Modulational instability in nonlocal Kerr-type media with random parameters
E.V. Doktorov∗ and M.A. Molchan†
B.I. Stepanov Institute of Physics, 68 F. Skaryna Ave., 220072 Minsk, Belarus
Modulational instability of continuous waves in nonlocal focusing and defocusing Kerr media with
stochastically varying diffraction (dispersion) and nonlinearity coefficients is studied both analyt-
ically and numerically. It is shown that nonlocality with the sign-definite Fourier images of the
medium response functions suppresses considerably the growth rate peak and bandwidth of insta-
bility caused by stochasticity. Contrary, nonlocality can enhance modulational instability growth
for a response function with negative-sign bands.
PACS numbers: 42.25.Dd, 42.70.-a, 42.65.Jx
I. INTRODUCTION
Modulational instability (MI) in nonlinear media is a
destabilization mechanism which produces a self-induced
breakup of an initially continuous wave into localized
(solitary wave) structures. This phenomenon was pre-
dicted in plasma [1, 2], nonlinear optics [3, 4], fluids [5]
and atomic Bose-Einstein condensates [6, 7, 8]. MI of
continuous waves can be used to generate ultra-high
repetition-rate trains of soliton-like pulses [9, 10, 11]. It is
common knowledge that MI is absent in the defocusing
Kerr medium and presents as the long-wave instability
with a finite bandwidth in the focusing Kerr medium [12].
The above results were obtained for media with de-
terministic parameters. Contrary, in realistic media the
characteristic parameters are not constants, as a rule,
but fluctuate randomly around their mean values. It was
shown in the setting of nonlinear optics that stochastic
inhomogeneities in a Kerr-type medium extend the do-
main of MI of continuous waves, as compared with deter-
ministic systems, over the whole spectrum of modulation
wavenumbers, even for the defocusing regime [13, 14, 15].
A comprehensive review of MI of electromagnetic waves
in inhomogeneous and in discrete media is given in
Ref. [16].
Another important aspect of a class of realistic nonlin-
ear media is concerned with their nonlocality. Nonlocal-
ity is typically a result of underlying transport processes
such as heat conduction in thermal nonlinear media [17],
diffusion of atoms in a gas [18], long-range electrostatic
interaction in liquid crystals [19], charge carrier transfer
in photorefractive crystals [20, 21], and many-body in-
teraction in Bose-Einstein condensates [22]. Nonlocality
can prevent the collapse of self-focused beams [23, 24] and
dramatically alter interaction between dark solitons [25].
MI in deterministic nonlocal Kerr-type media was stud-
ied in Refs. [26, 27], and it was shown that nonlocality
does not produce MI in the defocusing case for small and
moderate values of the product “modulation amplitude
∗Electronic address: doktorov@dragon.bas-net.by
†Electronic address: m.moltschan@dragon.bas-net.by
× nonlocality parameter”.
In the present paper we unite the two above lines of
study of nonlinear media and analyze MI in nonlocal me-
dia with stochastic parameters. Since nonlocality spreads
out localized excitations, it is reasonable to expect a par-
tial suppression of the stochasticity-induced MI gain. In-
deed, we demonstrate that the aforementioned situation
with MI in local stochastic media with the sign-definite
Fourier images of the response functions changes drasti-
cally, if nonlocality is taken into account. Namely, both
the growth rate peaks and bandwidths of instability are
considerably decreased. On the other hand, there can be
an “anomalous” behavior of nonlocality when the Fourier
image of the response function of a nonlocal medium al-
lows for sign-negative bands. In this case the MI gain of
a nonlocal medium can exceed that of a local stochastic
medium for some values of the modulation wavenumber.
We adopt the nonlocal nonlinear Schrödinger equation
with random coefficients as a model to reveal peculiar-
ities of MI of continuous waves. The results obtained
are illustrated by the white noise model for parameter
fluctuations and by response functions of several types.
II. MODEL
The propagation of an optical beam along the z axis in
a nonlocal medium with random parameters is governed
by the nonlinear Schrödinger equation
iuz +
d(z)uxx + g(z)u
dx′R(x− x′)|u|2(x′, z) = 0.
(2.1)
Here x is the transverse coordinate, u(x, z) is the complex
envelope amplitude and we use the standard dimension-
less variables. The group velocity dispersion (or diffrac-
tion) coefficient d(z) and nonlinearity coefficient g(z) are
considered as stochastic functions which fluctuate around
their mean values d0 (d0 > 0) and g0 (g0 ≷ 0):
d(z) = d0(1 +md(z)), g(z) = g0(1 +mg(z)). (2.2)
Here md and mg are independent zero-mean random pro-
cesses of the Gaussian white-noise type,
〈md〉 = 〈mg〉 = 0, 〈md(z)md(z′)〉 = 2σ2dδ(z − z′),
http://arxiv.org/abs/0704.1093v1
mailto:doktorov@dragon.bas-net.by
mailto:m.moltschan@dragon.bas-net.by
〈mg(z)mg(z′)〉 = 2σ2gδ(z − z′)〉,
and the angle brackets stand for the expectation with
respect to the distribution of the processes md(z) and
mg(z). The integral in equation (2.1) represents the field-
intensity dependent change of the refractive index char-
acterized by the normalized symmetric response function
R(x),
dxR(x) = 1. The delta-function response
function R(x) = δ(x) corresponds to the local limit of
the model. We will discriminate between the focusing
(g0 > 0) and defocusing (g0 < 0) media.
Eq. (2.1) possesses the homogeneous plane wave solu-
u0 = A exp
dz′g(z′)
, (2.3)
where A is a real amplitude. Now we perform the linear
stability analysis of the solution (2.3). Assume that
u(x, z) = (A+ v(x, z)) exp
dz′g(z′)
(2.4)
is a perturbed solution of Eq. (2.1) with v(x, z) being a
small complex modulation. Substituting Eq. (2.4) into
Eq. (2.1) and linearizing about the plane wave (2.3), we
get a linear equation for v(x, z):
ivz +
d(z)vxx + 2g(z)A
dx′R(x− x′)Re v(x′, z) = 0.
(2.5)
After decomposing v into real and imaginary parts, v =
r(x, z) + is(x, z), and performing the Fourier transforms
ρ(k, z) =
dx r(x, z)eikx,
σ(k, z) =
dx s(x, z)eikx,
R̂(k) =
dxR(x)eikx,
Eq. (2.5) is converted to a system of linear equations for
ρ and σ:
d(z)k2
d(z)k2 + 2g(z)A2R̂ 0
(2.6)
If we were deal with the deterministic system with the
parameters d0 and g0, Eq. (2.6) would be the main object
to study MI [26]. However, MI induced by the random
fluctuations is not captured by the analysis of the first
moments 〈ρ〉 and 〈σ〉 [15], and it is necessary to compute
the modulational intensity growth given by the higher-
order moments.
III. THE SECOND-ORDER MOMENT MI GAIN
We consider the second moments 〈ρ2〉, 〈ρσ〉 and 〈σ2〉
as constituents of the column vector
X(2) =
〈ρ2〉, 〈ρσ〉, 〈σ2〉
. (3.1)
The moment 〈ρσ〉 is added to close the equations for
the second-order moments. Then we should calculate z-
evolution of the vector X(2). Its first component gives
〈ρ2〉 = 2〈ρzρ〉 = d0k2〈ρσ〉 + d0k2〈md(z)ρσ〉,
in accordance with Eqs. (2.2) and (2.6). For decoupling
of the mean 〈md(z)ρσ〉 we apply the Furutsu-Novikov
formula [28, 29]
〈md(z)ρσ〉 =
dyσ2dB(z − y)
. (3.2)
Here B(z − y) = δ(z − y) for the white-noise Gaussian
random process, while the functional derivative (δ/δmd)
is calculated from Eq. (2.6). Indeed, writing ρ(z) as the
integral ρ(z) = (1/2)k2
dyd(y)σ(y) (and the similar
integral for σ(z)) and accounting for the explicit repre-
sentation (2.2) of d(z) in terms of md gives
δ(ρσ)
σ + ρ
2(σ2 − ρ2).
Therefore, 〈md(z)ρσ〉 = (1/2)σ2dd0k2(〈σ2〉 − 〈ρ2〉) and
finally
〈ρ2〉 = d0k2〈ρσ〉+
4(〈σ2〉 − 〈ρ2〉).
Just in the same way we can calculate z-derivatives of
the other components of the vector X(2). As a result, we
obtain the evolution equation (d/dz)X(2) = M (2)X(2)
with the 3× 3 matrix M (2) of the form
M (2) =
4 d0k
2R̂ − 1
2 −σ2dd20k4
16σ2gg
4R̂2 + σ2dd
2R̂− 1
. (3.3)
Eigenvalues of M (2) with positive real parts lead to in-
stabilities, and the largest positive value determines the
MI gain G2(k). The eigenvalues λj are easily found from
Eq. (3.3) but they are too cumbersome to be reproduced
here explicitly. Below we separately analyze the cases of
the defocusing (g0 < 0) and focusing (g0 > 0) nonlin-
earities. Following [26], we will use for illustration the
Gaussian response function
RG(x) =
, R̂G(k) = exp
(3.4)
and the exponential one
Re(x) =
, R̂e(k) =
1 + a2k2
(3.5)
as examples of the response functions with the sign-
definite Fourier images, as well as the rectangular re-
sponse function
Rr(x) =
for |x| ≤ a,
0 for |x| > a,
R̂r(k) =
sin(ak)
(3.6)
whose Fourier transform has negative-sign bands. Here a
is the nonlocality parameter, a → 0 means R(x) → δ(x)
and R̂(k) → 1.
A. Defocusing nonlinearity
For the defocusing nonlinearity g0 < 0 we obtain one
real eigenvalue λ1 and two complex conjugate ones λ2
and λ3. Numerical analysis shows that λ1 is positive for
all k2, while λ2 and λ3 have negative real parts for the
Gaussian and exponential response functions. Let us re-
mind that there is no MI for g0 < 0 for local deterministic
Kerr media, while randomness of the coefficients d(z) and
g(z) completely destroys stability of the continuous wave
solution. This situation considerably changes for nonlo-
cal media. Indeed, Fig. 1 clearly shows that nonlocality
with the sign-definite response functions suppresses both
the growth rate peak of G2(k) ≡ λ1 and MI bandwidth,
the latter being practically finite. When the nonlocality
parameter a grows, the suppression effect becomes more
pronounced. Somewhat different situation takes place
0 1 2 3 4
0 1 2 3 4
FIG. 1: Defocusing media. Plots of the MI gain G2(k) for
local stochastic medium (solid line), nonlocal stochastic media
with the Gaussian (dash-dotted line) and exponential (dashed
line) response functions. Here d0 = 2, |g0|A2 = 1, σ2d = σ2g =
0.1. Upper panel: a = 1; lower panel: a =
for the rectangular response function (3.6). For suffi-
ciently high nonlocality, MI gain maximum for a given
wavenumber k can exceed the corresponding value of G2
for a local random medium (Fig. 2). Besides, the MI
bandwidth becomes strictly finite in this limit.
B. Focusing nonlinearity
In the case of the focusing nonlinearity (g0 > 0) a local
deterministic medium produces the long-wave instability
with a finite bandwidth. Stochasticity of medium param-
eters extends the bandwidth to the whole spectrum of
modulation wavenumbers. Calculation of eigenvalues of
the matrix M2 (3.3) with g0 > 0 demonstrates that non-
locality suppresses the MI gain and bandwidth for media
with both sigh-definite (Fig. 3) and sign-indefinite (Fig.
0 1 2 3 4
FIG. 2: Defocusing media with the rectangular response func-
tion. Plots of the MI gain G2(k) for local stochastic medium
(solid line), nonlocal stochastic media with a = 2 (dashed
line), a = 6 (dash-dotted line), and a = 10 (dotted line).
Other parameters are the same as in Fig. 1.
0 1 2 3 4
0 1 2 3 4
FIG. 3: Focusing media. Plots of the MI gain G2(k) for a lo-
cal deterministic medium (solid line), local stochastic medium
(dotted line), nonlocal stochastic media with the Gaussian
(dash-dotted line) and exponential (dashed line) response
functions. Here d0 = 2, g0A
2 = 1, σ2d = 0.1, σ
g = 0.2.
Upper panel: a = 1; lower panel: a = 2.5.
4) response functions. Notice that stronger nonlocality
is needed for focusing media to achieve a reduction of
the MI gain, as compared with defocusing ones. Besides,
maximum positions of the MI gains shift toward smaller
wavenumbers k under nonlocality growth, producing fi-
nite bandwidth.
0 0.5 1 1.5 2 2.5 3
FIG. 4: Focusing media. Plots of the MI gain G2(k) for a local
deterministic medium (solid line), local stochastic medium
(dotted line), nonlocal stochastic media with the rectangular
response function: a = 2 (dash-dotted line), a = 8 (dashed
line). Other parameters are the same as in Fig. 3.
IV. HIGHER-ORDER MOMENTS
The second-order moments (3.1) do not provide an
analysis of the MI gain in stochastic media with sufficient
detail. In particular, it is important to see fluctuations
of the exponential growth of the modulation amplitude.
More deep insight into the problem demands to account
for higher-order moments
X(2n) =
< ρ(2n−j)σj >
, j = 0, . . . , 2n. (4.1)
In this section we study the interplay of nonlocality and
exponential growth of the higher momentsX(2n) in virtue
of stochasticity. As before, applying the Furutsu-Novikov
formula (3.2), we obtain a matrix M (2n) in the form
M (2n) = d0k
2A(2n) +
2R̂ (4.2)
B(2n) + d20k
4σ2dC
(2n) + 16g20A
4R̂2σ2gD
(2n).
Non-zero entries of the matrices A(2n), B(2n), C(2n) and
D(2n) are written as
j,j+1 = n−
j,j−1 = j; C
jj = −
(n+2nj−j2),
j,j+2 =
j + 1
, (4.3)
j,j−2 = D
j,j−2 =
j(j − 1), j = 0, . . . , 2n.
Then the maximal real part of roots of the characteristic
polynomial det |M (2n) − λI| will give nG2n(k). Since all
the matrix elements of M (2n) are real and the character-
istic polynomial is of the odd degree, at least one of the
eigenvalues of M (2n) is real and the others are mutually
complex conjugate. In what follows we will consider the
4-th and 6-th moments.
A. Defocusing nonlinearity
In Fig. 5 we show the results of calculating MI gains
G2, G4 and G6 for both the exponential and Gaussian re-
sponse functions and compare them with the same curves
for local stochastic media obtained in [15]. It is seen
that nonlocality suppresses the higher-order moments as
well. Notice that in defocusing media positions of MI
gain maxima for moments of different orders coincide [15]
(they are deterministic rather than random). Nonlocality
does not disturb this property. Fig. 6 demonstrate simi-
lar curves for the rectangular response function for differ-
ent values of the nonlocality parameter a. It is seen that
for sufficiently high a the medium demonstrates prac-
tically coinciding distributions of higher-moment growth
rates, their maxima being shifted to shorter wavelengths.
Evidently, higher-order moments for the rectangular re-
sponse function manifest the same “anomalous” enhance-
ment of the growth rate in a narrow region of modulation
wavenumbers, as compared with the local stochastic case.
0 1 2 3 4
0 1 2 3 4
FIG. 5: Defocusing media. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for a lo-
cal stochastic medium (upper three curves) and for nonlocal
stochastic media (lower three curves). Here d0 = 2, a
2 = 1,
|g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: exponential re-
sponse function; lower panel: Gaussian response function.
B. Focusing nonlinearity
For the focusing media the higher-order MI gains
demonstrate much the same behavior as for the defo-
cusing ones for both sign-definite and sign-indefinite re-
0 1 2 3 4 5
0 1 2 3
0 0.5 1 1.5
FIG. 6: Defocusing nonlocal stochastic media with the rect-
angular response function. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for d0 = 2,
|g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: a = 2; middle panel:
a = 6; lower panel: a = 8.
sponse functions. Fig. 7 shows the MI gains for local
and nonlocal stochastic media for the Gaussian response
function. Curves for the exponential and rectangular re-
sponse functions are qualitatively the same. With in-
creasing the nonlocality parameter a, curves for MI gains
of different orders become closer one the other, so high
nonlocality smoothes fluctuations of the modulation am-
plitude growth.
0 0.5 1 1.5 2 2.5 3
FIG. 7: Focusing media. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for a lo-
cal stochastic medium (upper three curves) and for nonlo-
cal stochastic medium with the Gaussian response function
(lower three curves). Here d0 = 2, a = 2, g0A
2 = 1,
d = σ
g = 0.1.
V. CONCLUSION
Within the limits of the linear stability analysis, we
have investigated the MI of a homogeneous wave in a
nonlocal nonlinear Kerr-type medium with random pa-
rameters. For the case of the white-noise model of pa-
rameter fluctuations, we derived the equations which gov-
ern the dependence of the MI gain on the modulation
wavenumber. As was expected from physical motiva-
tions, nonlocality causes considerable suppression of the
stochasticity-induced MI growth rate for media with the
sign-definite Fourier images of the response functions. At
the same time, nonlocal media with the sign-indefinite
Fourier images of the response functions can display a
somewhat different behavior leading to an increase, as
compared with local media, of the MI gain for some do-
mains of modulation wavenumbers.
Acknowledgments
The authors are very grateful to F. Abdullaev and J.
Garnier for constructive comments.
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  Modulational instability of continuous waves in nonlocal focusing and
defocusing Kerr media with stochastically varying diffraction (dispersion) and
nonlinearity coefficients is studied both analytically and numerically. It is
shown that nonlocality with the sign-definite Fourier images of the medium
response functions suppresses considerably the growth rate peak and bandwidth
of instability caused by stochasticity. Contrary, nonlocality can enhance
modulational instability growth for a response function with negative-sign
bands.

Modulational instability in nonlocal Kerr-type media with random parameters
E.V. Doktorov∗ and M.A. Molchan†
B.I. Stepanov Institute of Physics, 68 F. Skaryna Ave., 220072 Minsk, Belarus
Modulational instability of continuous waves in nonlocal focusing and defocusing Kerr media with
stochastically varying diffraction (dispersion) and nonlinearity coefficients is studied both analyt-
ically and numerically. It is shown that nonlocality with the sign-definite Fourier images of the
medium response functions suppresses considerably the growth rate peak and bandwidth of insta-
bility caused by stochasticity. Contrary, nonlocality can enhance modulational instability growth
for a response function with negative-sign bands.
PACS numbers: 42.25.Dd, 42.70.-a, 42.65.Jx
I. INTRODUCTION
Modulational instability (MI) in nonlinear media is a
destabilization mechanism which produces a self-induced
breakup of an initially continuous wave into localized
(solitary wave) structures. This phenomenon was pre-
dicted in plasma [1, 2], nonlinear optics [3, 4], fluids [5]
and atomic Bose-Einstein condensates [6, 7, 8]. MI of
continuous waves can be used to generate ultra-high
repetition-rate trains of soliton-like pulses [9, 10, 11]. It is
common knowledge that MI is absent in the defocusing
Kerr medium and presents as the long-wave instability
with a finite bandwidth in the focusing Kerr medium [12].
The above results were obtained for media with de-
terministic parameters. Contrary, in realistic media the
characteristic parameters are not constants, as a rule,
but fluctuate randomly around their mean values. It was
shown in the setting of nonlinear optics that stochastic
inhomogeneities in a Kerr-type medium extend the do-
main of MI of continuous waves, as compared with deter-
ministic systems, over the whole spectrum of modulation
wavenumbers, even for the defocusing regime [13, 14, 15].
A comprehensive review of MI of electromagnetic waves
in inhomogeneous and in discrete media is given in
Ref. [16].
Another important aspect of a class of realistic nonlin-
ear media is concerned with their nonlocality. Nonlocal-
ity is typically a result of underlying transport processes
such as heat conduction in thermal nonlinear media [17],
diffusion of atoms in a gas [18], long-range electrostatic
interaction in liquid crystals [19], charge carrier transfer
in photorefractive crystals [20, 21], and many-body in-
teraction in Bose-Einstein condensates [22]. Nonlocality
can prevent the collapse of self-focused beams [23, 24] and
dramatically alter interaction between dark solitons [25].
MI in deterministic nonlocal Kerr-type media was stud-
ied in Refs. [26, 27], and it was shown that nonlocality
does not produce MI in the defocusing case for small and
moderate values of the product “modulation amplitude
∗Electronic address: doktorov@dragon.bas-net.by
†Electronic address: m.moltschan@dragon.bas-net.by
× nonlocality parameter”.
In the present paper we unite the two above lines of
study of nonlinear media and analyze MI in nonlocal me-
dia with stochastic parameters. Since nonlocality spreads
out localized excitations, it is reasonable to expect a par-
tial suppression of the stochasticity-induced MI gain. In-
deed, we demonstrate that the aforementioned situation
with MI in local stochastic media with the sign-definite
Fourier images of the response functions changes drasti-
cally, if nonlocality is taken into account. Namely, both
the growth rate peaks and bandwidths of instability are
considerably decreased. On the other hand, there can be
an “anomalous” behavior of nonlocality when the Fourier
image of the response function of a nonlocal medium al-
lows for sign-negative bands. In this case the MI gain of
a nonlocal medium can exceed that of a local stochastic
medium for some values of the modulation wavenumber.
We adopt the nonlocal nonlinear Schrödinger equation
with random coefficients as a model to reveal peculiar-
ities of MI of continuous waves. The results obtained
are illustrated by the white noise model for parameter
fluctuations and by response functions of several types.
II. MODEL
The propagation of an optical beam along the z axis in
a nonlocal medium with random parameters is governed
by the nonlinear Schrödinger equation
iuz +
d(z)uxx + g(z)u
dx′R(x− x′)|u|2(x′, z) = 0.
(2.1)
Here x is the transverse coordinate, u(x, z) is the complex
envelope amplitude and we use the standard dimension-
less variables. The group velocity dispersion (or diffrac-
tion) coefficient d(z) and nonlinearity coefficient g(z) are
considered as stochastic functions which fluctuate around
their mean values d0 (d0 > 0) and g0 (g0 ≷ 0):
d(z) = d0(1 +md(z)), g(z) = g0(1 +mg(z)). (2.2)
Here md and mg are independent zero-mean random pro-
cesses of the Gaussian white-noise type,
〈md〉 = 〈mg〉 = 0, 〈md(z)md(z′)〉 = 2σ2dδ(z − z′),
http://arxiv.org/abs/0704.1093v1
mailto:doktorov@dragon.bas-net.by
mailto:m.moltschan@dragon.bas-net.by
〈mg(z)mg(z′)〉 = 2σ2gδ(z − z′)〉,
and the angle brackets stand for the expectation with
respect to the distribution of the processes md(z) and
mg(z). The integral in equation (2.1) represents the field-
intensity dependent change of the refractive index char-
acterized by the normalized symmetric response function
R(x),
dxR(x) = 1. The delta-function response
function R(x) = δ(x) corresponds to the local limit of
the model. We will discriminate between the focusing
(g0 > 0) and defocusing (g0 < 0) media.
Eq. (2.1) possesses the homogeneous plane wave solu-
u0 = A exp
dz′g(z′)
, (2.3)
where A is a real amplitude. Now we perform the linear
stability analysis of the solution (2.3). Assume that
u(x, z) = (A+ v(x, z)) exp
dz′g(z′)
(2.4)
is a perturbed solution of Eq. (2.1) with v(x, z) being a
small complex modulation. Substituting Eq. (2.4) into
Eq. (2.1) and linearizing about the plane wave (2.3), we
get a linear equation for v(x, z):
ivz +
d(z)vxx + 2g(z)A
dx′R(x− x′)Re v(x′, z) = 0.
(2.5)
After decomposing v into real and imaginary parts, v =
r(x, z) + is(x, z), and performing the Fourier transforms
ρ(k, z) =
dx r(x, z)eikx,
σ(k, z) =
dx s(x, z)eikx,
R̂(k) =
dxR(x)eikx,
Eq. (2.5) is converted to a system of linear equations for
ρ and σ:
d(z)k2
d(z)k2 + 2g(z)A2R̂ 0
(2.6)
If we were deal with the deterministic system with the
parameters d0 and g0, Eq. (2.6) would be the main object
to study MI [26]. However, MI induced by the random
fluctuations is not captured by the analysis of the first
moments 〈ρ〉 and 〈σ〉 [15], and it is necessary to compute
the modulational intensity growth given by the higher-
order moments.
III. THE SECOND-ORDER MOMENT MI GAIN
We consider the second moments 〈ρ2〉, 〈ρσ〉 and 〈σ2〉
as constituents of the column vector
X(2) =
〈ρ2〉, 〈ρσ〉, 〈σ2〉
. (3.1)
The moment 〈ρσ〉 is added to close the equations for
the second-order moments. Then we should calculate z-
evolution of the vector X(2). Its first component gives
〈ρ2〉 = 2〈ρzρ〉 = d0k2〈ρσ〉 + d0k2〈md(z)ρσ〉,
in accordance with Eqs. (2.2) and (2.6). For decoupling
of the mean 〈md(z)ρσ〉 we apply the Furutsu-Novikov
formula [28, 29]
〈md(z)ρσ〉 =
dyσ2dB(z − y)
. (3.2)
Here B(z − y) = δ(z − y) for the white-noise Gaussian
random process, while the functional derivative (δ/δmd)
is calculated from Eq. (2.6). Indeed, writing ρ(z) as the
integral ρ(z) = (1/2)k2
dyd(y)σ(y) (and the similar
integral for σ(z)) and accounting for the explicit repre-
sentation (2.2) of d(z) in terms of md gives
δ(ρσ)
σ + ρ
2(σ2 − ρ2).
Therefore, 〈md(z)ρσ〉 = (1/2)σ2dd0k2(〈σ2〉 − 〈ρ2〉) and
finally
〈ρ2〉 = d0k2〈ρσ〉+
4(〈σ2〉 − 〈ρ2〉).
Just in the same way we can calculate z-derivatives of
the other components of the vector X(2). As a result, we
obtain the evolution equation (d/dz)X(2) = M (2)X(2)
with the 3× 3 matrix M (2) of the form
M (2) =
4 d0k
2R̂ − 1
2 −σ2dd20k4
16σ2gg
4R̂2 + σ2dd
2R̂− 1
. (3.3)
Eigenvalues of M (2) with positive real parts lead to in-
stabilities, and the largest positive value determines the
MI gain G2(k). The eigenvalues λj are easily found from
Eq. (3.3) but they are too cumbersome to be reproduced
here explicitly. Below we separately analyze the cases of
the defocusing (g0 < 0) and focusing (g0 > 0) nonlin-
earities. Following [26], we will use for illustration the
Gaussian response function
RG(x) =
, R̂G(k) = exp
(3.4)
and the exponential one
Re(x) =
, R̂e(k) =
1 + a2k2
(3.5)
as examples of the response functions with the sign-
definite Fourier images, as well as the rectangular re-
sponse function
Rr(x) =
for |x| ≤ a,
0 for |x| > a,
R̂r(k) =
sin(ak)
(3.6)
whose Fourier transform has negative-sign bands. Here a
is the nonlocality parameter, a → 0 means R(x) → δ(x)
and R̂(k) → 1.
A. Defocusing nonlinearity
For the defocusing nonlinearity g0 < 0 we obtain one
real eigenvalue λ1 and two complex conjugate ones λ2
and λ3. Numerical analysis shows that λ1 is positive for
all k2, while λ2 and λ3 have negative real parts for the
Gaussian and exponential response functions. Let us re-
mind that there is no MI for g0 < 0 for local deterministic
Kerr media, while randomness of the coefficients d(z) and
g(z) completely destroys stability of the continuous wave
solution. This situation considerably changes for nonlo-
cal media. Indeed, Fig. 1 clearly shows that nonlocality
with the sign-definite response functions suppresses both
the growth rate peak of G2(k) ≡ λ1 and MI bandwidth,
the latter being practically finite. When the nonlocality
parameter a grows, the suppression effect becomes more
pronounced. Somewhat different situation takes place
0 1 2 3 4
0 1 2 3 4
FIG. 1: Defocusing media. Plots of the MI gain G2(k) for
local stochastic medium (solid line), nonlocal stochastic media
with the Gaussian (dash-dotted line) and exponential (dashed
line) response functions. Here d0 = 2, |g0|A2 = 1, σ2d = σ2g =
0.1. Upper panel: a = 1; lower panel: a =
for the rectangular response function (3.6). For suffi-
ciently high nonlocality, MI gain maximum for a given
wavenumber k can exceed the corresponding value of G2
for a local random medium (Fig. 2). Besides, the MI
bandwidth becomes strictly finite in this limit.
B. Focusing nonlinearity
In the case of the focusing nonlinearity (g0 > 0) a local
deterministic medium produces the long-wave instability
with a finite bandwidth. Stochasticity of medium param-
eters extends the bandwidth to the whole spectrum of
modulation wavenumbers. Calculation of eigenvalues of
the matrix M2 (3.3) with g0 > 0 demonstrates that non-
locality suppresses the MI gain and bandwidth for media
with both sigh-definite (Fig. 3) and sign-indefinite (Fig.
0 1 2 3 4
FIG. 2: Defocusing media with the rectangular response func-
tion. Plots of the MI gain G2(k) for local stochastic medium
(solid line), nonlocal stochastic media with a = 2 (dashed
line), a = 6 (dash-dotted line), and a = 10 (dotted line).
Other parameters are the same as in Fig. 1.
0 1 2 3 4
0 1 2 3 4
FIG. 3: Focusing media. Plots of the MI gain G2(k) for a lo-
cal deterministic medium (solid line), local stochastic medium
(dotted line), nonlocal stochastic media with the Gaussian
(dash-dotted line) and exponential (dashed line) response
functions. Here d0 = 2, g0A
2 = 1, σ2d = 0.1, σ
g = 0.2.
Upper panel: a = 1; lower panel: a = 2.5.
4) response functions. Notice that stronger nonlocality
is needed for focusing media to achieve a reduction of
the MI gain, as compared with defocusing ones. Besides,
maximum positions of the MI gains shift toward smaller
wavenumbers k under nonlocality growth, producing fi-
nite bandwidth.
0 0.5 1 1.5 2 2.5 3
FIG. 4: Focusing media. Plots of the MI gain G2(k) for a local
deterministic medium (solid line), local stochastic medium
(dotted line), nonlocal stochastic media with the rectangular
response function: a = 2 (dash-dotted line), a = 8 (dashed
line). Other parameters are the same as in Fig. 3.
IV. HIGHER-ORDER MOMENTS
The second-order moments (3.1) do not provide an
analysis of the MI gain in stochastic media with sufficient
detail. In particular, it is important to see fluctuations
of the exponential growth of the modulation amplitude.
More deep insight into the problem demands to account
for higher-order moments
X(2n) =
< ρ(2n−j)σj >
, j = 0, . . . , 2n. (4.1)
In this section we study the interplay of nonlocality and
exponential growth of the higher momentsX(2n) in virtue
of stochasticity. As before, applying the Furutsu-Novikov
formula (3.2), we obtain a matrix M (2n) in the form
M (2n) = d0k
2A(2n) +
2R̂ (4.2)
B(2n) + d20k
4σ2dC
(2n) + 16g20A
4R̂2σ2gD
(2n).
Non-zero entries of the matrices A(2n), B(2n), C(2n) and
D(2n) are written as
j,j+1 = n−
j,j−1 = j; C
jj = −
(n+2nj−j2),
j,j+2 =
j + 1
, (4.3)
j,j−2 = D
j,j−2 =
j(j − 1), j = 0, . . . , 2n.
Then the maximal real part of roots of the characteristic
polynomial det |M (2n) − λI| will give nG2n(k). Since all
the matrix elements of M (2n) are real and the character-
istic polynomial is of the odd degree, at least one of the
eigenvalues of M (2n) is real and the others are mutually
complex conjugate. In what follows we will consider the
4-th and 6-th moments.
A. Defocusing nonlinearity
In Fig. 5 we show the results of calculating MI gains
G2, G4 and G6 for both the exponential and Gaussian re-
sponse functions and compare them with the same curves
for local stochastic media obtained in [15]. It is seen
that nonlocality suppresses the higher-order moments as
well. Notice that in defocusing media positions of MI
gain maxima for moments of different orders coincide [15]
(they are deterministic rather than random). Nonlocality
does not disturb this property. Fig. 6 demonstrate simi-
lar curves for the rectangular response function for differ-
ent values of the nonlocality parameter a. It is seen that
for sufficiently high a the medium demonstrates prac-
tically coinciding distributions of higher-moment growth
rates, their maxima being shifted to shorter wavelengths.
Evidently, higher-order moments for the rectangular re-
sponse function manifest the same “anomalous” enhance-
ment of the growth rate in a narrow region of modulation
wavenumbers, as compared with the local stochastic case.
0 1 2 3 4
0 1 2 3 4
FIG. 5: Defocusing media. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for a lo-
cal stochastic medium (upper three curves) and for nonlocal
stochastic media (lower three curves). Here d0 = 2, a
2 = 1,
|g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: exponential re-
sponse function; lower panel: Gaussian response function.
B. Focusing nonlinearity
For the focusing media the higher-order MI gains
demonstrate much the same behavior as for the defo-
cusing ones for both sign-definite and sign-indefinite re-
0 1 2 3 4 5
0 1 2 3
0 0.5 1 1.5
FIG. 6: Defocusing nonlocal stochastic media with the rect-
angular response function. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for d0 = 2,
|g0|A2 = 1, σ2d = σ2g = 0.1. Upper panel: a = 2; middle panel:
a = 6; lower panel: a = 8.
sponse functions. Fig. 7 shows the MI gains for local
and nonlocal stochastic media for the Gaussian response
function. Curves for the exponential and rectangular re-
sponse functions are qualitatively the same. With in-
creasing the nonlocality parameter a, curves for MI gains
of different orders become closer one the other, so high
nonlocality smoothes fluctuations of the modulation am-
plitude growth.
0 0.5 1 1.5 2 2.5 3
FIG. 7: Focusing media. Plots of the MI gains G6 (solid
line), G4 (dashed line), and G2 (dash-dotted line) for a lo-
cal stochastic medium (upper three curves) and for nonlo-
cal stochastic medium with the Gaussian response function
(lower three curves). Here d0 = 2, a = 2, g0A
2 = 1,
d = σ
g = 0.1.
V. CONCLUSION
Within the limits of the linear stability analysis, we
have investigated the MI of a homogeneous wave in a
nonlocal nonlinear Kerr-type medium with random pa-
rameters. For the case of the white-noise model of pa-
rameter fluctuations, we derived the equations which gov-
ern the dependence of the MI gain on the modulation
wavenumber. As was expected from physical motiva-
tions, nonlocality causes considerable suppression of the
stochasticity-induced MI growth rate for media with the
sign-definite Fourier images of the response functions. At
the same time, nonlocal media with the sign-indefinite
Fourier images of the response functions can display a
somewhat different behavior leading to an increase, as
compared with local media, of the MI gain for some do-
mains of modulation wavenumbers.
Acknowledgments
The authors are very grateful to F. Abdullaev and J.
Garnier for constructive comments.
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