω∗δ , see Figure 5.
Condition (4.17) ensures that Lδ is away from the level line Re g(., ξ) = 0 and from any point
of the discrete spectrum.
0−1q−1
1q−1 T∗δ,j
Cδ
C∗δ
I∗
Ω∗δ
IJ
Ωδ
L∗δ
LδJ∗
Figure 5. Contour deformation of Step 2.
Let X(z), z ∈ Ωδ, be the continuation of χ(z) as described in Remark 2.4. Set
GF (z) =
(
F−1(z) −Π2(z)F (z)
X(z) e−2tg(z)
0 F (z)
)
, z ∈ Ωδ.
Define m(2)(z) by
m(2)(z) =
m(1)(z)GF (z), z ∈ Ωδ,
m(2)
(
z−1
)
σ1, z ∈ Ω∗δ ,
m(1)(z)(F (z))−σ3 , z ∈ C \ (Ωδ ∪ Ω∗δ).
24 I. Egorova and J. Michor
Theorem 4.6. For every ξ ∈ Iε, the vector function m(2)(z) = m(2)(z, ξ) is the unique solution
of the following RHP: to find a holomorphic function in the domain
C \ (Σδ ∪ Σ∗δ ∪ Lδ ∪ L∗δ ∪ J ∪ J∗)
which has continuous limits on the sides of the contour Σδ ∪Σ∗δ ∪Lδ ∪L∗δ ∪J ∪J∗ except possibly
at points J (2.20) and satisfies:
� the jump condition m
(2)
+ (z, n, t) = m
(2)
− (z, n, t)v(2)(z, n, t),
v(2)(z) =
vmod(z), z ∈ I ∪ I∗ ∪ J ∪ J∗,(
1 Π2(z)F 2(z)
X(z) e−2tg(z)
0 1
)
, z ∈ Lδ,
[F (z)]−σ3v(1)(z)[F (z)]σ3 , z ∈ Σδ \ I,
σ1
(
v(2)
(
z−1
))
σ1, z ∈ Σ∗δ ∪ J∗ ∪ L∗δ ,
(4.18)
where Σδ is defined by (2.13) and
vmod(z) =
iσ1, z ∈ I,
e−(nΛ+tU+i∆)σ3 , z ∈ J
σ1v
mod
(
z−1
)
σ1, z ∈ I∗ ∪ J∗,
(4.19)
� the symmetry condition m(2)
(
z−1
)
= m(2)(z)σ1,
� the normalization condition m
(2)
1 (0) ·m(2)
2 (0) = 1, m
(2)
1 (0) > 0,
� at points of the set (2.20), m(2)(z) has at most a fourth root singularity,
m(2)(z) = O(z − κ)−1/4, as z → κ ∈ J .
Proof. The proof of this theorem is completely analogous to the proof of [11, Theorem 3.6]
except for a small contour Iδ = [q1, q1 − δ] = Ωδ ∩ [q1,−1] ⊂ J , where the jump matrix
v(2)(z) =
[
GF−(z)
]−1
v(1)(z)GF+(z) should be evaluated. From Lemmas 4.1(f), and 4.5(ii), it
follows that
v(2) =
(
F−
Π2F−
X e−2tg−
0 F−1
−
)(
et(g+−g−) 0
0 et(g−−g+)
)(
F−1
+ −Π2F+
X e−2tg+
0 F+
)
=
(
F−
F+
et(g+−g−) Π2e−t(g++g−)
X (F+F− − F+F−)
0 F+
F−
et(g−−g+)
)
=
(
e−nΛ−tU−i∆ 0
0 enΛ+tU+i∆
)
.
On the symmetric contour Iδ,∗ (oriented from left to right) we get v(2) = vmod by the sym-
metry. �
Remark 4.7. According to (4.16) and (4.19) we see that
vmod(z) = vmod
j (z, n, t) =
e−(nΛ+tU+i∆j)σ3 , z ∈ J, n
t ∈ I
j
ε ,
iσ1, z ∈ I,
σ1v
mod
(
z−1
)
σ1, z ∈ I∗ ∪ J∗.
(4.20)
How Discrete Spectrum and Resonances Influence the Asymptotics 25
Let us label the jump contour which appears in Theorem 4.6 by
Kδ = Σδ ∪ Σ∗δ ∪ Lδ ∪ L∗δ ∪ J ∪ J∗. (4.21)
We extend the matrix vmod(z) to the whole contour Kδ by defining it as the identity matrix on
the remaining part Kδ \ (I ∪ I∗ ∪ J ∪ J∗).
Lemma 4.8. Uniformly with respect to ξ ∈ Iε,
sup
z∈Kδ
∥∥v(2)(z)− vmod(z)
∥∥ ≤ C3(ε)e−C4(ε)t, Ci(ε) > 0. (4.22)
Proof. Recall that
inf
z∈Lδ
Re g(z, ξ) > C5(ε) > 0 ∀ξ ∈ Iε.
This inequality verifies (4.22) on the contour Lδ ∪ L∗δ . Since the conjugation
[F (z)]−σ3v(1)(z)[F (z)]σ3 , z ∈ Σδ \ I,
does not impair estimates (4.9), (4.10) and since v(2)(z) = vmod(z) for z ∈ I ∪ I∗ ∪ J ∪ J∗,
then (4.22) is straightforward for the remaining part of Kδ. �
5 Solution of the model problem and conclusive analysis
In Section 3, Lemma 3.3, we constructed the vector-function m̃(p), which solves the jump
problem (3.13) for ∆ given by (3.12). One can treat this result in the following way: let ∆ be
an arbitrary real value and let p0 be the unique solution of the Jacobi inversion problem (3.12).
Consider this point as the initial Dirichlet divisor and let â(n, t,∆), b̂(n, t,∆) be the finite gap
solution associated with this divisor and with the spectrum S. In particular, we can construct
m̃(p) associated with ∆ given by (4.14). Being considered on the z-plane, the vector-function
m̃(z) = m̃(p(z)) has additional jumps on I ∪ I∗ due to (3.14) and solves the jump problem
m̃+(z) = m̃−(z)
{
σ1, z ∈ I ∪ I∗,
vmod(z), z ∈ J ∪ J∗,
with vmod(z) given by (4.19) (or by (4.20)) on J ∪ J∗. Introduce the function
H(z) = 4
√
(q1 − z)
(
q−1
1 − z
)
(q − z)
(
q−1 − z
) ,
which satisfies
H
(
z−1
)
= H(z), z ∈ C \ (I ∪ I∗); H+(z) = iH−(z), z ∈ I ∪ I∗; H(0) = 1.
Thus mmod(z) := H(z)m̃(z) is the unique solution of the following
Model RH problem. Find a vector-function mmod(z) holomorphic in C \
[
q, q−1
]
, continuous
up to the boundary except of points of the set (2.20), which satisfies the jump condition
mmod
+ (z) = mmod
− (z)vmod(z), z ∈ I ∪ I∗ ∪ J ∪ J∗,
with vmod(z) given by (4.19), and the symmetry and normalization conditions
mmod
(
z−1
)
= mmod(z)σ1, mmod
1 (0) =
[
mmod
2 (0)
]−1
> 0.
26 I. Egorova and J. Michor
At points of the set (2.20) it has a fourth root singularity
mmod(z) = O(z − κ)−1/4, as z → κ ∈ J .
Uniqueness of the solution of such a problem was established in [12].
The dependence of mmod(z) on n, t and ∆
(
n
t
)
is due to the jump exp
((
−nΛ−tU+i∆
(
n
t
))
σ3
)
.
For large n and t, if n
t ∈ I
j
ε , then n+1
t ∈ I
j
ε . Recall that ∆(ξ) has constant values ∆j on Ijε
(cf. (4.16)). Therefore,
mmod(z, n, t, j) := mmod(z, n, t,∆j) and mmod(z, n+ 1, t, j) := mmod(z, n+ 1, t,∆j)
are well defined for n
t ∈ I
j
ε . Here mmod(z, n + 1, t, j) is the solution of the jump problem with
the jump exp (−((n+ 1)Λ + tU + i∆j)σ3) on J and associated jumps on J∗ ∪ I ∪ I∗. Since
H2(z) =
(
1− z
2q1
− zq1
2
+O
(
z2
))(
1 +
z
2q
+
zq
2
+O
(
z2
))
= 1− 2z(2a) +O
(
z2
)
and λ = 1
2z (1 + o(1)), then Theorem 3.6 and (4.15) imply
Lemma 5.1. For all n→∞, t→∞ and n
t ∈ I
j
ε ,
lim
z→0
mmod
1 (z, n, t, j)
mmod
1 (z, n+ 1, t, j)
= â(n, t,∆j)ã
−1,
lim
z→0
1
2z
(
mmod
1 (z, n, t, j)mmod
2 (z, n, t, j)− 1
)
= b̂(n, t,∆j). (5.1)
Here the phase ∆j is defined by (4.16), (4.6) and ã = CapS.
Our next task is to prove the following approximation
Theorem 5.2. For all n → ∞, t → ∞ such that n
t ∈ I
j
ε , the following asymptotic holds
as z → 0
m(2)(z, n, t) = mmod(z, n, t, j) +O
(
e−C(ε)t
)
(1 +O(z)), C(ε) > 0. (5.2)
Moreover,
m1(z)m2(z) = mmod
1 (z, n, t, j)mmod
2 (z, n, t, j) + βj1(ξ, t) + βj2(ξ, t)z + βj3(ξ, t)O
(
z2
)
,
z → 0, (5.3)
where βjk(ξ, t) = O
(
e−C(ε)t
)
, k = 1, 2, 3, uniformly with respect to ξ ∈ Ijε .
The proof of Theorem 5.2 essentially repeats the arguments used in [11, Section 7]. But since
this theorem provides the justification of our asymptotic analysis, we will briefly describe the
key points of the final analysis as applied to our case.
According to the standard approach (cf. [3, 16]), to perform the conclusive analysis we first
have to evaluate in C the “error vector”
merr(z, n, t, j) := m(2)(z, n, t)
[
Mmod
j (z, n, t)
]−1
, (5.4)
where Mmod
j (z, n, t) is a matrix solution of the model RHP. Recall that by Remark 4.7, the
model jump problem
Mmod
j,+ (z, n, t) = Mmod
j,− (z, n, t)vmod
j (z, n, t), z ∈ I ∪ I∗ ∪ J ∪ J∗, n
t
∈ Ijε , (5.5)
How Discrete Spectrum and Resonances Influence the Asymptotics 27
has a piecewise constant (with respect to ξ) jump matrix vmod(z) given by (4.20). The choice of
such a matrix solution is not unique. However, a proper solution Mj(z) := Mmod
j (z, n, t) should
be invertible and satisfy at least the symmetry Mj
(
z−1
)
= σ1Mj(z)σ1, because the vector
merr(z, n, t, j) should fulfill the standard symmetry property. Moreover, the singularities of merr
should be removable outside the jump contour and should not be more than L2-integrable on the
contour. By construction (5.4), merr does not have a jump on I ∪ I∗ ∪ J ∪ J∗. Since m(2)(z) has
singularities of order O(z−q̃)−1/4 at q̃ ∈ J (2.20), the matrix [Mj(z)]
−1 should have singularities
of order not exceeding o(z − q̃)−3/4 at q̃ ∈ J and should be less than poles at its other singular
points outside the contour. As it is shown in [13] for the KdV shock wave (and the same is true
for the Toda shock case) such an invertible solution with weak singularities does not exist for
certain arbitrary large t at any direction ξ ∈ Iδ. Instead, we choose a matrix solution with poles
at points 1,−1, such that its determinant is equal to 1 and merr has removable singularities
at these points.
To explain the construction of our solution in more detail, recall that the quasimomen-
tum (3.1) has the jump on J ∪ J∗ defined by (3.10). Introduce the function
G(z) := exp
(∫ p(z)
b−2a
ω∞+,∞−
)
, z ∈ C \
[
q, q−1
]
.
Recall that for z ∈ D we have p(z) =
(
z+z−1
2 ,+
)
∈ M (upper sheet), and for z ∈ D∗ we have
p(z) =
(
z+z−1
2 ,−
)
. Then G(z) has the following properties [11, Section 5]:
� G(z) is holomorphic on C \
[
q, q−1
]
and satisfies G
(
z−1
)
= G−1(z).
� Its jumps are given by
G+(z) = G−(z)e−Λ, z ∈ J,
G+(z) = G−(z)eΛ, z ∈ J∗,
G±(z) =
[
G±
(
z−1
)]−1
, z ∈ I ∪ I∗.
� The following asymptotic expansion is valid,
G(z) = − ã
2z
(
1 + 2b̃z +O
(
z2
))
, G(z−1) = −2z
ã
(
1− 2b̃z +O
(
z2
))
,
where ã = CapS and b̃ ∈ R.
Lemma 5.3. Introduce the following vector function holomorphic in C \ ([q, q−1] ∪ {0}),
m#(z, n, t, j) = mmod(z, n+ 1, t, j)[G(z)]−σ3 .
Then m#(z) solves the jump problem
m#
+(z, n, t, j) = m#
−(z, n, t, j)vmod
j (z, n, t), z ∈ I ∪ I∗ ∪ J ∪ J∗, n
t
∈ Ijε ,
where vmod
j (z, n, t) is given by (4.20). The vector m#(z) satisfies the symmetry condition
m#
2
(
z−1
)
= m#
1 (z). The normalization condition is not fulfilled, instead we have
m#
1 (z, n, t, j) = −2z
ã
mmod
1 (0, n+ 1, t, j)(1 +O(z)),
m#
2 (z, n, t, j) = − ã
2z
mmod
1 (∞, n+ 1, t, j)(1 +O(z)), as z → 0.
28 I. Egorova and J. Michor
Proof. The proof follows immediately from the properties of G and mmod above. �
The function
ρ(z) = ρ(z, n, t, j) =
mmod
2 (0, n, t, j)mmod
2 (∞, n+ 1, t, j)
2ã
(
z−1 − z
)
is defined for all z 6= ±1 and is odd, ρ(z) = −ρ
(
z−1
)
. The function ρ does not have jumps, there-
fore the vector ρ(z)m#(z) solves the same jump problem with (4.20). However, it is bounded
as z → 0, z →∞ and has simple poles at 1 and −1 instead. In conclusion, the vector
Ψ(z, n, t, j) =
1
2
mmod(z, n, t, j) + ρ(z)m#(z, n, t, j)
solves our vector model RHP, and the same is true for Ψ
(
z−1, n, t, j
)
σ1. From here on, we fix
the parameters n, t, j and omit them to shorten notations when necessary. In particular, the
symmetry conditions for mmod and m# and oddness of ρ imply for the components of Ψ that
Ψ1(z) =
1
2
mmod
1 (z) + ρ(z)m#
1 (z), Ψ2(z) =
1
2
mmod
2 (z) + ρ(z)m#
2 (z),
Ψ1
(
z−1
)
=
1
2
mmod
2 (z)− ρ(z)m#
2 (z), Ψ2
(
z−1
)
=
1
2
mmod
1 (z)− ρ(z)m#
1 (z). (5.6)
Lemma 5.4 ([11, Lemma 5.2]).
(i) The matrix Mmod(z) = Mmod
j (z, n, t)
Mmod(z) =
(
Ψ1(z) Ψ2(z)
Ψ2(z−1) Ψ1(z−1)
)
, z ∈ C \
[
q−1, q
]
,
is a meromorphic matrix solution for the model jump problem
Mmod
+ (z) = Mmod
− (z)vmod(z), z ∈ I ∪ I∗ ∪ J ∪ J∗,
with vmod(z) given by (4.19). It has simple poles at z = ±1.
(ii) Mmod(z) satisfies the symmetry
Mmod(z−1) = σ1M
mod(z)σ1. (5.7)
(iii) The following equality is valid,
mmod(z, n, t) = (1, 1)Mmod(z, n, t). (5.8)
(iv) The determinant of Mmod(z) is a constant with respect to n, t, j, z,
detMmod(z) = 1, z ∈ C. (5.9)
Using this lemma, we can establish the properties of the error vector function (5.4), which
we denote here by merr(z). We recall the definition of Kδ from (4.21).
Theorem 5.5. The vector function merr(z) is holomorphic in C \ Ξδ, where
Ξδ := Lδ ∪ L∗δ ∪ Cδ ∪ C∗δ ∪
⋃
zk∈σd
(
Tδ,k ∪ T∗δ,k
)
= Kδ \ (I ∪ I∗ ∪ J ∪ J∗),
and satisfies the following properties:
How Discrete Spectrum and Resonances Influence the Asymptotics 29
(i) merr(z) has removable singularities at 0, 1, −1, ∞ and on J .
(ii) It solves the jump problem
merr
+ (z) = merr
− (z)(I +W (z)), z ∈ Ξδ, (5.10)
where
W (z) = Mmod(z)
(
v(2)(z)− I
)[
Mmod(z)
]−1
. (5.11)
(iii) It satisfies the symmetry condition
merr(z) = merr
(
z−1
)
σ1, z ∈ C \ Ξδ. (5.12)
In particular,
merr
−
(
z−1
)
= merr
− (z)σ1, z ∈ Ξδ, (5.13)
merr
2 (0) = merr
1 (∞) =
1
2
(
m
(2)
1 (0)
mmod
1 (0)
+
mmod
1 (0))
m
(2)
1 (0)
)
:= τ > 0. (5.14)
Here mmod
1 (0) = mmod
1 (0, n, t, j) and m(2)(0) = m(2)(0, n, t).
Proof. The absence of singularities at the points 0, 1, −1,∞ and on the set J was proven in [11,
Lemmas 5.4 and 5.5]. The jump (5.10) and the symmetry property (5.12) follow from (4.18),
(4.19), (5.5), (5.7) and (5.9). Property (5.13) holds due to the mutual orientation of the sym-
metric parts of the contour Ξδ. Last, (5.14) follows from (5.4), (5.6), (5.9) and the definition
of ρ which implies
lim
z→0
ρ(z)m#
2 (z) = − 1
mmod
1 (0)
, lim
z→0
ρ(z)m#
1 (z) = 0. �
Now we are ready to prove Theorem 5.2. We follow the well-known approach via singular
integral equations (see, e.g., [9], [16, Chapter 4], [19]). A peculiarity of this approach applied to
the Toda equation is generated by the type of normalization condition of the vector RHP and the
symmetry condition. In particular, if we want to preserve the symmetry condition (5.12) in the
Cauchy-type formula for merr(z), we should use a matrix Cauchy kernel (cf. [19, equation (B.8)],
Ω̂(s, z) =
( 1
s−z 0
0 1
s−z −
1
s
)
ds, s ∈ Ξδ, z /∈ Ξδ.
Since [19, equation (B.9)]
Ω̂
(
s, z−1
)
= σ1Ω̂
(
s−1, z
)
σ1,
and W (s−1) = σ1W (s)σ1, s ∈ Ξδ, this implies with (5.13) and the orientation of Ξδ that the
symmetry property holds:∫
Ξδ
merr
− (s)W (s)Ω̂(s, z) =
∫
Ξδ
merr
− (s)W (s)Ω̂
(
s, z−1
)
σ1.
Note that the 1, 1-entry of the Cauchy kernel Ω̂(s, z) has a zero at z = ∞ while the 2, 2-entry
has a zero at z = 0. From (5.10) and (5.14) it follows that
merr(z) =
(
merr
1 (∞),merr
2 (0)
)
+
1
2πi
∫
Ξδ
merr
− (s)W (s)Ω̂(s, z)
= τ(1, 1) +
1
2πi
∫
Ξδ
merr
− (s)W (s)Ω̂(s, z).
30 I. Egorova and J. Michor
Recall that according to Lemma 4.8,
‖W (z)‖ = O
(
e−C(δ)t
)
, uniformly with respect to z ∈ Ξδ and n, t, j :
n
t
∈ Ijε .
In turn, it implies the following
Lemma 5.6. Uniformly with respect to n
t ∈ Iε,∥∥zkW (z)
∥∥
Lp(Ξδ)
= O
(
e−C(δ)t
)
, p ∈ [1,∞], k = 0, 1. (5.15)
Now we are ready to apply the technique of singular integral equations. Let C denote the
Cauchy operator associated with Ξδ,
(Ch)(z) =
1
2πi
∫
Ξδ
h(s)Ω̂(s, z), s ∈ C \ Ξδ,
where h =
(
h1 h2
)
∈ L2(Ξδ) and satisfies the symmetry h(s) = h
(
s−1
)
σ1. Let (C+h)(z)
and (C−h)(z) be the non-tangential limiting values of (Ch)(z) from the left and right sides
of Ξδ, respectively. As usual, we introduce the operator CW : L2(Ξδ) ∩ L∞(Ξδ) → L2(Ξδ) by
CWh = C−(hW ), where W is the error matrix (5.11). Then
‖CW ‖ = ‖CW ‖L2(Ξδ)→L2(Ξδ) ≤ C‖W‖L∞(Ξδ) = O
(
e−C(ε)t
)
,
as well as∥∥(I− CW )−1
∥∥ =
∥∥(I− CW )−1
∥∥
L2(Ξδ)→L2(Ξδ)
≤ 1
1−O
(
e−C(ε)t
) (5.16)
for sufficiently large t. Consequently, for t� 1, on Ξδ we define a vector function
µ(s) = (τ, τ) + (I− CW )−1CW
(
(τ, τ)
)
(s),
with τ given by (5.14). Then by (5.15) and (5.16)
‖µ(s)− (τ, τ)‖L2(Ξδ) ≤
∥∥(I− CW )−1
∥∥‖C−‖‖W‖L∞(Ξδ) = O
(
e−C(ε)t
)
. (5.17)
With the help of µ, the vector function merr(z) can be represented as
merr(z) = (τ, τ) +
1
2πi
∫
Ξδ
µ(s)W (s)Ω̂(s, z),
and by virtue of (5.17) and Lemma 5.6 we obtain as z → 0
merr(z) = (τ, τ) +
1
2πi
∫
Ξδ
(τ, τ)W (s)Ω̂(s, z) + E(z). (5.18)
Here E(z) is a vector function holomorphic in a vicinity of z = 0 which admits the estimate
‖E(z)‖ ≤ ‖W‖L2(Ξδ)‖µ(s)− (τ, τ)‖L2(Ξδ)(1 +O(z)) = O
(
e−C(ε)t
)
(1 +O(z)),
and O(z) is uniformly bounded for n
t ∈ I
j
ε . From (5.18) and (5.8) we get
m(2)(z) = merr(z)Mmod(z) = τmmod(z) + τO
(
e−C(ε)t
)
E1(z),
How Discrete Spectrum and Resonances Influence the Asymptotics 31
where E1(z) is a holomorphic vector function in a vicinity of z = 0, uniformly bounded with
respect to n, t, j as n
t ∈ I
j
ε . The normalization conditions for m(2) and mmod imply that
τ2
(
1 +O
(
e−C(ε)t
))
= 1, that is, τ = 1 +O
(
e−C(ε)t
)
.
With (5.14) this yields (5.2) and at the same time (5.3), which proves Theorem 5.2.
To finish the proof of Theorem 1.1 recall that in a vicinity of z = 0 the initial vector-
function (2.14) and the transformed function m(2)(z) are connected by
m(2)(z, n, t) = m(z, n, t)
[
Π(z, ξ)et(Φ(z,ξ)−g(z,ξ))F (z, ξ)
]−σ3 , ξ =
n
t
. (5.19)
The results of Theorem 5.2 and Lemmas 2.2 and 5.1 imply
b(n, t) = b̂(n, t,∆j) +O
(
e−C(ε)t
)
,
n
t
∈ Ijε .
On the other hand, from (2.16), (5.19) and Lemma 4.1(e), it follows
2a(n, t) =
m1(0, n, t)
m1(0, n+ 1, t)
=
m
(2)
1 (0, n, t)
m
(2)
1 (0, n+ 1, t)
et(K(nt )−K(n+1
t )),
because Π(z) and F (z) are the same for n and n + 1. By (4.1), K(ξ) = − log(2ã)(ξ − const),
that is,
t
(
K
(
n
t
)
−K
(
n+ 1
t
))
= log(2ã).
Using (5.1) we finally get
2a(n, t) =
mmod
1 (0, n, t)
mmod
1 (0, n+ 1, t)
e2ã +O
(
e−C(ε)t
)
= 2â(n, t,∆j) +O
(
e−C(ε)t
)
.
This finishes the proof of our main result, Theorem 1.1.
Acknowledgements
This research was supported by the Austrian Science Fund (FWF) under Grant No. P31651.
We thank the referees for their careful reading and their recommendations.
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1 Introduction
2 Notations and statement of the initial holomorphic RHP
3 g-function as an Abel integral and its connection with the Baker–Akhiezer function
4 Reduction of the initial RHP to the model RHP
5 Solution of the model problem and conclusive analysis
References
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| id | nasplib_isofts_kiev_ua-123456789-211304 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T02:46:08Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Egorova, Iryna Michor, Johanna 2025-12-29T11:06:38Z 2021 How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave. Iryna Egorova and Johanna Michor. SIGMA 17 (2021), 045, 32 pages 1815-0659 2020 Mathematics Subject Classification: 37K40; 35Q53; 37K45; 35Q15 arXiv:2012.12371 https://nasplib.isofts.kiev.ua/handle/123456789/211304 https://doi.org/10.3842/SIGMA.2021.045 We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically a finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184]. This research was supported by the Austrian Science Fund (FWF) under Grant No. P31651. We thank the referees for their careful reading and their recommendations. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave Article published earlier |
| spellingShingle | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave Egorova, Iryna Michor, Johanna |
| title | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave |
| title_full | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave |
| title_fullStr | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave |
| title_full_unstemmed | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave |
| title_short | How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave |
| title_sort | how discrete spectrum and resonances influence the asymptotics of the toda shock wave |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211304 |
| work_keys_str_mv | AT egorovairyna howdiscretespectrumandresonancesinfluencetheasymptoticsofthetodashockwave AT michorjohanna howdiscretespectrumandresonancesinfluencetheasymptoticsofthetodashockwave |