General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright
owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
You may not further distribute the material or use it for any profit-making activity or commercial gain
You may freely distribute the URL identifying the publication in the public portal
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from orbit.dtu.dk on: Aug 09, 2022
Collapse arrest and soliton stabilization in nonlocal nonlinear media
Bang, Ole; Krolikowski, Wieslaw; Wyller, John; Juul Rasmussen, Jens
Published in:
Physical Review E. Statistical, Nonlinear, and Soft Matter Physics
Link to article, DOI:
10.1103/PhysRevE.66.046619
Publication date:
2002
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Bang, O., Krolikowski, W., Wyller, J., & Juul Rasmussen, J. (2002). Collapse arrest and soliton stabilization in
nonlocal nonlinear media. Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 66(4), 046619.
https://doi.org/10.1103/PhysRevE.66.046619
Collapse arrest and soliton stabilization in nonlocal nonlinear media
Ole Bang,
1
Wieslaw Krolikowski,
2
John Wyller,
3
and Jens Juul Rasmussen
4
1
Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
2
Australian Photonics Cooperative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering,
Australian National University, Canberra Australian Capital Territory 0200, Australia
3
Department of Mathematical Sciences, Agricultural University of Norway, P. O. Box 5035, N-1432 Ås, Norway
4
Riso” National Laboratory, Optics and Fluid Dynamics Department, OFD-128 P. O. Box 49, DK-4000 Roskilde, Denmark
共Received 9 January 2002; revised manuscript received 12 July 2002; published 24 October 2002兲
We investigate the properties of localized waves in cubic nonlinear materials with a symmetric nonlocal
nonlinear response of arbitrary shape and degree of nonlocality, described by a general nonlocal nonlinear
Schro
¨
dinger type equation. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlin-
earity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions.
The nonlocal nonlinear response must be symmetric and have a positive definite Fourier spectrum, but can
otherwise be of completely arbitrary shape and degree of nonlocality. We use variational techniques to find the
soliton solutions and illustrate the stabilizing effect of nonlocality.
DOI: 10.1103/PhysRevE.66.046619 PACS number共s兲: 42.65.Tg, 42.65.Jx, 42.65.Sf
I. INTRODUCTION
Collapse is a fundamental physical phenomenon well
known in the theory of waves in nonlinear media. It refers to
the situation when strong contraction or self-focusing of a
wave leads to a catastrophic increase or blowup of its ampli-
tude after a finite time or propagation distance 共see 关1–3兴 for
reviews兲. Wave collapse has been observed in plasma waves
关4兴共the famous Langmuir wave collapse兲, electromagnetic
waves or laser beams 关5兴共also called self-focusing兲, Bose-
Einstein condensates 共BEC’s兲 or matter waves 关6兴, and even
capillary-gravity waves on deep water 关7兴. The effect of col-
lapse appears also in astrophysics, where the gravitational
attraction plays the same role as the self-focusing of electro-
magnetic waves, tending to compress stars of sufficient
mass, eventually leading to their collapse into a black hole
关8兴.
Typically the contraction must be able to act freely in two
or more dimensions to be strong enough to generate a col-
lapse. Moreover, the so-called norm, which is the power for
electromagnetic and plasma waves, the atom density for
BEC’s, and the mass for stars, must be above a certain criti-
cal value for a collapse to occur. Most commonly the col-
lapse has been discussed in the context of the nonlinear
Schro
¨
dinger 共NLS兲 equation, which is a universal model for
dispersive 共or diffractive兲 weakly nonlinear physical systems
关9兴. The NLS equation models, e.g., all systems mentioned
above, in which a wave collapse has been predicted and veri-
fied experimentally.
The collapse singularity is an artifact of the model and
signals the limit of its validity. Close to the singularity, when
the amplitude is extremely high and the temporal and spatial
scales are extremely short, different physical processes will
come into play 关1–3兴. A common effect is nonlinear dissipa-
tion, such as two-photon absorption of electromagnetic
waves and inelastic two- and three-body recombination for
matter waves, which efficiently absorbs the collapsing part of
the wave. Thus collapse acts as an effective nonlinear loss
mechanism, as is well known in, e.g., Langmuir turbulence
关11兴 and BEC’s 关12兴. Effects such as discreteness 关13兴, non-
paraxiality 关14兴, and saturation of the nonlinearity 共see 关3兴
for references to all the different types of saturation, such as
exponential, threshold, logarithmic, etc.兲 will also com-
pletely eliminate the possibility of a collapse singularity ap-
pearing. In contrast, effects such as weak linear loss 关3兴,
temperature fluctuations 关15兴, and spatial incoherence 关16兴,
cannot eliminate the collapse, only change the critical value
of the norm. In any case the collapse effect represents a
strong mechanism for energy localization, which it is impor-
tant to study to understand the properties of a given physical
system.
The inherent nonlocal character of the nonlinearity has
attracted considerable interest as a means of eliminating col-
lapse and stabilizing multidimensional solitary waves. Non-
locality appears naturally in optical systems with a thermal
关17兴 or diffusive 关18兴 type of nonlinearity. Nonlocality is also
known to influence the propagation of electromagnetic
waves in plasmas 关19–22兴 and plays an important role in the
theory of BEC’s, where it accounts for the finite-range many-
body interaction 关12,23–25兴.
In this work we consider NLS equations with a general
nonlocal form of the nonlinearity. Turitsyn proved the ab-
sence of collapse for three particular shapes of the nonlocal
nonlinear response 关26兴. The analysis of the collapse condi-
tions for general response functions is difficult and has been
carried out only numerically 关24兴. However, in many sys-
tems, such as BEC’s, one has no knowledge of the particular
response function. Furthermore, the degree of nonlocality is
the relative width of the response function and the wave
packet and thus it changes dynamically when the wave
packet spreads or contracts. Therefore it is important to
maintain the generality of the nonlocal response function in
the model. Here we prove rigorously that nonlocality elimi-
nates collapse in all physical dimensions for arbitrary shapes
of the nonlocal response, as long as the response function is
symmetric and has a positive definite Fourier spectrum.
PHYSICAL REVIEW E 66, 046619 共2002兲
1063-651X/2002/66共4兲/046619共5兲/$20.00 ©2002 The American Physical Society66 046619-1
II. GENERAL MODEL
We consider the evolution of a wave field u⫽u(r
ជ
)
⫽ u(r
ជ
,
) described by the general nonlocal NLS equation
i
u
⫹ ⵜ
2
u⫺V
共
r
ជ
兲
u⫹N
共
I
兲
u⫽0, 共1兲
where V⫽V(r
ជ
) is an external 共linear兲 confining potential, I
⫽ I(r
ជ
)⫽I(r
ជ
,
)⫽
兩
u
兩
2
,
is the evolution coordinate and r
ជ
⫽ (r
1
,r
2
,r
3
) spans a D-dimensional ‘‘transverse’’ coordinate
space. By virtue of being a confining potential, V(r
ជ
) has a
finite global minimum, which can be set to zero without loss
of generality due to the gauge invariance of Eq. 共1兲. Thus
V(r
ជ
)⬎0. The nonlinear term N⫽ N(I) is represented in the
general nonlocal form
N
共
I
兲
⫽
冕
R
共
r
ជ
⬘
⫺ r
ជ
兲
I
共
r
ជ
⬘
兲
dr
ជ
⬘
, 共2兲
where the integral
兰
dr
ជ
is over all transverse dimensions. We
consider only response functions R(r
ជ
) that are real 共i.e., no
nonlinear loss or gain兲 and symmetric 共e.g., excluding the
asymmetric Raman response—see 关28兴 and references
therein兲. We further assume the response function to be lo-
calized or L
1
integrable like all physically reasonable re-
sponse functions. In this case Eq. 共1兲 may always be rescaled
so that
冕
R
共
r
ជ
兲
dr
ជ
⫽ 1 共3兲
without any lack of generality. In media with, e.g., a thermal
or diffusive type of nonlinearity the response profile is an
exponential function, R(r
ជ
)⫽(1/2
)exp(
兩
r
ជ
兩
/
) 关17,18兴, where
determines the degree of nonlocality.
Because the response function is real Eq. 共1兲 conserves
the power 共in optics兲 or number of atoms 共for BEC’s兲 P,
P⫽
冕
Idr
ជ
共4兲
for localized waves. Because the response function is also
symmetric Eq. 共1兲 conserves the Hamiltonian H,
H⫽
冕
冋
兩
ⵜu
兩
2
⫹ VI⫺
1
2
NI
册
dr
ជ
. 共5兲
In optics u is the envelope of the electric field with inten-
sity I and V represents a guiding structure 共waveguide兲. Here
Eq. 共2兲 represents a general phenomenological model for
self-focusing Kerr-like media, in which the change in the
refractive index induced by an optical beam involves a trans-
port process. This includes heat conduction in materials with
a thermal nonlinearity 关17兴 or diffusion of molecules or at-
oms in atomic vapors 关18兴. A nonlocal response in the form
共2兲 appears naturally due to many-body interaction processes
in the description of BEC’s 关12,24,25,27兴, if the assumption
of a zero-range interaction potential is relaxed 关27兴. For
BEC’s with a negative scattering length Eq. 共1兲 is the nonlo-
cal Gross-Pitaevskii 共GP兲 equation for the collective wave
function u(
is time兲, with I representing the density of at-
oms and V representing the magnetic trap.
III. SIMPLE KNOWN LIMITS
In the limit when the response function is a delta function,
R(r
ជ
)⫽
␦
(
兩
r
ជ
兩
), the nonlinear response is local 关see Fig. 1共a兲兴
and simply given by
N
共
I
兲
⫽ I, 共6兲
as in local optical Kerr media described by the standard NLS
equation and in BEC’s described by the standard GP equa-
tion. In this local limit multidimensional optical beams with
a power higher than a certain critical value will experience
unbounded self-focusing and collapse after a finite propaga-
tion distance 关1–3兴. It is also well known that BEC’s will
collapse when the total number of atoms is larger than a
critical number 关12兴.
With increasing width of the response function R(r
ជ
) the
wave intensity in the vicinity of the point r
ជ
also contributes
to the nonlinear response at that point. In case of weak non-
locality, when R(r
ជ
) is much narrower than the width of the
beam 关see Fig. 1共b兲兴, one can expand I(r
ជ
⬘
) around r
ជ
⬘
⫽ r
ជ
and
obtain the simplified model
FIG. 1. Degrees of nonlocality, as given by the relative width of
the response function R and the intensity profile I in the x plane.
Shown is the local 共a兲, the weakly nonlocal 共b兲, the general 共c兲, and
the strongly nonlocal 共d兲 response.
BANG, KROLIKOWSKI, WYLLER, AND RASMUSSEN PHYSICAL REVIEW E 66, 046619 共2002兲
046619-2
N
共
I
兲
⫽ I⫹
␥
ⵜ
2
I,
␥
⫽
1
2
冕
r
2
R
共
r
兲
dr
ជ
, 共7兲
where the small positive definite parameter
␥
measures the
relative width of the nonlocal response. The diffusion type
model 共7兲 of the nonlocal nonlinearity is a model in its own
right in plasma physics, where
␥
can take any sign 关19,20兴.It
was also applied to BEC’s 关25兴, nonlinear optics 关29兴, and
energy transfer in biomolecules 关30兴. In weakly nonlocal me-
dia with N(I)⫽ I⫹
␥
ⵜ
2
I it is straightforward to show that
collapse cannot occur. This was first done for plasmas 关20兴,
and later for BEC’s 关25兴.
In the limit of a strongly nonlocal response much broader
than the characteristic width of the wave function 关see Fig.
1共d兲兴, one can instead expand the response function and ob-
tain 共to lowest order兲
N
共
I
兲
⬇P
共
R
0
⫹ R
2
r
2
兲
, 共8兲
where R
0
⫽ R(0
ជ
) and R
2
⫽
1
2
ⵜ
2
R(0
ជ
). The evolution of opti-
cal beams in such a strongly nonlocal medium was consid-
ered in 关31兴. Since this relation is linear, the highly nonlinear
effect of collapse cannot occur.
So in the two extreme limits of a weakly and highly non-
local nonlinear response the collapse is prevented. For arbi-
trary degree of nonlocality it is difficult to prove anything
rigorously. Just saying that the dynamics is described by ei-
ther the weakly nonlocal model 共7兲 or the linear oscillator
model 共8兲, which both have no collapse, is not enough. First
of all the degree of nonlocality is the relative width of the
response function and the wave packet and thus it changes
dynamically when the wave packet spreads or contracts.
Thus, the system may dynamically switch state, e.g., from
being in the local limit 共6兲 to the highly nonlocal limit 共8兲.
Furthermore, as is well known from studies of general NLS
equations, the typical singularity is a so-called blowup fea-
turing the amplitude locally going to infinity on a broad
background localized structure 关1兴. Such a two-scale field
distribution, which was also recently observed in BEC’s
关24兴, is clearly described by neither of the two simple limit-
ing systems.
IV. PROOF OF ABSENCE OF COLLAPSE
AND SOLITON STABILITY
The stabilizing effect of nonlocality of an arbitrary degree
was proven by Turitsyn for three specific examples, includ-
ing Coulomb interaction
关
R(r
ជ
)⫽1/
兩
r
ជ
兩
兴
关26兴. Turitsyn
bounded the Hamiltonian from below for fixed power, which
proves that a collapse cannot occur and that the soliton so-
lutions are stable in the Liapunov sense. Here we consider
the general case of arbitrarily shaped, nonsingular response
functions and prove rigorously that the Hamiltonian is
bounded from below in all dimensions.
Introducing the D-dimensional Fourier transform 共de-
noted with a tilde兲
I
˜
共
k
ជ
兲
⫽
冕
I
共
r
ជ
兲
exp
共
ik
ជ
•r
ជ
兲
dr
ជ
共9兲
and its inverse
I
共
r
ជ
兲
⫽
1
共
2
兲
D
冕
I
˜
共
k
ជ
兲
exp
共
⫺ ik
ជ
•r
ជ
兲
dk
ជ
, 共10兲
it is straightforward to show that for N(I) given by Eq. 共2兲
the following relations hold:
兩
I
共
k
ជ
兲
兩
⫽
冏
冕
I
共
r
ជ
兲
e
ik
ជ
•r
ជ
dr
ជ
冏
⭐
冕
Idr
ជ
⫽ P, 共11兲
冕
NIdr
ជ
⫽
1
共
2
兲
D
冕
R
˜
共
k
ជ
兲
兩
I
˜
共
k
ជ
兲
兩
2
dk
ជ
. 共12兲
With these relations we can bound the Hamiltonian by con-
served quantities, which is necessary for employing standard
Liapunov stability theory 关32兴共first applied by Rosen 关33兴
for the standard NLS equation兲. For response functions with
a finite degree of nonlocality R
0
⬍ ⬁ and a positive definite
spectrum R
˜
(k
ជ
)⭓0, we obtain the following bound of the
Hamiltonian:
R
0
⬍ ⬁, R
˜
共
k
ជ
兲
⭓0: H⭓
兩兩
ⵜu
兩兩
2
2
⫺
R
0
2
P
2
, 共13兲
where
兩兩
u
兩兩
p
p
⬅
兰
兩
u
兩
p
dr
ជ
⬎ 0 and we have used that V(r
ជ
)⬎0.
In the local limit when R(r
ជ
)⫽
␦
(r
ជ
) and thus R
0
⫽ ⬁, the
well-known properties of the standard NLS equation apply
关1,2,9兴.
The inequality 共13兲 is the main result of this article. It
shows that, for all symmetric response functions with a posi-
tive definite Fourier spectrum and a finite value at the center,
the Hamiltonian is bounded from below by the conserved
quantity ⫺
1
2
R
0
P
2
, or conversely, that the gradient norm
兩兩
ⵜu
兩兩
2
2
is bounded from above by the conserved quantity
H⫹
1
2
R
0
P
2
. According to standard Liapunov theory this rep-
resents a rigorous proof that a collapse with the wave-
amplitude locally going to infinity cannot occur in BEC’s,
plasma, or optical Kerr media with a nonlocal nonlinear re-
sponse, for any physically reasonable response function with
a positive definite spectrum.
V. ILLUSTRATION THROUGH THE VARIATIONAL
APPROACH
The stabilizing effect of the nonlocality can be further
illustrated by the properties of the stationary solutions of Eq.
共1兲. As a simple example we consider nonlocal optical bulk
Kerr media with a Gaussian response
R
共
r
ជ
⬘
⫺ r
ជ
兲
⫽
冉
1
2
冊
D/2
exp
冉
⫺
兩
r
ជ
⬘
⫺ r
ជ
兩
2
2
冊
. 共14兲
The ground-state stationary solutions are then radially sym-
metric bell shaped, nodeless solutions of the form u(r
ជ
,z)
COLLAPSE ARREST AND SOLITON STABILIZATION . . . PHYSICAL REVIEW E 66, 046619 共2002兲
046619-3
⫽
(r)exp(iz), where the profile
(r) is found from the
Euler-Lagrange equations for the Lagrangian
L⫽
冕
冋
2
⫹
兩
ⵜ
兩
2
⫺
1
2
N
共
2
兲
2
册
dr
ជ
. 共15兲
To capture the main physical effects we use the approximate
variational technique with a Gaussian trial profile
(r)
⫽
␣
exp
关
⫺(r/

)
2
兴
, in view of the fact that the Gaussian pro-
file is an exact solution in the strongly nonlocal limit with
N(I) given by the parabolic potential 共8兲. Inserting this an-
satz into the Lagrangian 共15兲, with N given by the general
expression 共2兲, the Euler-Lagrange equations give the ampli-
tude
␣
2
⫽ (⫹D/

2
)(2⫹ 2
2
/

2
)
D/2
and width

2
⫽
关
4
⫺ D⫹
冑
(4⫺ D)
2
⫹ 16
2
兴
/(2). In Fig. 2 we show the
power P
s
⫽ (
/2)
D/2
␣
2

D
and Hamiltonian of the stationary
solutions in two dimensions 共2D兲. The dashed lines give the
results of the weakly nonlocal approximation with N given
by Eq. 共7兲, from which
␣
2
⫽ 4 and

2
⫽ 2/⫹2
2
is found,
resulting in the power
P
s
⫽ 4
共
1⫹
2
兲
, 共16兲
where 4
is the (-independent兲 power of the Gaussian ap-
proximation to the soliton solution of the standard 2D NLS
equation, recovered in the local limit
⫽ 0.
In the 2D NLS equation the collapse is critical and the
stationary solutions are ‘‘marginally stable’’ with dP
s
/d
⫽ 0 关1,2,9兴. Typically, perturbations act against the self-
focusing, with several effects, such as nonparaxiality and
saturability, completely eliminating collapse 关3兴. This is also
the case with nonlocality, as evidenced from Fig. 2 and the
simplified expression 共16兲, which shows that any finite width
of the response function 共nonzero value of
) implies that
dP
s
/d becomes positive definite. According to the 共neces-
sary兲 Vakhitov-Kolokolov 共VK兲 criterion 关34兴 the soliton so-
lutions therefore 共possibly兲 become linearly stable.
For small the soliton width

decreases as 1/. Thus
the assumption of weak nonlocality, i.e., that the soliton is
much wider than the response function, applies only to suf-
ficiently small values of satisfying
2
Ⰶ 1, which is also
clearly seen from Fig. 2. The accuracy of the assumption of
weak nonlocality is further discussed in Ref. 关35兴 in terms of
modulational instability.
The 3D case shown in Fig. 3 is more interesting, because
the nonlinearity is much stronger than in 2D. The collapse in
the local 3D NLS equation is so-called supercritical
关1,2,9,10兴. Again the soliton width

decreases as 1/,soa
threshold width should exist, below which the nonlocality is
not strong enough to stabilize the soliton. This is exactly
what is observed in Fig. 3: For ⬍
th
the solitons are still
linearly unstable with dP
s
/d⬍0, but above threshold the
nonlocality is strong enough to bend the curve and make
dP
s
/d⬎0, i.e., the solitons become linearly stable accord-
ing to the VK criterion. From the definition dP
s
(
th
)/d
⫽ 0 the variational results give
th
⫽ 1/(2
2
), corresponding
to a threshold in the soliton power 共dashed curve in Fig. 3兲
and width
P
s
th
⫽
共
5
兲
3/2
5
/4,

th
⫽ 2
, 共17兲
which are both proportional to the degree of nonlocality
.
Thus, sufficiently broad and high-power solitons are stable.
In the Hamiltonian versus power diagram in Fig. 3 the lower
共upper兲 branches correspond to stable 共unstable兲 solutions
while the threshold is represented by the cusp 关36兴. This
stable solution branch was recently found numerically in the
context of BEC’s with a nonlocal negative scattering poten-
tial 关25兴. It corresponds to high-density, self-bound states of
the condensate.
VI. CONCLUSION
In conclusion we studied the properties of localized wave
packets in nonlocal NLS equations. We have presented a
simple, but rigorous proof that nonlocality of an arbitrary
shape eliminates collapse in all physical dimensions. The
only requirement is that the nonlocal response function
should have a positive definite Fourier spectrum, as do most
physically reasonable response functions.
We also demonstrated that multidimensional soliton solu-
FIG. 2. 2D variational results with Gaussian response and trial
function. Left: Soliton power 共solid兲 versus eigenvalue for differ-
ent degrees of nonlocality,
⫽ 0, 0.2, 0.4, and 0.6. Dashed lines
show the weakly nonlocal approximation. Right: Corresponding
Hamiltonian versus power diagrams.
FIG. 3. 3D variational results with Gaussian response and trial
function. Left: Soliton power 共solid兲 versus eigenvalue for differ-
ent degrees of nonlocality,
⫽ 0, 0.2, 0.4, and 0.6. Dashed lines
show the threshold power 共17兲. Right: Corresponding Hamiltonian
versus power diagrams.
BANG, KROLIKOWSKI, WYLLER, AND RASMUSSEN PHYSICAL REVIEW E 66, 046619 共2002兲
046619-4
