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PHYSICAL REVIEW E 87,052915(2013)
Phase shielding soliton in parametrically driven systems
Marcel G. Clerc,*M´
onica A. Garcia ˜
Nustes,†and Yair Z´
arate
Departamento de F´
ısica, Facultad de Ciencias F´
ısicas y Matem´
aticas, Universidad de Chile, Casilla 4873, Santiago, Chile
Saliya Coulibaly‡
Laboratoire de Physique des Lasers, Atomes et Mol´
ecules, CNRS UMR No. 8523, Universit´
edesSciencesetTechnologiesdeLille,
59655 Villeneuve d’Ascq Cedex, France
(Received 13 September 2012; revised manuscript received 7 February 2013; published 24 May 2013)
Parametrically driven extended systems exhibit dissipative localized states. Analytical solutions of these states
are characterized by a uniform phase and a bellshaped modulus. Recently, a type of dissipative localized state with
a nonuniform phase structure has been reported: the phase shielding solitons. Using the parametrically driven
and damped nonlinear Schr¨
odinger equation, we investigate the main properties of this kind of solution in
one and two dimensions and develop an analytical description for its structure and dynamics. Numerical
simulations are consistent with our analytical results, showing good agreement. A numerical exploration
conducted in an anisotropic ferromagnetic system in one and two dimensions indicates the presence of phase
shielding solitons. The structure of these dissipative solitons is well described also by our analytical results. The
presence of corrective higherorder terms is relevant in the description of the observed phase dynamical behavior.
DOI: 10.1103/PhysRevE.87.052915 PAC S n u m b e r ( s ) : 89.75.Kd, 05.45.Yv
I. INTRODUCTION
Particletype solutions or macroscopic localized states
arising in systems out of equilibrium have been observed in a
wide range of physical systems. Examples include magnetic
materials, liquid crystals, gas discharge systems, chemical
reactions, ﬂuids, granular media, and nonlinear optics media
(see [1–3] and references therein). The variety of systems
exhibiting these solutions confers them a universal nature.
Given their particlelike properties, one can characterize them
by a family of continuous parameters such as position, ampli
tude, and size. A prototypical model that exhibits dissipative
localized states or dissipative solitons in the quasireversible
limit—systems under the assumption of small injection and
dissipation of energy [4–8]—is the parametrically driven and
damped nonlinear Schr¨
odinger (PDNLS) equation [9,10]
∂tψ=−iνψ −iψ2ψ−i∂xxψ−µψ+γ¯
ψ,(1)
where ψ(x,t) is a complex ﬁeld that accounts for the envelope
of the oscillation in the system under study. The variable ¯
ψ
stands for the complex conjugate of ψ;{x,t}denote the spatial
and temporal coordinates, respectively; νis the detuning
parameter, which is proportional to the difference between
half the forcing frequency and the natural frequency of the
oscillator ﬁeld; µis the damping parameter, which accounts
for the energy dissipation processes; and γis the amplitude of
the parametric forcing. It is important to note that Eq. (1)
describes an oscillatory focusing medium with dispersive
coupling [11]sincethenonlinearandspatialcouplingterms
have the same sign. For ν<0, γ∼µ#1, and µ2<γ2<
ν2+µ2,thePDNLSmodeladmitsanalyticallocalizedstates
characterized by a uniform phase and a belllike shape
for the modulus of the amplitude—uniform phase solitons
*marcel@dﬁ.uchile.cl
†mgarcianustes@ing.uchile.cl
‡saliya.coulibaly@phlam.univlille1.fr
(UPSs) [9,10]. Figure1shows a typical dissipative soliton
in a polar coordinate representation [ψ(x,t)=R(x,t)eiϕ(x,t)]
observed in the parametrically driven and damped nonlinear
Schr¨
odinger equation.
The PDNLS model has been derived in various physical
contexts. Indeed, the PDNLS equation can be deduced from
the amplitude equation approach in the parametrically driven
pendulum chain [12–14]. Using the same approach in the con
text of magnetic systems for an easyplane ferromagnetic spin
chain exposed to both a constant and a timeperiodic external
magnetic ﬁeld perpendicular to the hard axis, the PDNLS
equation was obtained by means of the LandauLifshitz
Gilbert equation [15,16]. Additional physical scenarios where
the PDNLS model can be derived include surface waves
in vertically oscillating layers of water [17–19], localized
structures in nonlinear lattices [20], and the Kerrtype optical
parametric oscillator [21].
Recently, we have shown that the phase of dissipative
solitons exhibits an unexpected dynamical behavior in the
PDNLS equation [22]. More precisely, we have found that, for
alargerangeofparameters,PDNLSsolitonsolutionsshow
dynamical phase fronts that, after some transient behavior,
reach a stationary state surrounding the soliton core. Due
to its shieldlike phase structure, this type of dissipative
localized state has been denominated a phase shielding soliton
(PSS). Using the asymptotic expression for the amplitude,
valid far from the core of the soliton, we have determined
analytically the shape of the phase fronts and their dynamics.
The above analysis allows us to characterize the different type
of shieldlike structures of the phase. Performing a numerical
stability analysis and using the size of the system Las a
control parameter, we have proved that UPS solutions lose
their stability through a AndronovHopf bifurcation followed
by the appearance of the PSS as the stable solution of the
system.
The emergence of phase fronts on dissipative solitons repre
sents an alternative perspective in the study of parametrically
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0 100 200
0
0.1
0.2
R(x)
(x)
Space
x0
δ−1/2
(2δ)1/2
ϕ
FIG. 1. (Color online) Modulus R(x)(reddashedline)and
phase φ(x)(bluesolidline)ofaUPSintheparametricallydriven
and damped nonlinear Schr¨
odinger equation with µ=0.1000, ν=
−0.0122, γ=0.1002, δ=0.0185, and L=200. Here x0stands for
the position of the maximum of the soliton amplitude. The soliton
width and height are indicated.
driven systems. Until now, the phase of single solitons was
considered uniform in most of the aforementioned physical
systems. However, a more complete analysis could demon
strate the existence of PSS solutions in such physical scenarios.
In the present paper we extend the analytical and numerical
studies of phase shielding solitons in onedimensional (1D)
and 2D cases. We also analyze the appearance of PSS solutions
in an easyplane ferromagnetic spin system in both 1D and 2D
conﬁgurations. For this purpose, we organize the paper in the
following way. In Sec. II we review the main features of the
PDNLS equation and the analytical solution of the UPS and
its stability. We then introduce the phase shielding soliton
solutions. An analytical description of PSS solutions and their
dynamics is presented. A numerical stability analysis is also
performed to establish the connection between the UPS and
PSS solutions. To show the existence of the PSS in physical
systems, we study in Sec. III an easyplane ferromagnetic
classical spin chain exposed to an external magnetic ﬁeld. The
existence, stability properties, and dynamical evolution of the
phase shielding solitons in twodimensional extended systems
are analyzed in Sec. IV.InSec.Vwe consider a physical
example of a parametrically driven system in two dimensions:
aforcingmagneticlayer.Weconcludewithasummaryin
Sec. VI.
II. DISSIPATIVE SOLITONS IN THE PDNLS MODEL
IN ONE DIMENSION
For µ=γ=0, Eq. (1) becomes the wellknown nonlinear
Schr¨
odinger equation [23], which describes the envelope
of an oscillatory coupled system. This model is a time
reversible Hamiltonian system with the reﬂection symmetry
{t→−t,ψ→¯
ψ}.However,thetermsproportionaltothe
energy dissipation µand the injection γbreak this symmetry.
The higherorder terms in Eq. (1) are ruled out by a scaling
analysis where µ#1, ν∼µ∼γ,ψ∼µ1/2,∂x∼µ1/2,
and ∂t∼µ1/2.
AtrivialsolutionofEq.(1) is the homogeneous (quiescent)
state ψ0=0. For ν<0, ψ0becomes unstable through a
subcritical stationary bifurcation at γ2=µ2+ν2(the Arnold
tongue) [11]. Inside this region the system has three uni
form solutions ψ0=0, and ψ±=σ±i√(µ−γ)(µ+γ)σ,
where σ=!(γ−µ)(−ν+"γ2−ν2)/2γ.Thesethree
states merge through a pitchfork bifurcation at γ2=µ2+ν2
when ν>0. However, for positive detuning, ψ0is stable only
when γ<µbecause this state exhibits a spatial instability at
γ=µ[24], which gives rise to a spatially periodic state with
awavenumberkc=√ν.
A. Solitons with constant phase
For negative detuning, the PDNLS equation exhibits local
ized states or dissipative solitons supported asymptotically by
the quiescent state. In order to obtain the localized states, we
introduce the Madelung transformation ψ=R(x,t)eiϕ(x,t)in
Eq. (1).Separatingtheimaginaryandrealparts,weobtainthe
set of equations
∂tR=2∂xR∂xϕ+R∂xxϕ−µR +γRcos(2ϕ),(2)
∂tϕ=−ν−R2−∂xxR
R+(∂xϕ)2−γsin(2ϕ),(3)
where Rand ϕdenote the amplitude and phase of the ﬁeld
ψ,respectively.Intheparameterregionγ!µand −ν±
"γ2−µ2!0, we get nontrivial steady homoclinic solutions
of the form [9,10]
Rs(x,x0)="2δ±sech("δ±[x−x0]),(4)
cos(2ϕs)=µ/γ,(5)
where the parameter δ±≡−ν±"γ2−µ2and x0stands for
the position of the maximum of the soliton, which will be
called the core of the soliton in what follows. The modulus
width and height are given by √2δ±and 1/√δ±,respectively
(see Fig. 1). Hence Eqs. (4) and (5) show that such states have
abellshapedmodulusandauniformphase(UPS).
Equation (5) shows that the system can generate dissipative
solitons when the injection of energy exceeds its dissipation.
As a consequence of the spatial translational invariance of
Eq. (1),thedissipativesolitonsconstituteafamilyofstates
parametrized by a continuous parameter x0corresponding
to their Goldstone mode [25]. This parameter stands for the
position of the core of the localized state (see Fig. 1). In brief,
the above model admits soliton solutions in the {µ,ν,γ}region
bounded by ν<0, µ"γ,andγ2<µ
2+ν2. Under these
conditions, the relation cos(2ϕs)=µ/γadmits four equilibria
in the interval [π,−π]. As a consequence of the symmetry
ψ→−ψof Eq. (1),ifϕsis a solution then ϕs±πis also
asolution.Figure2illustrates this relation and the respective
stability regions of the different particletype solutions. From
this ﬁgure one can infer that the localized states appear or
disappear by simultaneous saddlenode bifurcations when
the injection and dissipation of energy are equal (γ=µ).
The stable solutions are characterized by Re(ψ)Im(ψ)<0
[Re(ψ)Im(ψ)>0] for γ>0(γ<0), thus both ﬁelds have
different signs when γ>0[15]. In contrast, the solution with
a bellshaped modulus ψ[Eq. (4)] also appears through
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PHASE SHIELDING SOLITON IN PARAMETRICALLY ... PHYSICAL REVIEW E 87,052915(2013)
3 2 1 1 2 3
1.0
0.5
0.5
1.0
120
0
0.3
0.6
80
80 120
0
0.2
0.4
80 120
0.6
0.3
0
80 120
0.4
0.2
0
cos(2ϕ)= µ
_
γ
Re(ψ)
Im(ψ)
Re(ψ)
Im(ψ)
Re(ψ)
Im(ψ)
Re(ψ)
Im(ψ)
FIG. 2. (Color online) Schematic representation of the different stability regions (colored shadow) of dissipative solitons. The circles
represent the different solutions of cos(2ϕs)=µ/γ: closed (open) circles correspond to stable (unstable) localized states. The insets depict the
different types of dissipative solitons with µ=0.10, γ=0.13, ν=−0.12, and L=200, where δ+=0.203 and δ−=0.037.
asaddlenodebifurcationatγ=µwhere δ±=−ν.The
solutions of Eq. (4) with δ−are unstable.
B. Phase shielding soliton
Previous works have reported additional amplitude bifur
cations when νis increased far from the tip of the Arnold
tongue where the maximum of ψundergoes period doubling,
quasiperiodicity, and ﬁnally chaos [26]. Usually, the phase
ﬁeld of the dissipative soliton state is considered uniform.
However, further numerical analysis reveals unexpected and
rich phase dynamics of single solitons in parametrically driven
systems [22], which reaches a nonuniform steady phase.
Asolitoniscreatedbyslightlyperturbinganinitial
homogenous state. The parameters are chosen to fulﬁll the
conditions for the appearance of UPS solutions, i.e., ν<0,
γ2<µ
2+ν2,andµ!γ. Initially, the perturbation quickly
evolves to a welldeﬁned bellshaped amplitude. At the
same time, the phase becomes uniform around the core of
the localized state followed by some intricate transient that
rapidly goes away from the system. The front propagation
is characterized by a rather slow motion (see Fig. 3), which
suddenly reaches a steady state. From this initial behavior a
pair of counterpropagative fronts emerges, which propagates
in a rather slow motion, reaching suddenly a steady state
(see Fig. 3). Near the soliton core, the phase takes the
value corresponding to the stable uniform soliton (UPS),
i.e, the value −ϕsor −ϕs+π.Conversely,farfromthe
soliton position, the asymptotic values in (−∞,∞) tend to
the unstable UPS phase values either ϕsor ϕs−π.Hencethe
number of conﬁgurations is given by different combinations
of equilibrium connections (unstablestableunstable). There
are eight stationary conﬁgurations that connect different phase
equilibria. Figure 4displays all the different phase shielding
solitons. These steady phase structures depend strongly on
initial conditions and are equally likely to appear in the same
region of parameters.
1. Analytical approach at dominant order
To provide an analytical background to these phase struc
ture dynamics, we take advantage of the x→−xsymmetry
of the PDNLS model [Eq. (1)]. We consider a semiinﬁnite
domain whose origin corresponds to the core position of the
soliton x0.Numericalresultsshowthatthephasepresentsa
single front that emerges at a position xf$1/√δ+,where
xfrepresents the point with the highest spatial variation of
the phase. In the region x$1/√δ+,thesolitonmodulusR
decays exponentially [see Fig. 6(a)].
Deﬁning '≡1/√δ+,whichaccountsforthewidthofthe
soliton bellshaped amplitude (see Fig. 1), we propose the
FIG. 3. (Color online) Phase propagation of the dissipative
soliton. Spatiotemporal diagram of phase ϕ(x,t)forγ=0.123,
µ=0.100, and ν=−0.093.
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0 100
0
0 100 200
R(x) R(x)
R(x) R(x)
ϕ(x) ϕ(x)
ϕ(x) ϕ(x)
−
ϕs
ϕs
ϕs  π
− ϕs
ϕs
ϕs  π
π/2
−π/2
−π/2
π/2
−π
−π
0
0
0 100
0
0100 200
R(x)
R(x)
R(x)
R(x)
ϕ(x) ϕ(x)
ϕ(x) ϕ(x)
π − ϕs
π − ϕs
ϕs
ϕs
π
π
π/2
π/2
FIG. 4. (Color online) Different phase shielding soliton states in the parametrically driven damped nonlinear Schr¨
odinger equation with
µ=0.10, ν=−0.12, γ=0.14, and L=200. The PSS states supported by the inner uniform phases −ϕsand π−ϕsare shown on the right
and left, respectively. Dashed (red) and solid (blue) lines account for the modulus and phase of the complex ﬁeld ψ,respectively.
following ansatz for the modulus and the phase of the soliton:
R(x"&,x0)=2!2δ+e−f(x,x0)(6)
and
ϕ(x)=ϕF(x−xf),(7)
respectively. At dominant order, we consider f(x,x0)≈
√δ+(x−x0). In this approximation, R(x,x0)coincideswith
the asymptotic exponential decay of the stable UPS modulus.
Substituting the former ansatz in (2) and (3), we obtain two
different equations. The ﬁrst one allows us to get analytically
the dominant proﬁle of the phase front. The second one
describes the phase front dynamical behavior. Accordingly,
the phase proﬁle is characterized by
∂xxϕF=2!δ+∂xϕF+µ−γcos(2ϕF).(8)
Introducing the effective potential energy U(ϕF)≡−µϕF+
(γ/2) sin(2ϕF), Eq. (8) can be written as a Newtontype
equation that describes a particle moving in a tilted periodic
potential with an injection of energy proportional to the speed
∂xϕF,
∂xxϕF=−∂U
∂ϕF+2!δ+∂xϕF.(9)
Hence the solutions of the above equation correspond to
stationary phase fronts. The uniform equilibrium states of
Eq. (8) coincide with the phase equilibria of cos(2ϕs)=µ/γ
in the range from −πto π. Therefore, the phase front solutions
represent heteroclinic orbits in the {ϕ,ϕx}space that interpolate
from one equilibrium to another of the Newtontype equation
(9) (see Fig. 5).
Deﬁning the change of variable x=2√δ+x&in Eq. (8),
we can perform an asymptotic series ϕF(x)=ϕ0+)ϕ1(x)+
)2ϕ2(x)+···,with)≡1/4δ'1, which at ﬁrst order has
the analytical solution
ϕF(x,xf)≈ϕ0=
fsol −πfor [−π,−π/2)
fsol for (−π/2,−π/2)
fsol +πfor (π,π/2],
(10)
where
fsol =arctan %&γ±µ
γ∓µtanh !γ2−µ2(x−xf)
2√δ+'.(11)
Note that the phase front solutions are also parametrized by
the continuous parameter xf. Figure 6shows the numerically
computed phase front proﬁles, which present a difference
of 1% with respect to expression (10). If one considers
the dominant correction ϕF≈ϕ0+∂xϕ0/4√δ,thisdifference
decreases to 0.8%.
Considering the complete soliton domain, we obtain the
eighth possible shelllike conﬁguration that we have previously
observed in numerical simulations (see Fig. 4).
ϕ(x)
R(x)
1.0
0.5
0.5
1.0
cos2
ϕ=µ
γ
π

π
/2
π
/2
π
R(x)
ϕ
(x)
0
δ
1/2
AA
B
C
ABC
FIG. 5. (Color online) Schematic representation of the stationary
phase front structure that connects different equilibrium solutions
(A–C) of the Newtontype equation (9). The inset is a schematic
representation of the potential U(ϕF).
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0 100 200 300 400
−1
0
1
R(x)
ϕ(x) ϕ(x)
x
50 150 250 350
100
300
500
700
900
−1
−0.5
0
0.5
1
(a) (b)
time
Space
FIG. 6. (Color online) Phase propagation: (a) phase front proﬁle considering half of the dissipative soliton and (b) spatiotemporal diagram
of the phase front obtained from Eq. (1) with γ=0.083, ν=−0.063, and µ=0.058. The dashed curve is the numerical solution obtained
using Eq. (17).
2. Higherorder corrections
AdeeperanalysisofnumericalsimulationsrevealsthatPSS
solutions are composed of two qualitatively different regions:
inner and outer regions. The inner and outer regions stand for
the central and asymptotic parts of the PSS, respectively. Note
that the asymptotic phase of the PSS in the inner and outer
regions coincides with the phase of the stable and unstable
UPSs, respectively (see Fig. 4). Therefore, the PSS can be
understood as a soliton buildup by the stable (inner region)
and unstable (outer region) UPS solutions. To illustrate this
statement, Fig. 7shows the logarithm of the PSS modulus as
afunctionofthespace.Clearly,thereisacrossoverregion
0 1
00
200
300
400
80
60
40
20
0
20
Outer
Re
g
io
n
O
ute
r
Re
g
io
n
Space
ln[R(x)]
0.2
0
0.2
ϕ(x)
x
f
FIG. 7. (Color online) Phase ϕ(x) as a function of the space (top)
and logarithm of the PSS modulus R(x)(bottom)withµ=0.1,
ν=−0.09, γ=0.12, and L=400. Inner and outer regions are
deﬁned. The exponential decay value changes from √δ+=0.3946
(theoretical √δ+=0.3954) inside the inner region to √δ−=0.1508
(theoretical √δ+=0.1538) in the outer region. The transition point
between both regions coincides with the phase front position xf.
between both exponential decay rates of the UPS solution that
is characterized by a transition point. Such a point outlines
the border transition between the inner and outer regions and
corresponds to the phase front core position xf.Therefore,the
PSS exponential decay rate f(x,x0)mustbeamendedby
f(x,x0)≈!δ+(x−x0)+B(x,xf),(12)
with
B(x,xf)≡[!δ−−!δ+]%(x−xf)(x−xf),(13)
where %(x−xf) denotes the Heaviside function. Note that
the function f(x,x0) is a smooth function. However, its
approximation (12) is continuous but not differentiable at
x=xf.
Using the amended ansatz (12),Eqs.(8) and (9) can be
reobtained in the inner and outer regions. In the inner region,
Eqs. (8) and (9) remain the same. In contrast, these equations
are modiﬁed by substituting the value δ+for δ−in the outer
region. Following the same procedure showed in Sec. II B1,
the amended front phase is obtained
ϕ0(x)=arctan "#γ±µ
γ∓µtanh !γ2−µ2(x−xf)
2δ(x,xf)$,(14)
with
δ(x,xf)≡[!δ++(!δ−−!δ+)%(x−xf)].(15)
In this approximation the phase fronts are continuous but
not differentiable at x=xf,emphasizingthatthePSSis
composed of the stable and unstable UPS solutions.
It is important to note that ansatz (6) considers a uniform
exponential decay rate of the modulus. Such an assumption
leads, at dominant order, to the obtention of phase front
solutions [Eq. (10)]. Higherorder corrections allow us to get
an improved description of the phase shield soliton where the
modulus also exhibits an amplitude shielding structure (see
Fig. 7). However, this structure is exponentially suppressed
in comparison to the soliton height √2δ+.Incontrast,the
phase shielding structure is order one. Therefore, a possible
experimental characterization of the PSS must be achieved by
means of phase measurements.
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C. Phase front dynamics
As discussed in the previous section, the transient preceding
the formation of the phase proﬁle is governed by the fronts
dynamics. Here we propose an analytical study of the dynam
ical evolution of these fronts. For this purpose let us consider
the typical evolution of a soliton in a semiinﬁnite system, as
shown in Fig. 6.Ascanbeseenfromtheﬁgure,thefront
displays a dynamical behavior characterized by a nontrivial
motion. For the sake of simplicity, let us consider the ansatz
(6) and (7) at dominant order where f(x,x0)≈√δ(x−x0)
and δ≡δ+.SubstitutinginEq.(3), we obtain the equation for
position of the core
−˙
xf∂xϕF=−(ν+δ)−8δe−2√δx+(∂xϕF)2−γsin(2ϕF).
(16)
In order to account for the front dynamics xf(t)hasbeen
promoted to a timedependent function. The time derivative of
xfis given by ˙
xf.
Multiplying the above equation by ∂zϕF(z)withz≡x−xf
and introducing the inner product %fg&≡!fgdz, we obtain,
after straightforward calculations, an ordinary differential
equation for the core of the phase front
˙
xf=A+Be−2√δxf,(17)
where
A≡%[ν+δ+γsin(2ϕF)−(∂zϕF)2]∂zϕF&
%∂zϕF∂zϕF&
and
B≡8δ"e−2√δz∂zϕF#
%∂zϕF∂zϕF&
are real numbers, which can be either positive or negative,
depending on the shape of the phase front. For example, when
one considers a front that increases monotonically with the
spatial coordinate, A(B)isanegative(positive)constant.
The term proportional to Aaccounts for the constant speed at
which the larger phase value invades the smaller one, giving
rise to a phase front that propagates towards the position of the
soliton x0.Thisspeedcanbeunderstoodasaconsequenceof
the effective potential energy U(ϕF)differencebetweenboth
equilibria. In contrast, the term proportional to Baccounts
for the effect of spatial variation of the tail of the amplitude
soliton, which induces a force that leads to phase fronts
moving away from the position of the soliton. Consequently,
the superposition of these two antagonistic forces generates a
stable equilibrium for the position of the phase front, which
is consistent with the dynamical behavior illustrated by the
spatiotemporal diagram of Fig. 6(b).SolvingEq.(17),weget
an analytical solution for the typical trajectory
xf(t)=ln $B
A%
2√δ+ln(e−2√δA(t−t0)−1)
2√δ−A(t−t0).(18)
The dashed curve shown in Fig. 6(b) is obtained using the
above formula wherein Aand Bare used as ﬁtting parameters.
Note that the constant ln(B/A)/2√δaccounts for the steady
equilibrium position of the front, which corresponds to the
characteristic size of the shell structure in the phase. For higher
order corrections of the phase, we obtain a similar expression
for the dynamics of the front.
D. Stability analysis for the uniform phase soliton
As we have already shown, the uniform phase and phase
shielding solitons are solutions of the PDNLS model (1).Thus
a natural question arises: What are the bifurcation scenarios of
these solutions? Here we examine this question, performing a
numerical linear stability analysis based on Ref. [15]. Given
the complexity of the linear operator, an analytical stability
analysis is not affordable. We consider small perturbations ρ
and 'around the solutions Rs(x)andϕ0,respectively,i.e.
R=Rs(x)+ρ(x,t),ϕ=ϕ0+'(x,t),(19)
where ρ,''1. Substituting in (2) and (3) and linearizing,
we obtain
∂tρ=2∂xRs∂x'+Rs∂xx'+2&γ2−µ2'Rs(20)
and
Rs∂t'=δρ −3R2
sρ−∂xxρ−2µRs',(21)
respectively. Equations (20) and (21) represent an eigenvalue
problem that can be written in the matrix representation
˙
'ρ
'(=M'ρ
'(,(22)
where
M≡'02∂xRs(x)∂x−Rs(x)∂xx −2Rs(x)&γ2−µ2
1
Rs(x)[δ−3Rs(x)2−∂xx]−2µ(.(23)
An analytical solution to Eq. (22) is a difﬁcult task [15].
Therefore, to solve it we adopt a numerical strategy. In order
to obtain the spectrum, a set of eigenvalues associated with
the linear stability analysis, we proceed to discretize in space
with grid points x→j(x,F(x,t)→F(j(x,t)≡Fj(t) with
j=1,...,N, where Nis the number of points of the system
and L=N(x.Insuchacase,thedifferentialoperatorMwith
spatiotemporal coefﬁcients turns into a matrix of rank 2N.We
also consider µ=µ0and x0=L/2fordifferentvaluesof
{γ,ν}in the region of existence of solitons, i.e., γ2!ν2+µ2
and ν<0.
The Lparameter controls the size effect. Changing Nwith
(xﬁxed, we can easily vary it. In previous reports, this param
eter was not considered as a relevant parameter system, being
usually a small constant. We shall see that the parameter L
plays a main role in the stability properties of dissipative states.
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150 250 350 450
0
4
8
12
−0.2 −0.1 0 0.1
−4
−2
0
2
4
−0.2 −0.1 0 0.1
−4
−2
0
2
4
−0.2 −0.1 0 0.1
−4
−2
0
2
4
(a)
(b)
(c)
λ = 0
max[Re(λ)]
(a) (b) (c)
System size L
Re(λ)
Re(λ)
Re(λ)
Im(λ)
Im(λ)
Im(λ)
Lc
x10
2
FIG. 8. (Color online) Real part of the largest eigenvalue
max[Re(λ)] (red circles) and the eigenvalue related to the Goldstone
mode (blue triangles) as a function of system size. The stability of
solitons is shown in the spectra of the soliton with constant phase
(a) before (system size L=284), (b) during (L=304), and (c) after
(L=324) the bifurcation for γ=0.105, µ=0.1, and ν=−0.05.
Hence let us consider Las a control parameter with the
system parameters {µ,ν,γ}ﬁxed. When Lis small enough
the spectrum is characterized by being centered on an axis
parallel to the imaginary one, where every single eigenvalue
has a negative real part. Such behavior of the eigenvalues is
typical of quasireversible systems [5,7]. Increasing L, the set
of eigenvalues begins to collide, creating a continuum set.
Up to a critical value of Lc,wheresomeofthemcrossthe
imaginary axis at a nonzero frequency, the set exhibits an
AndronovHopf bifurcation [27,28]. The inset Figs. 8(a)–8(c)
outline the spectrum before, during, and after the bifurcation,
respectively. The main plot of Fig. 8illustrates the real part
of the largest eigenvalue max[Re(λ)] (red circles) and the
eigenvalue related to the Goldstone mode (blue triangles) as a
function of the system size L.Asaresultofthetranslational
invariance, the eigenvalue related to the Goldstone mode is at
the origin of the complex plane [25]. For γ=0.105, µ=0.1,
and ν=−0.05, we observe that [see Fig. 8(a), inset] below
the critical value Lc=304, the largest eigenvalue corresponds
to the Goldstone mode. Close to the bifurcation, the largest
conjugate pair of eigenvalues crosses the real axis destabilizing
the uniform phase solution [see Figs. 8(b) and 8(c), insets].
The numerical stability analysis of UPS solutions reveals
astrongdependenceonthesystemsize.Sucharesultisin
accordance with the inner and outer region crossover. The
inner region has a deﬁnite length for a given set of system
parameters {µ,ν,γ}. If the system size is small enough (Lis
less than the length of the inner region), the crossover does
not occur. Then the PSS solution cannot appear and the UPS
is a stable solution. For Lgreater than the length of the inner
region, the UPS destabilizes, generating the PSS solution.
Given that the exponential decay of the stable UPS, and
therefore the length of the inner region, is a function of the
system parameters {µ,ν,γ}it is natural to infer that by varying
such parameters with Lﬁxed, the UPS destabilization will take
place as well. Indeed, following the above strategy, we perform
a numerical stability analysis of the UPS varying γfor Lﬁxed
10.05 10.45 10.85 11.25
0
4
8
12
16
x 10
−3
−0.2 −0.1 0 0.1
−4
−2
0
2
4
−0.2 −0.1 0 0.1
−4
−2
0
2
4
−0.2 −0.1 0 0.1
−4
−2
0
2
4
Im(λ)
Im(λ)
Im(λ)
Re(λ)
Re(λ)
Re(λ)
λ = 0
max[Re(λ)]
(a)
(b)
(c)
(a) (b) (c)
γc
x 10
−2
Forcing Parameter γ
FIG. 9. (Color online) Real part of the largest eigenvalue
max[Re(λ)] (red circles) and the eigenvalue related to the Goldstone
mode (blue triangles) as a function of the forcing parameter γ.
The stability of solitons is shown in the spectra of the soliton with
constant phase (a) before (γ=0.1065), (b) during (γ=0.1090), and
(c) after (γ=0.1115) the bifurcation for µ=0.1andν=−0.05
with L=280 ﬁxed.
with µ=0.1 and ν=−0.05. We choose the same parameter
region {ν,γ}with L=280 (before bifurcation; see Fig. 8)to
ensure an initial stable UPS solution. Figure 9displays the
eigenvalue spectrum evolution as γvaries. As before, up to a
critical γc,thesystemexhibitsanAndronovHopfbifurcation,
which leads to the appearance of a PSS solution similar to the
one observed in Figs. 8(a)–8(c).
In brief, the above instability mechanism is a robust
phenomenon. Figure 10 displays the UPS stability over a wide
parameter region {ν,γ}for µ=0.05 and L=400 ﬁxed. For a
system size smaller than the critical one, we observe that for the
parameters 0 <γ−µ"1, the soliton with constant phase is
stable. Notwithstanding, increasing the forcing amplitude γor
the detuning parameter ν,thesolitonbecomesunstableagain
by an AndronovHopf bifurcation.
In the case that the system size is large enough, the UPS
solution exhibits an AndronovHopf bifurcation that leads to a
PSS solution. Further increasing the system parameters {ν,γ},
asecondarybifurcationleadstoaperiodicsolitonlikethose
 0.20  0.15  0.10  0.05
0.05
0.10
0.15
0.20
0.25 γ
ν
γ=µ
γ = µ + ν
22
2
Quiescent State
UPS unstable
UPS stable
FIG. 10. (Color online) The PSS bifurcation diagram in the γν
space obtained by solving (22) numerically for µ=0.050 and L=
400.
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observed in Ref. [26]. Conversely, for small Lthe Andronov
Hopf bifurcation leads directly to localized periodic solitons
without a secondary one.
To verify these results, we have also performed a stability
analysis of solutions in a Cartesian representation of the ﬁeld
ψ.Weintroducethelineartransformationψ=X+iYin
Eq. (1).Separatingintorealandimaginaryparts,weobtain
∂tX=νY+(X2+Y2)Y+∂xxY−µX+γX,(24)
∂tY=−νX−(X2+Y2)X−∂xxX−µY−γY,(25)
respectively. The solution for this set of equations (24) and
(25) is
Xs=Rscos(ϕ0),(26)
Ys=Rssin(ϕ0),(27)
where Rsand ϕ0are given by formulas (4) and (5) with δ+=δ.
We follow t h e s a m e p r ocedure show n a bov e a n d t a k e
into account small perturbations (δX,δY"1) around the
solutions (X,Y). Linearizing and using relations (4) and (5),
we get the dynamical system
˙
!δX
δY"=!−µ+γ−R2
s#γ2−µ2/γν+R2
s(2γ−µ)/γ+∂xx
−ν−R2
s(2γ+µ)/γ−∂xx −µ−γ+R2
s#γ2−µ2/γ"!δX
δY".(28)
This system (28) yields results similar to those already
observed using the polar representation.
III. PHASE SHIELDING SOLITON IN PHYSICAL
SYSTEMS
In Sec. Iwe emphasized the universality of the PDNLS
equation. Through an amplitude equation approach it can be
shown that the equation is present in different physical systems.
Based of this statement, one expects that results obtained in
this context can be transposed to the original systems. In the
following section we conduct numerical studies in an easy
plane ferromagnetic system in order to explore the observation
of PSS solutions in such a system.
A. Forced magnetic wire
Solitons in magnetism have been intensively studied in
past decades due to their possible technological applications.
It is known that an easyplane ferromagnetic spin chain in
the presence of both a constant and a timeperiodic external
magnetic ﬁeld perpendicular to the hard axis exhibits localized
structures. Such structures are commonly refer to as localized
precession states in a forced magnetic wire. Furthermore,
experimental realizations of the model have been already
achieved [29,30].
The forced magnetic wire is described phenomenologically
by the LandauLifshitzGilbert (LLG) equation. Following
an amplitude equation approach, it can be proved that, in
the quasireversible limit, the system can be described by
the parametrically driven and damped nonlinear Schr¨
odinger
equation [16].
Let us consider a onedimensional anisotropic Heisenberg
ferromagnetic chain formed by Nclassical spins or a magnetic
moment subject to an external magnetic ﬁeld. The direction of
the chain is described by the zcoordinate ˆ
z=(0,0,1) and the
external magnetic ﬁeld is orthogonal to this direction, denoted
by ˆ
x=(1,0,0).
When the quantum effects are small enough, the vector
Sican be treated as a classical spin or a magnetic moment
[16]. According to this latter assumption, the dynamics of the
magnetic moment Siis governed by ˙
Si=−γSi×(∂H/∂Si),
where γis the gyromagnetic constant and the Hamiltonian H
has the form
H=−J
N
$
i=1
SiSi+1+2D
N
$
i=1%Sz
i&2−gµHx
N
$
i=1
Sx
i.(29)
Here Jis the exchange coupling constant and Hxand D
stand for the external magnetic ﬁeld and the anisotropy energy,
respectively.
To study the continuum limit of this set of ordinary
differential equations, which accounts for a magnetic wire,
we can assume that
Si(t)→S(z,t)(30)
and
Jdz
2
γ−1!Si+1−2Si+Si−1
dz2"→lex∂2
zS(z,t),(31)
where lex denotes the characteristic interaction length. More
over, introducing a phenomenologically dissipative source,
the Gilbert damping, the motion of the magnetization ﬁeld is
governed by the wellknown LandauLifshitzGilbert equation
[31]
∂τM=M×[Mzz −β(M·ˆ
z)+He−α∂τM],(32)
where M≡S/Msstands for the unit vector of the magneti
zation and Msis the saturation magnetization. We have also
introduce the normalization {τ→γMst,β→4D/γ,He→
gµM/γMs}.Hereβ>0accountsfortheanisotropyconstant
(easyplane magnetization) and αthe damping parameter. For
several types of magnetic materials, this parameter is small
[31]. When the magnetic ﬁeld is time dependent, the above
model (32) is a timereversible system perturbed with injection
and dissipation of energy, i.e., a quasireversible system, as long
as this perturbation remains small.
As a result of the anisotropy and constant external ﬁeld
(He=H0ˆ
x), the natural equilibrium of the previous model
(32) corresponds to the magnetization ﬁeld lying in the
direction of the external magnetic ﬁeld M=ˆ
x. When spatial
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300 400 500 600 700 800
100
200
300
400
500
600
700
800
900 −0.15
−0.1
−0.05
0
0.05
0.1
0.15
Time
Space
mz (z,t)
y
z
x
(b)
(a)
FIG. 11. (Color online) (a) Schematic representation of a soliton in a onedimensional anisotropic ferromagnetic chain (magnetic wire).
(b) Spatiotemporal evolution of a soliton for L=1024, H0=4.800, β=0.200, h1=0.042, α=0.019, and ν=−0.030.
coupling is ignored, it is easy to show that the dynamics around
this equilibrium is described by a nonlinear oscillator with
natural frequency ω2
0=H0(β+H0)[16]. It is worth noting
that in Eq. (32) the magnetization components are proportional
to the external magnetic ﬁeld, which therefore acts as a
parametric forcing. Then if this ﬁeld combines a constant and
atimeperiodicpart(He=[H0+h1]ˆ
x,whereh1=%cos(ωt)
oscillates about twice the natural frequency, ω≡2(ω0+ν),
and νis the detuning parameter), the system exhibits a
parametric resonance at %2(β/4ω0)2=α2(H0+β/2)2+ν2
for small parameters {α,ν,H0,%}.Intheparameterspace
(%,ν), the region above this curve corresponds to the Arnold
tongue. Dynamically speaking, this resonance corresponds
to an undamped precession of the magnetization unit vector
around the direction of the external magnetic ﬁeld with angular
velocity ω0[Fig. 11(a)]. Thus, rewriting Eq. (32) for one of
the components, for instance, mz,aftersomecalculations,
considering
m2
x≈1−m2
y+m2
z
2(33)
and
my≈1
H0!1+%
H0"˙
mz,(34)
one can obtain, in the weakly nonlinear regime [16],
¨
mz=−ω2
0mz+(β+2H0)∂2
zmz−µ˙
mz−˙
h1
H0
˙
mz
+(β+2H0)h1mz+β(H0+h1)
2#m2
y+m2
z$mz,(35)
where µ≡α(H0β/2) and γ=β%/4ω0are, respectively, the
effective driving strength and the detuning parameter.
Now, close to the parametric resonance, we can introduce
the following ansatz [14,16]intoEq.(32):
mz=4%ω0H0
β#ω2
0+3H2
0$Re[A(z,t)]ei(ω0+ν)t+W(z,t ,A),(36)
where W(z,t,A) stands for a small correction. Linearizing and
imposing the solvability condition for W(z,t,A), one obtains
the amplitude equation of the oscillations at dominant order
(the parametrically driven and damped nonlinear Schr¨
odinger
equation)
∂tA=−iνA−iA2A−i∂ZZA−µA +γ¯
A, (37)
where Z≡√2ω0/(β+2H0)z. The terms proportional to
{ν,γ,µ}stand for the detuning, effective driving, and damping
of the magnetic system.
This equation has different homogeneous states, where the
simplest one is A=0, representing a constant magnetization
along the external ﬁeld direction (M=ˆ
x). Single solitons are
among the nontrivial steady states of Eq. (37) [15]. Other
stationary state solutions of the PDNLS in the magnetic context
can be found in Refs. [14,16,32,33].
1. The PSS in the magnetic wire
The chain of classical spins or magnetic wire subject to an
external magnetic ﬁeld represents an adequate physical system
to study the formation of a phase shielding soliton. Different
works have studied these magnetic localized states by means
of the LLG model. Direct numerical simulations of Eq. (32)
close to the Arnold tongue for negative detuning and for small
values of dissipation and damping (µ∼γ&1) reveal the
formation of a localized precession state. Figure 11(b) shows
the spatiotemporal evolution of the mzcomponent for this
soliton.
However, phase shielding solitons are characterized for
their phase structure. Therefore, we must extract information
of the instantaneous phase angle φzfrom the knowledge of
the oscillatory ﬁeld mz(z,t). With this aim, we compute the
instantaneous phase using the Hilbert transform technique for
signal processing [34]. Let us recall that the original real ﬁeld
mz(z,t), can be express as
mz(x,t)=Re[R(x,t)eiφ(x,t)],(38)
where Re(·)representtherealpart,R(x,t)themodulusofthe
envelope, and φ(x,t)thephase.
The Hilbert transform technique reconstructs a complex
variable θz=θr+iθifrom the original real data θrand its
onesided Fourier transform Sr(ω),
Sr(ω)=&∞
−∞
θr(x,t)*(ω)e−iωtdt, (39)
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0 200 400 600 800 1000
−0.85
−0.8
−0.75
−0.7
−0.65
−0.6
−0.55
−0.5
φz
Space
FIG. 12. (Color online) Reconstructed phase angle φzfor L=
1024, H0=4.800, β=0.200, h1=0.042, α=0.019, and ν=
−0.030. The appearance of two opposite fronts can be observed.
where %(ω)istheHeavisidestepfunction.Toobtaintheone
sided Fourier transformer of the imaginary part Si(ω), we used
the relation
Si(ω)=H(ω)Sr(ω),(40)
where
H(ω)=!−i, 0!ω<π
i, −π!ω<0(41)
is a 90◦phase shifter or Hilbert transformer [35].
Calculating the inverse Fourier transform of Si(ω), we get
the imaginary part θi(x,t)ofthecomplexvariable.Hencethe
modulus R(x,t) and phase φ(x,t) can be easily rebuilt from the
real data. This method gives a good description of the modulus
and phase for bandlimited signals.
Using the above described Hilbert procedure, we analyze
the spatial structure of the phase at different times. Figure 12
displays the phase φz(x)atagivenvaluetfor L=1024. The
presence of two stationary but opposite fronts, quite similar to
those seen in previous numerical analysis of the parametrically
driven and damped nonlinear Schr¨
odinger equation, is clear.
The phase also exhibits a slot or hole at the center of
the spatial phase angle. The appearance of this hole is a
consequence of the nonlinear terms present at the original
model. As we show above, the amplitude equation is an
approximation, at dominant order, of the original system.
Higherorder corrections are not taken into account [36]. Since
the simulations of the magnetic system are directly from
the LLG model, they include all the nonlinear corrections.
Usually such corrections are not discernible in the modulus,
but become relevant in the phase, especially close to the center
of the soliton. Away from the core, the nonlinear corrections
decreases exponentially, hence the formation of the slot close
to the center.
Another important consequence of the nonlinear correc
tions is related to the phase front dynamics. Numerical
simulations reveal that the phase fronts reach their steady state
closer to the core of the soliton than in the usual parametrically
driven and damped nonlinear Schr¨
odinger equation. Since
the magnetic dissipative soliton is a precession state, the
phase angle φzexhibits a modulo 2πtemporal periodicity.
Therefore, there is a continuous change in the phase values
500 1000 1500 2000
5
10
15
20
25
30 −0.09
−0.07
−0.05
−0.03
−0.01
0
500 1000 1500 2000
5
10
15
20
25
30
35
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Space
Space
Time Time
Envelope (φz)
Phase (φz)
FIG. 13. (Color online) Stroboscopic spatiotemporal diagrams of
the reconstructed envelope (top) and the phase angle φz(bottom) for
L=2048, H0=4.800, β=0.200, h1=0.042, α=0.019, and ν=
−0.030. The spatial slot displayed by the phase has been removed.
close to the ends (0,L) that introduces a ﬁctitious periodic
motion of the phase front around the center. To evaluate
the phase dynamical behavior and the front positions, we
take stroboscopic snapshots with the same periodicity as that
of the soliton oscillation. Figure 13 displays the stroboscopic
phase evolution of the phase structure. It is clear that a magnetic
phase shielding soliton is formed. In order to emphasize
the phase fronts, we have removed the slot in the vicinity
of the center. The amplitude of the envelope is shown to
stress the soliton position. Different numerical simulations
with L=512,1024, and 2048 support this observation. It is
important to note that all the observed phase shielding solitons
are symmetric. Asymmetric solitons have not been observed
so far in the magnetic wire.
In brief, the phase fronts in the magnetic wire reach a steady
state, allowing the formation of a phase shielding soliton;
the shielding phase is established closer to the soliton core
position, in comparison with the PSS observed in the PDNLS
equation; and the phase shield structure is always symmetric
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PHASE SHIELDING SOLITON IN PARAMETRICALLY ... PHYSICAL REVIEW E 87,052915(2013)
[Fig. 4(a)]. The presence of nonlinear corrections plays an
important role in the structure and dynamical behavior of the
magnetic PSS.
IV. PHASE SHIELDING SOLITON IN TWO DIMENSIONS
Localized structures presented in the previous sections
are considered only in one spatial dimension. In the present
section we will study the existence, stability properties, and
dynamical evolution of the twodimensional extension of the
phase shielding soliton. Such extensions are not evident.
Conservative solitons observed in the nonlinear Schr¨
odinger
equation in one spatial dimension collapse in two spatial
dimensions [37], i.e., these solutions are unstable in two
spatial dimensions. Most of the experimental observations of
localized structures in parametrically driven systems have been
reported in two spatial dimensions, for instance, in ﬂuid surface
waves [38], oscillons in granular media [39], and isolated states
in thermal convection [40]. Note that all these observations
have been realized in dissipative systems. Furthermore, the
greatest difﬁculty in characterizing theoretically localized
states in two spatial dimensions is the lack of analytical
expressions of these states.
In the context of conservative systems in two spatial
dimensions perturbed with energy injection through paramet
rically temporal modulation and dissipation—quasireversible
systems [4–8]—the prototypical model is the parametrically
driven and damped nonlinear Schr¨
odinger equation
∂tψ=−iνψ −iψ2ψ−i∇2
⊥ψ−µψ+γ¯
ψ,(42)
where ψ(ρ,θ,t)isacomplexﬁeldthataccountsforthe
envelope of the oscillation for the system under study, ∇2
⊥≡
(1/ρ)[∂ρ(ρ∂ρ)] +(1/ρ2)∂θθ is the Laplacian operator in polar
coordinates, ρ>0, and θ∈[0,2π]. Equation (42) has been
derived in twospatialdimensional physical systems such as
the parametrically driven magnetic layer [32]andKerrtype
optical parametric oscillators [21]. In the conservative limit
(µ=γ=0), the above equation is the nonlinear Schr¨
odinger
equation. This model is widely applied to understand wave
phenomena in hydrodynamics, nonlinear optics, nonlinear
acoustics, quantum condensates, heat pulses in solids, and
various other nonlinear instability phenomena [41]. The non
linear Schr¨
odinger equation is a universal model for weakly
dispersive and nonlinear media.
It is well known that Eq. (42) exhibits stable nonpropagative
dissipative solitons in two spatial dimensions [42]. In contrast,
in the conservative limit, Eq. (42) has unstable soliton
solutions, which exhibit blowup in ﬁnite time [37]. The above
phenomenon disappears in the model (42) as a result of the
balance between injection and energy dissipation. In order to
understand the existence, stability properties, and dynamical
evolution of dissipative solitons shown by Eq. (42),letusto
consider the ansatz
ψ=Rs(ρ,t)eiφ(ρ,t).(43)
Note that the above ansatz presents axial symmetry; that is,
there is not an explicit dependence on the angle θ.This
assumption is based on the numerical simulations, where we
do not observe signiﬁcant angular dependence. Inserting (43)
in Eq. (42),weobtain
∂tRs=2∂ρRs∂ρφ+Rs
ρ∂ρφ+Rs∂ρρ φ−µRs
+γRscos(2φ),(44)
Rs∂tφ=−νRs−R3
s−∂ρRs
ρ−∂ρρ Rs+Rs(∂ρφ)2
−γRssin(2φ).(45)
Analogously to the onedimensional problem, let us assume
aconstantphaseφ=φ0;thus
cos(2φ0)=µ
γ,(46)
∂ρρ Rs=δRs−R3
s−∂ρRs
ρ,(47)
where δ≡−ν+!γ2−µ2.Aswehavementionedbefore,
there is not an analytical solution of the localized state in two
dimensions [42]. However, using the variational method, one
can obtain a good approximation [32,43]
Rs(ρ)≈A0√δsech"B0#δ
2ρ$,(48)
where A0=2.166 and B0=1.32. However, this approxi
mation does not describe the asymptotic behavior of the
dissipative soliton. The asymptotic behavior of the dissipative
soliton is of the form
Rs(ρ→∞)→e−√δρ
√ρ.(49)
The interaction of a pair of dissipative solitons that is of
exponential type as a function of the distance between solitons
has been characterized in Ref. [32]. Numerical simulations
of an easyplane ferromagnetic layer submitted to a magnetic
ﬁeld that combines a constant and an oscillating part show
good agreement with this interaction law.
A. Numerical observation of phase shielding solitons
in two dimensions
The existence of 2D solitons described by the PDNLS
(42) raises the question of whether it is possible to observe a
shielding phase in this case. Based on the previous simulations
carried out in one dimension, we explore a similar parameter
region in the 2D case, close to the Arnold tongue, in order to
observe the possible formation of phase shielding solitons.
Indeed, we have characterized two type of phase shielding
conﬁgurations. The ﬁrst type consists of a phase front with
axial symmetry in the range [0,2π]. The connected uniform
states are, analogously to the 1D case, given by the phase
equilibria determined by relation (46). Figure 14 shows the
typical structure of this symmetrical state around the soliton
position. If we consider a 1D stationary front solution in
asemiinﬁnitedomain(ρ>0), the 2D symmetric phase
solution corresponds to a 2πrotation around an axis whose
origin is placed at the position of the dissipative soliton. It
is important to note that the process of formation of this
symmetrical state is complex since one needs special initial
conditions close to the equilibrium state.
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FIG. 14. (Color online) Front view of a stationary 2D symmetric
phase shielding soliton observed in twodimensional numerical
simulations of the parametrically driven and damped nonlinear
Schr¨
odinger equation with γ=0.560, ν=−0.068, and µ=0.250.
The phase and amplitude ﬁeld are represented simultaneously.
Colored shadow renders the phase shelllike structure ϕ(x,y)that
surrounds the amplitude soliton localized at the center R(x,y).
The second type is characterized by a phase front axially
symmetric from [0,π]andananalogousfrontwithdifferent
asymptotic states from [π,2π], i.e., each phase front has semi
axial symmetry. The process of formation of this conﬁguration
starts with a wellformed 2D soliton that is slightly perturbed.
After some complex phase transient state, the system exhibits
the appearance of a circular phase front that spreads rather
slowly. Figure 15(a) displays this primary stage. However,
asymptotically, the circular structure becomes asymmetrical,
giving rise to a new semicircular front that still propagates in
the range [π,2π][seeFig.15(b)]. Finally, the whole structure
becomes stationary, creating a 2D asymmetric phase shielding
soliton. Unlike the symmetric case, the steady phase solutions
coincide only with a πrotation around the soliton position as
the center of rotation [see Fig. 15(c)]. Additionally, numerical
simulations performed in a close region of parameters show
the same dynamical behavior. Figure 16 give us a comparison
between the stationary conﬁguration of this shieldlike phase
and the soliton size for a different set of parameter values
{µ,ν,γ}.
It is noteworthy that this second type of twodimensional
state is characterized by being composed of all the solutions
found in one dimension. Indeed, if one performs different cuts
containing the center (soliton position), one can recognize
the observed solutions in one dimension (see Fig. 4). Another
interesting property is the following: If one calculates the phase
change on a path that connects two opposite points with respect
to the soliton position (!%"
∇ϕd"s) within the region close to the
position of the soliton one ﬁnds that this is zero. Nevertheless,
if one takes this type of path far away from the soliton position,
one ﬁnds !%$"
∇ϕd"s=±π.
As an additional remark we would like to point out the
stability of the 2D phase shielding solitons. Axially symmetric
PSSs are attained only by setting up special initial conditions
close to the steady state. A slight perturbation in its modulus
leads to a symmetry breaking where the axial symmetry is lost
and a semiaxial symmetry appears. Hence, numerically, the
second type of PSS solitons has a large basin of attraction.
B. Analytic approach to PSSs in two dimensions
In this section we will discuss an analytical approach for the
shielding phase solitons in two dimensions. From numerical
simulations (see Fig. 16), we can observe that the phase
emerges far from the soliton position. In the same manner
as in Sec. II B1, to obtain the dominant phase correction we
consider the exponential asymptotic decay of the modulus.