Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data
The long-time asymptotic behavior of the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of the stimulated Raman scattering is studied. Considered is the case of zero initial condition and single-phase boundary data. By using the stee...
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| Cite this: | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data / E.A. Moskovchenko, V.P. Kotlyarov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 386-395. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data / E.A. Moskovchenko, V.P. Kotlyarov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 386-395. — Бібліогр.: 10 назв. — англ. |
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| description | The long-time asymptotic behavior of the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of the stimulated Raman scattering is studied. Considered is the case of zero initial condition and single-phase boundary data. By using the steepest descent method for oscillatory matrix Riemann{Hilbert problems it is shown that the solution of the IBV problem has different asymptotic behavior in different regions
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Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 4, pp. 386–395
Long-Time Asymptotic Behavior of an Integrable Model
of the Stimulated Raman Scattering with Periodic
Boundary Data
E.A. Moskovchenko and V.P. Kotlyarov
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv, 61103, Ukraine
E-mail:kuznetsova@ilt.kharkov.ua;
kotlyarov@ilt.kharkov.ua
Received July 6, 2009
The long-time asymptotic behavior of the initial-boundary value (IBV)
problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations
of the stimulated Raman scattering is studied. Considered is the case of
zero initial condition and single-phase boundary data (peiωt). By using the
steepest descent method for oscillatory matrix Riemann–Hilbert problems
it is shown that the solution of the IBV problem has different asymptotic
behavior in different regions, namely:
• the selfsimilar vanishing (as t →∞) wave, when x > ω2t;
• the modulated elliptic wave of finite amplitude, when ω2
0t < x < ω2t;
• the plane wave of finite amplitude, when 0 < x < ω2
0t.
The similar results are true for the same IBV problem with nonzero initial
condition vanishing as t →∞.
Key words: nonlinear equations, Riemann–Hilbert problem, asymptotics.
Mathematics Subject Classification 2000: 37K15, 35Q15, 35B40.
1. Introduction
We consider the initial boundary value problem for integrable model of the
stimulated Raman scattering (SRS equations):
2iqt = µ, µx = 2iνq, νx = i(q̄µ− qµ̄), x ∈ (0,∞), t ∈ (0,∞), (1)
c© E.A. Moskovchenko and V.P. Kotlyarov, 2009
Long-Time Asymptotic Behavior of the SRS Model
with the vanishing (as x →∞) initial function and periodic boundary conditions:
q(x, 0) = u(x), µ(0, t) = peiωt, p > 0, ν(0, t) = l = const. (2)
Since (1) implies ∂
∂x
(
ν2(x, t) + |µ(x, t)|2
)
= 0, in what follows we assume that
ν2(x, t) + |µ(x, t)|2 ≡ 1 and, particularly, p2 + l2 = 1. For definiteness we assume
that p = |µ(0, t)| > 0 and ω > 0, while l < 0. The case ω < 0 l > 0 is obtained
by passing to the complex conjugated SRS equations.
The phenomenon of the stimulated Raman scattering is described by three
coupled PDEs [1]. Initial boundary value problems for these equations in the
domain x ∈ (0, L), t ∈ (0, T ) are well posed [2] for any L > 0 and T > 0.
The SRS equations (1) are integrable reduction of them in a special case of the
transient limit [1, 3]. In other words, the SRS equations admit the Lax pair,
and the inverse scattering transform can be applied. We will use the version
[4] of this transform when simultaneous spectral analysis of both the Lax equa-
tions is involved. The IBV problem for the SRS equations is a nice model of
PDEs, which can be solved by using the method of simultaneous spectral ana-
lysis and the matrix Riemann–Hilbert problem without a restriction caused by
the so-called global relation [4, 5] between spectral functions. Such a restriction
takes place for the most of integrable equations because the method [4] involves
more boundary values than in the corresponding well-posed IBV problem. Such
an overdetermination of the boundary data implies the mentioned above global
relation.
If q(x, t) is real and 2q = vx, µ = i sin v, ν = cos v, then the SRS equations
are reduced to the sine-Gordon equation: vxt = sin v. The long-time asymptotic
behavior of the rapidly decreasing (as |x| → ∞) solution of this equation was
studied in [6].
The IBV problem in the finite domain [0, L]x[0, T] was studied in [1], where
the long-distance behavior of the system was established via the third Painleve
transcendent. The problem in the finite domain was also considered in [7], where
rigorous analysis of the Riemann–Hilbert problem was done. In the present paper,
the IBV problem for the SRS equations is studied in the domain (x > 0, t > 0)
with zero initial function and simple periodic boundary data. The similar problem
with nonzero initial function, vanishing at infinity, was studied in [8]. Using the
steepest descent method of P. Deift and X. Zhou [9] for the oscillatory matrix RH
problem, introduced in [8], there was obtained the asymptotics of the solution
of the IBV problem in the form of a selfsimilar vanishing wave travelling in the
region x > ω2t. By using the ideas of [10] we obtained the explicit formula for
the asymptotics of the solution of the IBV problem in the complementary region
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 387
E.A. Moskovchenko and V.P. Kotlyarov
0 < x < ω2t. In the region ω2
0t < x < ω2t, where
ω2
0 =
−8l3ω2
27− 18l2 − l4 +
√
(1− l2)(9− l2)3
, −1 < l < 0,
the solution takes the form of a modulated elliptic wave of finite amplitude while
in the region 0 < x < ω2
0t it takes the form of a plane wave. To make the asymp-
totic analysis more transparent, we consider the case when the initial function
u(x) ≡ 0.
2. Riemann–Hilbert Problem
To formulate the Riemann–Hilbert problem, related to the IBV problem
(1)–(2), we introduce the spectral functions corresponding to initial and boundary
conditions. We consider the case u(x) ≡ 0. Therefore spectral functions are
defined by boundary data only. The boundary values µ(0, t) = peiωt and ν(0, t) =
l (p2 + l2 = 1) give the t-equation from the Lax pair:
E(t, k)
dt
=
i
4k
(
l ipeiωt
−ipe−iωt −l
)
E(t, k). (3)
We choose the solution of (3) in the form
E(t, k) =
1
2
eiωσ3t/2
κ(k) +
1
κ(k)
κ(k)− 1
κ(k)
κ(k)− 1
κ(k)
κ(k) +
1
κ(k)
e−iΩ(k)σ3t, (4)
where
κ(k) = 4
√
k − Ē
k − E
, Ω(k) =
ω
2k
X(k), X(k) :=
√
(k − E)(k − Ē),
and
E =
l + ip
2ω
= E1 + iE2, Ē = E1 − iE2.
To fix the branches of the roots, we choose the cut in the complex k-plane along
the curve γ ∪ γ̄, where Im Ω(k) = 0, and define κ(k) and Ω(k) in such a way that
κ(k) = 1 + O(k−1), Ω(k) =
ω
2
+ O(k−1) as k →∞.
The set Σ := {k ∈ C| ImΩ(k) = 0} (Fig. ) consists of the real line Im k = 0 and
the circle arc γ̂ = γ ∪ γ̄, which is defined by
(
k1 − |E|2
2E1
)2
+ k2
2 =
( |E|2
2E1
)2
, k2
1 + k2
2 ≥ |E|2.
388 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Long-Time Asymptotic Behavior of the SRS Model
Let us define the oriented contour Γ as follows: Γ = R ∪ γ ∪ γ̄. Denoting
Ω±(k), κ±(k) the boundary values of Ω(k), κ(k) on the cut γ̂ from the right (+)
and left (-) sides of the cut (see Fig. 1), we have
Ω+(k) = −Ω−(k), κ−(k) = iκ+(k).
The matrix-valued function E(t, k) in (4) is analytic in C \ {γ̂ ∪ {0}} and it has
an essential singularity at the point 0. The spectral data corresponding to the
boundary values µ(0, t) = peiωt and ν(0, t) = l (p2 + l2 = 1) are defined as follows:
A(k) =
1
2
(
κ(k) +
1
κ(k)
)
, B(k) =
1
2
(
κ(k)− 1
κ(k)
)
k ∈ C \ γ̂. (5)
Im k
Re k
E
0
4wl
1__
g
-
-
g
E
+
+
-
-
Fig. The set Σ and contour Γ.
Define the following functions:
ρ(k) :=
B(k)
A(k)
, k ∈ C \ γ̂; f(k) :=
i
A+(k)A−(k)
, k ∈ γ, (6)
where the sign ± denotes boundary value of A(k) from the left (+) and from the
right (−) of the arc γ̂ = γ ∪ γ̄.
Consider the matrix Riemann–Hilbert problem on the contour Γ proposed
in [8].
Find a 2x2 matrix-valued function M(x, t, k) such that:
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 389
E.A. Moskovchenko and V.P. Kotlyarov
• M(x, t, k) is sectionally analytic for k ∈ C \ Γ;
• M(x, t, k) is bounded in neighborhoods of the end points E and Ē;
• M(x, t, k) = I + O(k−1), k →∞;
• M(x, t, k) = m̃0(x, t) + O(k), k → 0;
• M−(x, t, k) = M+(x, t, k)J(x, t, k), k ∈ Γ, where
J(x, t, k) =
(
1 ρ(k)e−2itθ(k)
−ρ(k)e2itθ(k) 1− ρ2(k)
)
, k ∈ R,
(
1 0
f(k)e2itθ(k) 1
)
, k ∈ γ,
(
1 f(k)e−2itθ(k)
0 1
)
, k ∈ γ̄,
(7)
with θ(k) = θ(k, ξ) = 1/4k − k/4ξ2, ξ2 := t/4x.
Theorem 2.1. Let ρ(k) and f(k) be given as (6) and (5). Then the Riemann–
Hilbert problem (7) has a unique solution M(x, t, k). The functions q(x, t), µ(x, t)
and ν(x, t), defined by the equations
q(x, t) := 2i lim
k→∞
[kM(x, t, k)]12,
(
ν(x, t) iµ(x, t)
−iµ(x, t) −ν(x, t)
)
:= −M(x, t, 0)σ3M
−1(x, t, 0),
are the solution of the IBV problem (1)-(2) with zero initial function u(x).
This theorem is a corollary of the more general (u(x) 6= 0) theorem proved
in [8].
3. Asymptotic Behavior of the Solution of the IBV Problem
In this section we study the long-time asymptotic behavior of the solution to
the IBV problem (1)–(2). We show that there exist three different asymptotic
formulas which describe the long-time behavior of the solution q(x, t) of the IBV
problem in the three different regions of the domain x > 0, t > 0.
The asymptotics of the solution in the region x > ω2t was obtained in [8]:
390 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Long-Time Asymptotic Behavior of the SRS Model
Theorem 3.1. Let q(x, t), µ(x, t) and ν(x, t) be the solution of the IBV prob-
lem (1)–(2). Then in the region x > ω2t the solution takes the form
q(x, t) = 2
√
ξ3η(ξ)
t
exp
{
2i
√
xt− iη(ξ) log
√
xt + iϕ(ξ)
}
+2
√
ξ3η(−ξ)
t
exp
{
−2i
√
xt + iη(−ξ) log
√
xt + iϕ(−ξ)
}
+ o(t−1/2), t →∞,
where the functions η(k) and ϕ(k) are given by the equations:
η(k) =
1
2π
log
(
1− ρ2(k)
)
, ξ2 =
t
4x
,
ϕ(k) =
π
4
− 3η(k) log 2− arg ρ(k)− arg Γ(−iη(k)) +
1
π
ξ∫
−ξ
log |s− k|d log[1− ρ2(s)].
Here Γ(−iη(k)) is the Euler gamma-function, and ρ(k) =
κ2(k)− 1
κ2(k) + 1
.
In the region ω2
0t < x < ω2t the solution of the IBV problem takes the form of
a modulated elliptic wave. In this region we use the new phase function instead
of the function θ(k, ξ) (7)
g(k, ξ) =
k∫
E
+
k∫
Ē
(
1− λ−(ξ)
z
)(
1− λ+(ξ)
z
) √
(z − d(ξ))(z − d̄(ξ))
(z −E)(z − Ē)
dz
8ξ2
,
where real λ−(ξ), λ+(ξ) and complex d(ξ) = d1(ξ) + id2(ξ) parameters are deter-
mined by the equations
λ− = −λ+ + E1
1− ξ4λ−2
− λ−2
+
1− ξ4|E|2λ−3
− λ−3
+
, λ+ =
I0(λ−, d1, d2)
I−1(λ−, d1, d2)
,
where
d1 = E1 − λ− − λ+, d2 =
√
ξ4|E|2
λ2−λ2
+
− (E1 − λ− − λ+)2
and
Im(λ−, d1, d2) =
d1+id2∫
d1−id2
(
1− λ−
z
) √
(z − d1)2 + d2
2
(z − E1)2 + E2
2
dz
zm
, m = 0,−1.
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 391
E.A. Moskovchenko and V.P. Kotlyarov
Let w(k) =
√
(k − E)(k − Ē)(k − d(ξ))(k − d̄(ξ)) define the Riemann surface
and U(k) be the normalized Abelian integral
U(k) =
1
c
k∫
E
dz
w(z)
, c = 2
d∫
d̄
dz
w(z)
, τ =
2
c
d∫
E
dz
w(z)
.
Introduce the following Riemann theta functions:
Θ11(t, ξ, k) =
1
2
[
κ̃(k) +
1
κ̃(k)
]
θ3[U(k) + U(E−
0 )− τ/2− α(ξ)t− β(ξ)]
θ3[U(k) + U(E−
0 )− 1/2− τ/2]
,
Θ12(t, ξ, k) =
1
2
[
κ̃(k)− 1
κ̃(k)
]
θ3[U(k)− U(E−
0 ) + τ/2 + α(ξ)t + β(ξ)]
θ3[U(k)− U(E−
0 ) + 1/2 + τ/2]
,
Θ22(t, ξ, k) =
1
2
[
κ̃(k) +
1
κ̃(k)
]
θ3[U(k) + U(E−
0 ) + τ/2 + α(ξ)t + β(ξ)]
θ3[U(k) + U(E−
0 ) + 1/2 + τ/2]
.
Here
θ3(z) =
∑
m∈Z
eπiτm2+2πimz, Im τ = Im τ(ξ) > 0,
is theta function. The branch of κ̃(k) =
[
(k − Ē)(k − d̄(ξ))
(k −E)(k − d(ξ))
]1/4
is fixed by its
asymptotics
κ̃(k) = 1− d2(ξ) + E2
2ik
+ O(k−2), k →∞.
The point E−
0 is the preimage of the complex number
E0 =
Ed(ξ)− Ēd̄(ξ)
E − Ē + d(ξ)− d̄(ξ)
on the second sheet of the Riemann surface of the function w(k). Parameters
α = α(ξ) and β = β(ξ) are periods of the normalized Abelian integrals of the
second kind g(k) and
ζ(k) =
1
2
k∫
E
+
k∫
Ē
z2 − e1z + e0
w(z)
dz,
i.e.,
α(ξ) =
1
π
d(ξ)∫
E
dg(k),
d(ξ)∫
d̄(ξ)
dg(k) = 0, β(ξ) =
1
π
d(ξ)∫
E
dζ(k),
d(ξ)∫
d̄(ξ)
dζ(k) = 0.
392 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Long-Time Asymptotic Behavior of the SRS Model
The definition of the Abelian integral ζ(k) implies
e1 =
E + Ē + d(ξ) + ¯d(ξ)
2
, e0 = −
d̄∫
d
(z2 − e1z)
dz
w(z)
d̄∫
d
dz
w(z)
−1
.
The large k expansion of ζ(k) is of the form
ζ(k) = k + ζ∞(ξ) + O(k−1), k →∞,
where
ζ∞ =
1
2
∞∫
E
+
∞∫
Ē
[
z2 − e1z + e0
w(z)
− 1
]
dz + E2
is a real function of ξ.
Theorem 3.2. If ω2
0t < x < ω2t, then for t → ∞ the solution of the IBV
problem (1)–(2) takes the form of a modulated elliptic wave:
q(x, t) =2i
Θ12(t, ξ,∞)
Θ11(t, ξ,∞)
exp[2itg∞(ξ)− 2iφ(ξ)] + O(t−1/2),
ν(x, t) =− 1 + 2
Θ11(t, ξ, 0)Θ22(t, ξ, 0)
Θ11(t, ξ,∞)Θ22(t, ξ,∞)
+ O(t−1/2),
µ(x, t) =2i
Θ11(t, ξ, 0)Θ12(t, ξ, 0)
Θ2
11(t, ξ,∞)
exp[2itg∞(ξ)− 2iφ(ξ)] + O(t−1/2),
where
g∞(ξ) =
∞∫
E
+
∞∫
Ē
×
[(
1− λ−(ξ)
z
)(
1− λ+(ξ)
z
)√
(z − d(ξ))(z − d̄(ξ))
(z − E)(z − Ē)
− 1
]
dz
8ξ2
− l
8ωξ2
,
and the phase shift φ(ξ) is defined by
φ(ξ) =
∫
γd∪γ̄d
(k − k0(ξ)) log
[
h(k)
δ2(k, ξ)
]
dk
2πw+(k)
, h(k) =
{
[A−(k)A+(k)]−1, k ∈ γd
A−(k)A+(k), k ∈ γ̄d;
δ(k) = exp
1
2πi
λ+(ξ)∫
λ−(ξ)
log(1− ρ2(s))ds
s− k
, k ∈ C\[λ−(ξ), λ+(ξ)], |E| ≤ ξ ≤ 1
2ω0
,
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 393
E.A. Moskovchenko and V.P. Kotlyarov
where A(k), ρ(k) are spectral functions (5), (6), and k0(ξ) = e1(ξ) + ζ∞(ξ),
λ±(ξ) are stationary points of the phase functions g(k, ξ), and γd (γ̄d) is an arc
connecting E and d(ξ) (Ē and d̄(ξ)), where Im g(k) = 0.
In the region 0 < x < ω2
0t the phase g-function takes the form:
ĝ(k, ξ) :=
(
ω
4k
+
1
4ξ2
) √
(k −E)(k − Ē).
This function allows us to prove the following
Theorem 3.3. The solution of the IBV problem (1)-(2) for t →∞ takes the
form of a plane wave:
q(x, t) = − p
2ω
exp
[
iωt− i
l
ω
x− 2iφ̂(ξ)
]
+ O(t−1/2),
µ(x, t) = p exp
[
iωt− i
l
ω
x− 2iφ̂(ξ)
]
+ O(t−1/2),
ν(x, t) = l + O(t−1/2),
where
φ̂(ξ) =
1
2π
λ−(ξ)∫
−∞
+
∞∫
λ+(ξ)
log A2(k)
dk
X(k)
,
λ±(ξ) are the stationary points of the function ĝ(k, ξ) (
1
2ω0
≤ ξ ≤ ∞), and A(k)
is defined by (5). Here the contour γg∪ γ̄g connects E and Ē along the arc, where
Im ĝ(k) = 0.
R e m a r k 3.1. If x = 0 (ξ = ∞), then λ−(∞) = −∞ and λ+(∞) = +∞.
Hence
φ̂(∞) = 0.
Therefore the plane wave µ(0, t) and ν(0, t) match with the boundary conditions.
Since g∞(ξ0) = ĝ∞(ξ0) = ω/2 − l/8ωξ2
0, φ(ξ0) = φ̂(ξ0), Im d(ξ0) = 0,where
ξ0 =
1
2ω0
, and theta-function θ3(∗)|ξ=ξ0 ≡ 1, we have that elliptic waves match
with the plane waves at ξ = ξ0.
394 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Long-Time Asymptotic Behavior of the SRS Model
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Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 395
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| id | nasplib_isofts_kiev_ua-123456789-106550 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T18:10:00Z |
| publishDate | 2009 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Moskovchenko, E.A. Kotlyarov, V.P. 2016-09-30T08:24:47Z 2016-09-30T08:24:47Z 2009 Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data / E.A. Moskovchenko, V.P. Kotlyarov // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 386-395. — Бібліогр.: 10 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106550 The long-time asymptotic behavior of the initial-boundary value (IBV) problem in the quarter plane (x > 0, t > 0) for nonlinear integrable equations of the stimulated Raman scattering is studied. Considered is the case of zero initial condition and single-phase boundary data. By using the steepest descent method for oscillatory matrix Riemann{Hilbert problems it is shown that the solution of the IBV problem has different asymptotic behavior in different regions en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data Article published earlier |
| spellingShingle | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data Moskovchenko, E.A. Kotlyarov, V.P. |
| title | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data |
| title_full | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data |
| title_fullStr | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data |
| title_full_unstemmed | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data |
| title_short | Long-Time Asymptotic Behavior of an Integrable Model of the Stimulated Raman Scattering with Periodic Boundary Data |
| title_sort | long-time asymptotic behavior of an integrable model of the stimulated raman scattering with periodic boundary data |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106550 |
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