On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation
We show that the long-time behavior of solutions to the Korteweg{de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schrödinger operator) depends only on the size of...
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Egorova, I. Gladka, Z. Teschl, G. 2018-07-10T10:42:15Z 2018-07-10T10:42:15Z 2016 On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation / I. Egorova, Z. Gladka , G. Teschl // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 3-16. — Бібліогр.: 23 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag12.01.003 Mathematics Subject Classification 2000: 37K40, 35Q53 (primary); 33E05, 35Q15 (secondary) https://nasplib.isofts.kiev.ua/handle/123456789/140545 We show that the long-time behavior of solutions to the Korteweg{de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schrödinger operator) depends only on the size of the step of the initial data and on the direction, x/ t =const, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi inversion problem. Показано, что поведение при большом времени решений уравнения Кортевега-де Фриза с начальными данными типа ступеньки, соответствующими волне сжатия, в области эллиптической волны может быть описано слабо модулированным двухзонным решением. Модуль этой эллиптической функции, определяемый спектром фонового оператора, зависит от размера ступеньки в начальных данных и от направления, в котором исследуется асимптотическое поведение решения. В свою очередь фазовый сдвиг (то есть спектр задачи Дирихле) в этой эллиптической функции зависит также от данных рассеяния, и он посчитан с помощью проблемы обращения Якоби. We thank Alexei Rybkin and Johanna Michor for valuable discussions on this topic. I.E. is indebted to the Department of Mathematics at the University of Vienna for its hospitality and support during the fall of 2015, where part of this work was done. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation Article published earlier |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation Egorova, I. Gladka, Z. Teschl, G. |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation |
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On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation |
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on the form of dispersive shock waves of the korteweg-de vries equation |
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Egorova, I. Gladka, Z. Teschl, G. |
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Egorova, I. Gladka, Z. Teschl, G. |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We show that the long-time behavior of solutions to the Korteweg{de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schrödinger operator) depends only on the size of the step of the initial data and on the direction, x/ t =const, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi inversion problem.
Показано, что поведение при большом времени решений уравнения Кортевега-де Фриза с начальными данными типа ступеньки, соответствующими волне сжатия, в области эллиптической волны может быть описано слабо модулированным двухзонным решением. Модуль этой эллиптической функции, определяемый спектром фонового оператора, зависит от размера ступеньки в начальных данных и от направления, в котором исследуется асимптотическое поведение решения. В свою очередь фазовый сдвиг (то есть спектр задачи Дирихле) в этой эллиптической функции зависит также от данных рассеяния, и он посчитан с помощью проблемы обращения Якоби.
|
| issn |
1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/140545 |
| citation_txt |
On the Form of Dispersive Shock Waves of the Korteweg-de Vries Equation / I. Egorova, Z. Gladka , G. Teschl // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 1. — С. 3-16. — Бібліогр.: 23 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-24T02:35:57Z |
| last_indexed |
2025-11-24T02:35:57Z |
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1850409293558317056 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 1, pp. 3–16
On the Form of Dispersive Shock Waves of the
Korteweg–de Vries Equation
I. Egorova, Z. Gladka
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Nauki Ave., Kharkiv, 61103, Ukraine
E-mail: iraegorova@gmail.com
gladkazoya@gmail.com
G. Teschl
Faculty of Mathematics University of Vienna
Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
International Erwin Schrödinger Institute for Mathematical Physics
Boltzmanngasse 9, 1090 Wien, Austria
E-mail: Gerald.Teschl@univie.ac.at
Received October 12, 2015,
published online December 4, 2015
We show that the long-time behavior of solutions to the Korteweg–de
Vries shock problem can be described as a slowly modulated one-gap solution
in the dispersive shock region. The modulus of the elliptic function (i.e., the
spectrum of the underlying Schrödinger operator) depends only on the size
of the step of the initial data and on the direction, x
t =const, along which
we determine the asymptotic behavior of the solution. In turn, the phase
shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the
scattering data, and is computed explicitly via the Jacobi inversion problem.
Key words: KdV equation, steplike, dispersive shock wave.
Mathematics Subject Classification 2010: 37K40, 35Q53 (primary); 33E05,
35Q15 (secondary).
1. Introduction
The Korteweg–de Vries (KdV) shock problem is concerned with the long-time
behavior of the solution of the KdV equation
qt(x, t) = 6q(x, t)qx(x, t)− qxxx(x, t), (x, t) ∈ R× R,
Research supported by the Austrian Science Fund (FWF) under Grant V120.
c© I. Egorova, Z. Gladka, and G. Teschl, 2016
I. Egorova, Z. Gladka, and G. Teschl
with the steplike initial profile{
q(x, 0)→ 0, as x→ +∞,
q(x, 0)→ −c2, as x→ −∞. (1.1)
This behavior is well understood on a physical level of rigor for the pure-step ini-
tial data (i.e., when q(x, 0) = 0 as x > 0, q(x, 0) = −c2 as x < 0), and was studied
using a quasi-classical Whitham approach in [13, 15, 16] with further treatments
using the matched-asymptotic method in [1, 21]. Extensions to steplike finite-gap
backgrounds were given in [3–6] and [23]. This led to the following three main
regions with different asymptotical behavior of the solution:
1. Region x < −6c2t, where the solution is asymptotically close to the back-
ground −c2 up to a decaying dispersive tail.
2. Middle region −6c2t < x < 4c2t, also known as dispersive shock or elliptic
region, where the solution can asymptotically be described by a modulated
elliptic wave.
3. Soliton region 4c2t < x, where the solution is asymptotically given by a
sum of solitons.
We refer to our paper [10] for further details and more on the history of this
problem. In this note we want to revisit the middle region which is the most
interesting and challenging one from a mathematical point of view. In particular,
in the case of pure-step initial data, Gurevich and Pitaevskii ([15, 16], see also
[21]) derived the following large-time asymptotical formula in the elliptic region:
q(x, t) ∼ qGP(x, t) = −2c2dn2
(
2tc(6ξ − c2(1 + m2(ξ)), m(ξ)
)
+ c2(1−m2(ξ)),
(1.2)
where ξ = x
12t , and the modulus m(ξ) is determined implicitly by
c2
6
(
1 + m2(ξ)− 2m2(ξ)(1−m2(ξ))K(m(ξ))
[E(m(ξ))− (1−m2(ξ))K(m(ξ))]
)
= ξ. (1.3)
Here dn(s,m) is the Jacobi elliptic function and K(m), E(m) are the standard
complete elliptic integrals. The function qGP(x, t) is a stationary running wave
of the KdV equation if the parameter ξ is a constant.
On the other hand, in [10], under the assumption
+∞∫
0
eC0x(|q(x)|+ |q(−x) + c2|dx <∞, C0 > c > 0, (1.4)
4 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
we derived the formula (for precise conditions, see our paper)
q(x, t) ∼ qRH(x, t) =
Γ(ξ)B′(ξ)
6π
d2
dv2
log θ3
(
tB(ξ) + ∆(ξ)
2π
+ v
) ∣∣∣
v=0
− z′(ξ)
6
,
(1.5)
where all quantities are associated with the elliptic Riemann surface y2 = λ(λ+
c2)(λ+ a2(ξ)) depending on the parameter a(ξ) > 0 for ξ ∈ (−c2/2, c2/3) implic-
itly determined by
0∫
a(ξ)
(
ξ +
c2 − a2(ξ)
2
− s2
)√
s2 − a2(ξ)
c2 − s2
ds = 0. (1.6)
As is shown in [18, 19], the point a(ξ) increases monotonically from 0 to c > 0
as ξ changes from −c2/2 to c2/3. The quantities B, Γ, z and ∆ are defined in
(2.9)–(2.12) below.
Formula (1.5) is very reminiscent of the usual Its–Matveev formula for the
one-gap solution of the KdV equation. Hence this raises two natural questions,
namely, whether (1.5) agrees with (1.2) and whether they reduce to the Its–
Matveev formula for the constant ξ, that is, whether (1.5) can be viewed as a
slowly modulated one-gap solution of the KdV equation. The purpose of the
present note is to give a positive answer to both questions.
2. Comparison between qRH(x, t) and the Its–Matveev Formula
We begin by recalling the well-known Its–Matveev formula for the finite-
gap solution of the KdV equation in the case of one gap in the spectrum (see,
for example, [14, 20, 22, 17]). To facilitate further comparison of this formula
with qRH(x, t), we suppose that the spectrum of this one-gap solution is the
set σ = [−c2,−a2(ξ)] ∪ [0,∞). Let M = M(ξ) be the elliptic Riemann surface
associated with the function
R(λ) := R(λ, ξ) =
√
λ(λ+ c2)(λ+ a2(ξ)),
where the cuts are taken along the spectrum σ, and R(λ) takes positive values
on the upper side of the cut along the interval [0,∞). A point on M is denoted
by p = (λ,±), λ ∈ C, and the projection onto C ∪ {∞} is denoted by π(p) = λ.
The sheet exchange map is given by p∗ = (λ,∓) for p = (λ,±). The sets
ΠU = {(λ,+) | λ ∈ C \ σ} ⊂M, ΠL = {p∗ | p ∈ ΠU},
are called the upper, the lower sheets, respectively. Introduce the canonical bases
of a and b cycles. The cycle b surrounds the interval [−c2,−a2] clockwise on the
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 5
I. Egorova, Z. Gladka, and G. Teschl
upper sheet, and the cycle a supplements b passing along the gap from −a2 to 0
in the positive direction on the lower sheet and back from 0 to −a2 on the upper
sheet. Next, let dω be a holomorphic differential on M normalized by
∫
a dω = 2πi.
Evidently,
dω = Γ̃
dλ
R(λ)
, Γ̃ = 2πi
∫
a
dλ
R(λ)
−1
> 0, (2.1)
and
τ̃ =
∫
b
dω < 0. (2.2)
Define now the theta function of M by
θ(z) = θ(z | τ̃) =
∑
m∈Z
exp{1
2
τ̃m2 + zm}.
Recall that this function is even and takes real values for z ∈ iR ∪ R.
Next, following [20], introduce two meromorphic differentials of the second
kind dΩ1 and dΩ3 with the vanishing a periods and with the only pole at infinity
of the form
dΩ1 =
i
2
λ− h
R(λ)
dλ, dΩ3 = −3i
2
(λ− ν1)(λ− ν2)
R(λ)
dλ, (2.3)
where
h =
∫
a
λdλ
R(λ)
∫
a
dλ
R(λ)
−1
∈ (−a2, 0), (2.4)
and the points νj ∈ R are chosen such that
∫
a dΩ3 = 0, and
2ν1 + 2ν2 + c2 + a2 = 0. (2.5)
The last equality guarantees the absence of the term of order λ−1/2dλ in repre-
sentation for dΩ3. Note that at least one of the points νi lies in the gap (−a2, 0).
Denote
iV =
∫
b
dΩ1, and iW =
∫
b
dΩ3. (2.6)
The values V and W are called the wave number and the frequency. Evidently,
V,W ∈ R.
Next recall the Abel map A(p) =
∫ p
∞ dω. Let p0 be a point on M with the
projection in the gap, π(p0) ∈ [−a2, 0]. For this point, the value A(p0) + K is
pure imaginary (cf. [8]), where K = − τ̃
2 + πi is the Riemann constant. Then
6 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
the Its–Matveev formula for the one-gap solution with initial Dirichlet divisor p0
reads (cf. [20, 22]):
qIM(x, t) = −2
d2
dx2
log θ (iV x− 4iWt−A(p0)−K)− a2 − c2 − 2h, (2.7)
where h is defined by (2.4).
Now we will study in more details formula (1.5), where the quantities are
defined as follows: τ ∈ iR+ is the period given by
τ := τ(ξ) = −
−a2(ξ)∫
−c2
dλ√
λ(λ+ c2)(λ+ a2(ξ))
0∫
−a2(ξ)
dλ√
λ(λ+ c2)(λ+ a2(ξ))
−1
,
(2.8)
and
θ3(v) = θ3(v | τ) =
∑
m∈Z
exp{(m2τ + 2mv)πi}
is the associated theta function. Furthermore,
B(ξ) = 24
c∫
a(ξ)
(
ξ +
c2 − a2(ξ)
2
− s2
)√
s2 − a2(ξ)
c2 − s2
ds; (2.9)
Γ(ξ) = −1
2
a(ξ)∫
−a(ξ)
(
(c2 − s2)(a2(ξ)− s2)
)−1/2
ds
−1
; (2.10)
z(ξ) =
12ξ(c2 − a(ξ)2) + 3c4 + 9a(ξ)4 − 6a(ξ)2c2
2
; (2.11)
∆(ξ) = 2
c∫
a(ξ)
log |(T (is)T1(is)|√
(c2 − s2)(s2 − a(ξ)2)
ds
a(ξ)∫
−a(ξ)
ds√
(c2 − s2)(a2(ξ)− s2)
−1
;
(2.12)
T and T1 are the left and the right transmission coefficients of the initial data
(1.1). In formulas (2.9), (2.10), and (2.12) the positive value of square root is
taken.
Denote −a2(ξ) = γ(ξ) := γ ,
µ = −ξ − c2 + γ
2
, (2.13)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 7
I. Egorova, Z. Gladka, and G. Teschl
and put λ = −s2 in (1.6). Then (1.6) is equivalent to
0∫
γ
G(λ)dλ = 0, G(λ, ξ) :=
(λ− µ)(λ− γ)
R(λ)
, (2.14)
or
F (γ, ξ) :=
0∫
γ
(
λ+ ξ +
γ + c2
2
)√
λ− γ
λ(λ+ c2)
dλ = 0. (2.15)
Equation (2.15) determines the function γ(ξ) implicitly, and the function µ(ξ) by
(2.13).
Lemma 2.1. For ξ ∈ (−c2/2, c2/3) the function γ(ξ) decays monotonically
from 0 to −c2, and µ(ξ) ∈ (γ(ξ), 0). Moreover,
d
dξ
γ(ξ) = 4
h(ξ)− γ(ξ)
3γ(ξ) + 2ξ + c2
, (2.16)
where h(ξ) is defined by formula (2.4) with R(λ) =
√
λ(λ− γ(ξ))(λ+ c2).
P r o o f. The existence and uniqueness of γ(ξ) is proved in [18]. Formally,
differentiating (2.15) with respect to ξ gives dγ
dξ = −∂F
∂ξ (∂F∂γ )−1. In turn, this
implies
dγ
dξ
=
4S
3γ + 2ξ + c2
, S =
0∫
γ
λ− γ
R(λ)
dλ
0∫
γ
dλ
R(λ)
−1
,
which implies (2.16). Evidently, S > 0 for γ < 0. Therefore, to prove monotonic-
ity of γ(ξ), it is sufficient to prove that
f(ξ) = 3γ(ξ) + 2ξ + c2
is negative for ξ ∈ (−c2/2, c2/3), and that γ(ξ) is also negative there. We observe
that F (−c2, c2/3) = 0, therefore γ(c2/3) = −c2. Since f(c2/3) = −4c2/3 < 0,
then γ′(c2/3) < 0. Thus γ(ξ) grows continuously starting from −c2 as ξ decreases
starting from c2/3. The function f is a continuous function of ξ, and monotonicity
of γ can only stop if there is a change of sign of f . Let ξ0 be a point where
f(ξ0) = 0. Then ξ0 satisfies the system{
3γ(ξ0) + 2ξ0 + c2 = 0
2µ(ξ0)− 2ξ0 − c2 − γ(ξ0) = 0.
Thus, µ(ξ0) = γ(ξ0). But for γ < 0, formula (2.14) holds iff µ ∈ (γ, 0). This
means that µ(ξ0) = γ(ξ0) = 0, that is, ξ0 = −c2/2.
8 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
Lemma 2.2. Let γ(ξ) and µ(ξ) be as in Lemma 2.1 and h(ξ) as in (2.4).
Let G(λ, ξ) be defined by (2.14) and dΩj, j = 1, 3 by (2.3)–(2.5). Then, for any
ξ ∈
(
− c2
2 ,
c2
3
)
, the following representation is valid:
Gdλ =
2i
3
dΩ3 − 2iξ dΩ1. (2.17)
Moreover, the following formula holds:
∂
∂ξ
G(λ, ξ) =
λ− h(ξ)
R(λ, ξ)
. (2.18)
P r o o f. By (2.14), the differential Gdλ has the vanishing a period.
Moreover, using (2.5), (2.13), one checks that Gdλ− 2i
3 dΩ3 + 2iξ dΩ1 has no pole
at ∞ and hence must vanish. This proves (2.17).
To get (2.18), we evaluate
∂
∂ξ
G(λ, ξ)− λ− h(ξ)
R(λ, ξ)
=
∂
∂ξ
(
(λ− µ)
√
λ− γ√
λ(λ+ c2)
)
− λ− h
R(λ)
=
(−2µ′ − γ′)λ+ 2µ′γ + µγ′ − 2λ+ 2h
2R(λ)
.
Formula (2.13) implies
2µ′ + γ′ + 2 = 0. (2.19)
Thus, to justify (2.18), one has to prove the equality
2µ′γ + µγ′ + 2h = 0.
By virtue of (2.16), (2.19), and (2.13), we get
2µ′γ + µγ′ + 2h = (−γ′ − 2)γ − γ′
(
ξ +
c2 + γ
2
)
+ 2h
= −1
2
γ′(3γ + c2 + 2ξ)− 2γ + 2h = 0,
which proves (2.18).
Lemma 2.3. Let B(ξ), Γ(ξ), and z(ξ) be defined by (2.9), (2.10), and (2.11),
respectively. Let V (ξ) and W (ξ) be the wave number and the frequency, defined
by (2.6), and let h(ξ) be as in (2.4). Then the following identities hold:
tB(ξ) = −4W (ξ)t+ V (ξ)x, (2.20)
(a)
d
dξ
B(ξ) = 12V (ξ), (b) 4πΓ(ξ) = −V (ξ), (2.21)
1
6
d
dξ
z(ξ) = c2 + a2(ξ) + 2h(ξ). (2.22)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 9
I. Egorova, Z. Gladka, and G. Teschl
P r o o f. To get (2.20), we make change of variables λ = −s2 in (2.9), and
take into account (2.17), (2.13), ξ = x
12t , and the definition of the b period on the
Riemann surface M(ξ). Then
tB(ξ) = −12t
−c2∫
−a2
(λ− µ)
√
−a2 − λ
−λ(λ+ c2)
dλ = 12t
γ∫
−c2
G(λ, ξ)dλ
= 4it
∫
b
dΩ3 − ix
∫
b
dΩ1 = −4tW (ξ) + xV (ξ).
This proves (2.20). Formula (2.21), (a) follows from (2.17) and (2.18):
d
dξ
B(ξ) = 12
d
dξ
γ∫
−c2
G(λ, ξ)dλ = 12γ′(ξ)G(γ(ξ), ξ)− 12i
∫
b
dΩ1 = 12V (ξ).
Next, by definition of the a period, formula (2.10) reads:
Γ(ξ) =
2
−a2∫
0
dλ√
−λ(λ+ c2)(λ+ a2)
−1
=
i∫
a
dλ
R(λ)
=
Γ̃
2π
, (2.23)
where Γ̃ is the normalization constant from (2.1). On the other side, formulas
(2.1), (2.3), and the residue theorem [12] yield
∫
a
dω
∫
b
dΩ1 = 2πi
∫
b
dΩ1 = 2πi Res∞
dΩ1(p)
p∫
∞
dω
, p = (λ,+).
Since in the local parameter z = λ−1/2, z → 0, we have
dΩ1 =
(
i
2
√
λ
+O
(
1
λ3/2
))
dλ = − i
z2
(1 + o(1))dz,
and
λ∫
∞
dω = − 2Γ̃√
λ
+O
(
1
λ3/2
)
= −2Γ̃z(1 + o(1)),
then, by (2.23),
Res∞
dΩ1(p)
p∫
∞
dω
= −2iΓ̃ = −4πiΓ.
10 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
Together with (2.6), this proves (2.21), (b). To prove the remaining formula
(2.22), we represent z(ξ) via γ(ξ) and apply (2.16). Then we get
d
dξ
z(ξ) =
1
2
d
dξ
(
12ξ(c2 + γ(ξ)) + 3c4 + 9γ(ξ)2 + 6γ(ξ)c2
)
= 6(c2 + γ(ξ)) +
(
6ξ + 9γ(ξ) + 3c2
)
γ′(ξ)
= 6(c2 + γ(ξ)) + 12(h(ξ)− γ(ξ)) = 6(c2 + a2(ξ)) + 12h(ξ).
Now we are ready to compare formula (1.5) with (2.7). Comparing the periods
τ defined by (2.8) and τ̃ defined by (2.1), (2.2), we observe that 2πiτ = τ̃ , and
therefore
θ(z | τ̃) = θ3(
z
2πi
| τ).
Put
B(ξ) =
Γ(ξ)
6π
(
d
dξ
B(ξ)
)
, C(ξ) =
1
6
d
dξ
z(ξ). (2.24)
Then qRH(x, t) can be represented as
qRH(x, t) = B(ξ)
d2
dv2
log θ3
(
itB(ξ) + i∆(ξ) + 2iπv
2πi
)
|v=0 +C(ξ),
= −4π2B(ξ)
d2
dv2
log θ (itB(ξ) + i∆(ξ) + v) |v=0 +C(ξ). (2.25)
On the other hand, by (2.22), we have
qIM(x, t)− C(ξ)
= −2(iV (ξ))2
d2
dv2
log θ (iV (ξ)x− 4iW (ξ)t−A(p0, ξ)−K(ξ) + v) |v=0 .
Substituting (2.20) and (2.21) into (2.24) and then into (2.25), we get B(ξ) =
−V 2(ξ) and itB(ξ) = iV (ξ)x − 4iW (ξ)t. We conclude that qRH(x, t) = qIM(x, t)
iff there exists a point p0, π(p0) ∈ [−a2, 0] such that the following equality is
fulfilled:
−A(p0, ξ)−K(ξ) = i∆(ξ) (mod 2πi),
where ∆(ξ) is defined by (2.12). Since
A(−c2) = πi (mod 2πi), A(−a2)−A(−c2) =
τ̃
2
, K = − τ̃
2
+ πi,
and ∆(ξ) is a real value, then the point p0(ξ) can be found as the unique solution
of the Jacobi inversion problem (cf. [12]):
p0(ξ)∫
−a2(ξ)
dω = −i∆(ξ). (2.26)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 11
I. Egorova, Z. Gladka, and G. Teschl
In summary we have proved
Theorem 2.4. For any fixed ξ = x
12t ∈ (−c2/2, c2/3) the function qRH(x, t)
is the usual one-gap solution of the KdV equation:
qRH(x, t) = −2
d2
dx2
log θ (iV (ξ)x− 4iW (ξ)t−A(p0, ξ)−K(ξ))
− a(ξ)2 − c2 − 2h(ξ),
associated with the spectrum [−c2,−a2(ξ)]∪ [0,∞) and the Dirichlet divisor p0(ξ)
defined via the Jacobi inversion problem (2.26).
Note that since θ(z+ 2πi) = θ(z), we see that qRH(x, t) is a periodic function
with respect to x and t of the periods 2π
V (ξ) and π
2W (ξ) , respectively.
3. Copmarison between qRH(x, t) and the Gurevich–Pitaevskii
Formula
Our next aim is to find a connection between the two functions m(ξ) and a(ξ)
implicitly given by (1.3) and (1.6).
Lemma 3.1. The following is true:
m2(ξ) =
a2(ξ)
c2
.
P r o o f. Represent (1.6) as
0 =
(
−c
2 + a2(ξ)
2
+ ξ
) a(ξ)∫
0
√
a2(ξ)− s2
c2 − s2
ds
+
a(ξ)∫
0
√
(a2(ξ)− s2)(c2 − s2)ds =
(
ξ − c2 + a2(ξ)
2
)
I1(ξ) + I2(ξ). (3.1)
Put m := m(ξ) = a(ξ)
c . Then (cf. [9], formulas 781.61 and 781.22)
I1 = c(E(m)− (1−m2)K(m)), I2 =
c3
3
{(m2 − 1)K(m) + (m2 + 1)E(m)},
12 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
where K(m) and E(m) are the standard complete elliptic integrals for 0 < m < 1.
Substituting this into (3.1), we obtain
ξ = c2
{
m2 + 1
2
− 1
3
(m2 − 1)K(m) + (m2 + 1)E(m)
E(m)− (1−m2)K(m)
}
=
c2
6
E(m)(m2 + 1)− (1−m2)(m2 + 1)K(m) + 2m2(m2 − 1)K(m)
E(m)− (1−m2)K(m)
=
c2
6
(
(m2 + 1) +
2m2(m2 − 1)K(m)
E(m)− (1−m2)K(m)
)
.
Thus, (1.3) and (1.6) define the same function: m(ξ) = m(ξ).
Lemma 3.1 implies that the period (2.8) corresponds in the standard way (cf.
[2]) to the elliptic modulus m, and the following formula is valid:
dn2(s,m) =
d2
du2
log Θ(u | τ) +
E(m)
K(m)
,
where
Θ(u | τ) = θ3
(
u
2K(m)
+
1
2
| τ
)
= θ3
(
u
2K(m)
+
1
2
)
,
and θ3(s) is as in (1.5).
For the remainder, we will fix ξ and omit it from our notation. Fixing ξ, we
also fix m = a c−1 and, consequently, we will also omit m in the complete elliptic
integrals: K := K(m), E := E(m). Therefore,
qGP (x, t) = − c2
2K2
d2
dv2
log θ3
(
tc(6ξ − c2(1 +m2))
K
+
1
2
+ v
)
+ c2(1−m2)− 2E
K
. (3.2)
Thus, to convince ourselves that this formula coincides with (1.5) (possibly, up
to a phase shift), it is sufficient to show:
Lemma 3.2. The following equalities hold:
− c2
2K2
=
Γ
6π
dB
dξ
; (3.3)
−6cξ − c3(1 +m2)
K
=
B
2π
; (3.4)
c2 − E
K
= −h, (3.5)
where B, Γ, and h are defined by (2.9), (2.10), and (2.4), respectively.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1 13
I. Egorova, Z. Gladka, and G. Teschl
P r o o f. Formulas (2.21), (2.10), and 781.01 of [9] imply
Γ
6π
dB
dξ
= −8Γ2 = −1
2
a∫
0
ds√
(c2 − s2)(a2 − s2)
−2
= − c2
2K2
. (3.6)
Next, by (2.4) and formula 781.11 from [9], we have
−h =
a∫
0
s2ds√
(c2 − s2)(a2 − s2)
a∫
0
ds√
(c2 − s2)(a2 − s2)
−1
=
c(K − E)
1
cK
.
This gives (3.5). To prove (3.4), we use (2.20) and (2.21). Namely, we have
tB
2π
= −2Γx− 4Wt = −2(x− 2(c2 + a2)t)Γ.
Here we used equation (4.4.13) of [22] which implies the equality W = (c2 +a2)Γ.
By (3.6), we have Γ = c
4K . Thus,
B
2π
= − tc
K
(
6ξ − (c2 + a2)
)
,
which proves (3.4). Note that the opposite sign with respect to (3.2) of the first
summand in θ3 is not essential as θ3 is even.
We proved that formulas (1.5) and (1.2) represent the same function up to
the phase shift. Namely, instead of the summand (2π)−1∆ in the argument of
the theta function for qRH(x, t), in the same formula for qGP(x, t) we have the
summand 1
2 . Recall now that the transmission coefficients for the pure step
potential have the representation (cf. [7, 11])
T (k) =
2ik
w(k)
, T1(k) =
2i
√
k2 + c2
w(k)
, w(k) = ik + i
√
k2 + c2.
Since in (2.12) k = is, s ∈ [0, c], then
|w(k)|2 = | − s+ i
√
c2 − s2|2 = c2,
and
|T (is)T1(is)| = 4sc−2
√
c2 − s2, s ∈ [0, c].
Therefore, the value of the phase shift for the pure step case is given by
∆ps
2π
=
∫ c
a log
(
4sc−2
√
c2 − s2
) (
(c2 − s2)(s2 − a2)
)−1/2
ds
2π
∫ a
0 ((c2 − s2)(s2 − a2))−1/2 ds
=
1
2πK(m)
1∫
m
log
(
4s
√
1− s2
)
√
(1− s2)(s2 −m2)
ds.
14 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 1
On the Form of Dispersive Shock Waves of the Korteweg–de Vries Equation
Acknowledgments. We thank Alexei Rybkin and Johanna Michor for valu-
able discussions on this topic. I.E. is indebted to the Department of Mathematics
at the University of Vienna for its hospitality and support during the fall of 2015,
where part of this work was done.
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