Interactions of bright and dark solitons with localized $\mathcal{PT}$-symmetric potentials

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by  N. Karjanto

Interactions of bright and dark solitons with localized -symmetric potentials
PT
N. Karjanto,1 W. Hanif,2 B. A. Malomed,3 and H. Susanto4
1Department of Mathematics, School of Science and Technology,
Nazarbayev University, Astana, 01000, Kazakhstan
2School of Mathematical Sciences, University of Nottingham,
University Park, Nottingham, NG7 2RD, United Kingdom
3Department of Physical Electronics, School of Electrical Engineering,
Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
4Department of Mathematical Sciences, University of Essex,
Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom∗
We study collisions of moving nonlinear-Schr¨odinger solitons with a PT-symmetric dipole em-
bedded into the one-dimensional self-focusing or defocusing medium. Analytical approximations
4
are developed for both bright and dark solitons. In the former case, an essential aspect of the
1
approximation is that it must take into regard the intrinsic chirp of the soliton, thusgoing beyond
0
2 the bounds of the simplest quasi-particle description of the soliton’s dynamics. Critical velocities
separatingreflectionandtransmission oftheincidentbrightsolitonsarefoundbymeansofnumeri-
g
calsimulations,andintheapproximatesemi-analyticalform. Anexactsolutionforthedarksoliton
u
pinnedby thecomplex PT-symmetric dipole is produced too.
A

5
I. INTRODUCTION between gain and loss in the system [7]. On the con-

trary, in conservative settings, including various models
]
S of nonlinear optics [8], solitons exist in continuous fami-
Losses are a ubiquitous feature appearing in all kinds
P lies, rather than as isolated solutions.
of optical systems. In most cases, losses are considered
.
n as a detrimental feature, which must be compensated Morerecently,aspecialclassofdissipativesystemswas
li by a properly introduced gain or feeding beam, in in- identified,withexactlybalancedspatiallyseparated(an-
n ternally and externally driven systems, respectively [1]. tisymmetricallyset)dissipativeandamplifyingelements.
[ However, losses may play a positive role too, helping to SuchsystemsrealizetheconceptofthePT (parity-time)
2 stabilize modes which otherwise would not exist. An ex- symmetry. This concept was originally elaborated in
v ample is a possibility to stabilize dissipative solitons in the quantum theory [9] for settings described by non-
1 models of laser cavities which are described by complex Hermitian Hamiltonians, which contain spatially even
4 Ginzburg-Landau (CGL) equations. The simplest ver- and odd real and imaginary potentials, respectively. A
2 sion of the CGL equation with the spatially uniform lin- distinctivefeatureoftheHamiltonianswithcomplex -
4 PT
eargainandcubiclossgivesrisetoexactsolutionsinthe symmetric potentials is the fact that, up to a certain
.
1 form of chirped sech pulses [2], but they are unstable, as critical value of the strength of the imaginary (dissipa-
0 the linear gain destabilizes the zero background around tive) part, their spectrum remains purely real. Actually,
4 the solitons. A possibility to stabilize the solitons was such -symmetricnon-HermitianHamiltonians(oflin-
1 PT
: proposed in Ref. [3], making use of dual-core couplers, earsystems)canbetransformedintoHermitianones[10].
v with the linear gain acting in one core, and linear loss – Intermsofthequantumtheory,the -symmetryisa
i PT
X intheother. Inthatsystem,thestablepulseexists,asan theoreticalpossibility. Toimplementitinrealsettings,it
attractor,alongwithanunstablecounterpartofasmaller isnaturaltoresorttothefactthatthelinearpropagation
r
a amplitude, which plays the role of a separatrix between equationforopticalbeamsintheparaxialapproximation
attraction basins of the stable pulse and stable zero so- hasessentiallythesameformastheSchr¨odingerequation
lution. The use of similar settings for the generation of in quantum mechanics, hence the evolution of the wave
stable plasmonic solitons [4], and for the creation of sta- function of a quantum particle may be emulated by the
ble two-dimensional dissipative solitons and vortices in transmissionofan optical beam. This fact makes it pos-
lasersystemswiththefeedbackdescribedbythelinearly sibleto simulatemanyquantum-mechanicalphenomena,
coupledstabilizingequation[5], havebeen proposedtoo. some of which are difficult to observe in direct experi-
In this connection, it is relevantto stress a crucialdif- ments,bymeansofrelativelysimplesettings availablein
ference between dissipative solitons, which are found, in classical optics [11].
particular, in the linearly-coupled systems with the sep- The realization of the -symmetric settings in op-
PT
arated gain and loss [5, 6], and solitons in conservative tics,whichcombinesspatiallysymmetricrefractive-index
media. Stable dissipative solitons exist as isolated at- landscapes and mutually balanced spatially separated
tractors,selectedasmodeswhichprovideforthebalance gain and loss, was proposed in Ref. [12] and demon-
stratedinRef. [13]. Theseworkshaddrawnagreatdeal
of attention to models of optical systems featuring the
symmetry, see recent review [14]. In the most basic
PT
∗Electronicaddress: [email protected] case, the models, which may naturally include the Kerr
2
nonlinearity,amounttothenonlinearSchr¨odinger(NLS) whichcorrespondstotheattractivedefect,whilethehost
equation for the local amplitude of the electromagnetic mediummaybeeitherself-focusinganddefocusing,were
wave, ψ(x,z), with a complex potential, whose real and recently found in an analytical form, and their stability
imaginaryparts, V(x) andW(x) are,as saidabove,spa- was investigated numerically, in Ref. [25].
tially even and odd, respectively: Previously,severaltechniqueshavebeendevelopedfor
analyzing interactions of bright [8, 26, 27] and dark
∂ψ 1∂2ψ [8, 28, 29] solitons with inhomogeneities, such as those
i + +g ψ 2ψ =[V(x)+iW(x)]ψ. (1)
∂z 2 ∂x2 | | represented by the complex potential in Eq. (1). In this
work,weuseaperturbationmethodfortheconsideration
This equation is written in terms of the spatial-domain
of interactions of moving solitons with -symmetric
setting,withpropagationdistancez,thesecondtermac- PT
dipole (2), and report results of systematic numerical
countingforthetheparaxialdiffractioninthetransverse
simulations of such interactions.
direction, x. The nonlinear term in Eq. (1) represents
The paper is organized as follows. The analytical ap-
the self-focusing (g = +1) or defocusing (g = 1) non-
proximation for the bright and dark solitons are devel-
−
linearity. It was also proposedto realize the same model
oped in Section II, which also includes a solution of the
as the Gross-Pitaevskii equation in Bose-Einstein con-
scattering problem for plane wavesin the linear medium
densates,withthelineargainprovidedbyamatter-wave
with the embedded dipole. In that section, exact
laser [15]. PT
solutionsarederivedtoofor trappeddarksolitonsin the
The presence of the nonlinearity in Eq. (1) naturally
model with the self-defocusing spatially uniform nonlin-
leads to -symmetric solitons [16]. A crucially impor-
earity and the -symmetric defect (2). Numerical re-
PT
tantissueis the stability ofsolitonsinsuchsystems. For PT
sultsandtheircomparisonwiththeanalyticalpredictions
-symmetric couplers, and for models with periodic
are reported in Section III. Conclusions are presented in
PT
complex potentials, anaccuratestability analysisofsoli-
Section IV.
tons solutions was reported, respectively, in Refs. [17]
and [18].
Another relevant problem is wave scattering on - II. ANALYTICAL CONSIDERATIONS
PT
symmetric potentials. In particular, periodic structures
can act as unidirectionally transmitting media near the
A. The scattering problem in the linear model
-symmetry-breakingpoint,withreflectionsuppressed
PT
atone endandenhancedatthe other,aspredictedtheo-
In the linearized version of Eq. (1) and (2),
retically in Ref. [19] and demonstrated experimentally
in a metamaterial [20]. The most natural setting for ∂ψ 1∂2ψ
the study of the scattering of broadlinear and nonlinear i = +[ǫδ(x)+iγδ′(x)]ψ, (3)
∂z −2∂x2
wave packets (including solitons) is offered by localized
-symmetricpotentials(defects)[21]. Suchdefectscan it is natural to consider the scattering problemfor plane
PT
be induced, for instance, by nonlinear -symmetric wave solutions, in the form of ψ(x,z) = eikzU(x), with
PT
oligomers embedded into a linear lattice [22]. In the lat- k <0 and U(x) satisfying the following stationary equa-
ter context, stationarystates in the formofplane waves, tion:
theirreflectionandtransmissioncoefficients,andthecor-
1
respondingrectificationfactors,illustratingtheasymme- kU = U′′+[ǫδ(x)+iγδ′(x)]U. (4)
− −2
try between left and right propagation, were analyzed.
Reflectionandtransmissionofsolitonsby -symmetric The general solution of the scattering problem should
PT
scattering potentials was studied in Refs. [23], where it be looked for as
wasshownthat,underspecialconditions,onecanhavea
unidirectionalflowofsingleandmultiplesolitons. Unidi- U(x)= eiqx+(R1+iR2)e−iqx, (5)
rectional tunneling of plane waves through epsilon-near- (T1+iT2)eiqx,
(cid:26)
zero -symmetric bilayers was also reported in Ref.
[24]. PT where eiqx with q = √ 2k represents the incident wave
−
The subject of the present work is the interaction of with the amplitude normalized to 1, while (R1+iR2)
bright and dark NLS solitons with a strongly localized and (T1+iT2), with real T1,2 and R1,2, are complex re-
-symmetric potential, which may be represented by flection and transmission coefficients, respectively.
PthTe dipole: The boundary conditions following from Eq. (4) at
PT x=0 are
V(x)+iW(x)=ǫδ(x)+iγδ′(x), (2)
∆(U′)=2ǫU , ∆(U)=2iγU , (6)
0 0
where δ denotes the Dirac-delta function, δ′ stands for
where ∆(...) stands for the jump, and
the derivative of the delta-function, ǫ and γ being real
constants (positive or negative). Static solutions for 1
bright solitons pinned by the dipole with ǫ < 0, U0 [U(x=+0)+U(x= 0)] (7)
PT ≡ 2 −
3
isthemeanvalueofU aroundx=0. Thesubstitutionof B. Bright solitons
thegenericformofthesolutiontothescatteringproblem,
in the form of Eq. (5), into Eq. (6) yields, after doing ThefreebrightNLSsolitonwithamplitudeη,velocity
some linear algebra, the following final results: v (in fact, it is the beam’s slope in the spatial-domain
q(ǫγ+q) q(ǫ γq) setting), and coordinate ξ is taken in the usual form, as
T1 = ǫ2+q2 , T2 =− ǫ2+−q2 , the solution to Eq. (1) with the self-focusing sign of the
nonlinearity, and V =W =0:
ǫ(ǫ+γq) q(ǫ+γq)
R = , R = . (8)
1 − ǫ2+q2 2 − ǫ2+q2 ψ(x,z)=η sech[η(x ξ(z)]exp(ivx+iφ(z)), (16)
−
Inparticular,forthecaseofγ =0,theseexpressionsgo
overintothewell-knownsolutionfortherealδ-functional dφ 1
= η2 v2 ,
potential: dz 2 −
q2 qǫ ǫ2 dξ = v.(cid:0) (cid:1) (17)
T1 = ǫ2+q2, T2 =R2 =−ǫ2+q2, R1 =−ǫ2+q2 , dz
(9) It is well known that the soliton may be considered as a
which satisfies the unitarity condition: particle with effective mass
T2+T2+R2+R2 1. (10) +∞
1 2 1 2 ≡ M = ψ(x)2dx 2η (18)
On the other hand, in the particular case of ǫ = 0 ex- | | ≡
Z−∞
pressions (8) reduce to a simple but, apparently, novel
and momentum
result:
+∞ ∂ψ∗
T =1, T = R =γ, R =0. (11)
1 2 2 1 P =i ψ(x) dx. (19)
−
∂x
Note that the general expression (8) and the par- Z−∞
ticular one (11) do not obey unitarity condition (10), The substitution of the unperturbed soliton’s wave
as additional power (norm) may be generated or ab- form (16) yields
sorbed by the term γ. Indeed, expression (11) yields
T2+T2+R2+R2 =∼ 1+2γ2 > 1. In the general case, P0 =2ηv Mv. (20)
1 2 1 2 ≡
solution(8) producesthe following resultfor the relative
In the presence of Hamiltonian perturbation (2), with
change of the total power as the result of the scattering:
ǫ = 0 but γ = 0, the soliton may be treated, in the
6
2γq(ǫ+γq) adiabatic approximation [26], as a particle which keeps
T2+T2+R2+R2 1= . (12)
1 2 1 2− ǫ2+q2 the constant mass (dη/dz = 0) and moves under the
action of the effective potential, U(ξ) = ǫη2sech2(ηξ),
Thus,the resultmaybenegative(thescattering-induced
according to Newton’s equation of motion,
loss of the total power) in the following cases (note that
we fix q >0, while both γ and ǫ may have either sign): d dξ dU sinh(ηξ)
2η = =2ǫη3 . (21)
ǫ > 0, 0< γ <ǫ/q; dz dz −dξ cosh3(ηξ)
(cid:18) (cid:19)
−
γ > 0, ǫ< γq. (13) In the presence of the dissipative potential γ, the
− ∼
Otherwise,thescatteringleadstotheincreaseofthetotal mass of the particle does not remain constant, because
power. the total power (norm) of the soliton evolves according
The above analysis does not take into regard the pos- to the equation
sibility ofthe existenceofa localizedlinear mode pinned
d +∞ +∞
to the dipole, which can be found at ǫ < 0, in the ψ(x)2dx = 2 W(x) ψ(x)2dx
PT dz | | | |
form of Z−∞ Z−∞
∂
U =U eǫ|x|[1+iγsgn(x)], (14) = 2γ ψ(x)2 , (22)
0 − ∂x | | |x=0
witharbitraryamplitudeU ,andthesingleeigenvalueof (cid:16) (cid:17)
0 or, after the substitution of ansatz (16),
the propagationconstant:
k =k0 ǫ2/2. (15) dη = 2γη3 sinh(ηξ) . (23)
≡ dz − cosh3(ηξ)
Indeed, the above scattering solutions exist for k < 0,
whileeigenvalue(15)ispositive,hencethescatteringand Underthe actionofthe samedissipativepotential,the
trapped-mode states cannot coexist. totalmomentumofthewavefield,definedasinEq.(19),
Finally, the result given by Eq. (8) should describe suffers losses according to the equation
approximately the scattering for broad pulses with ql
1,whereq isthe centralcarrierwavenumber,andl ist≫he dP = +∞W(x) ∂ ψ(x)2 dx. (24)
spatial width of the pulse. dz ∂x | |
(cid:18) (cid:19)γ Z−∞ h i
4
Substituting here expression (2) for W(x) and combin- Then,thesubstitutionofthechirpedansatz(28)intoEq.
ing it with Newton’s equation (21), one arrives at the (24), and the subsequent substitution of the respective
following evolution equation: correctiontodP/dz intoEq.(21),yields,insteadof(25),
a nonzero acceleration:
d dξ sinh(ηξ)
η =ǫη3 +γη4 3sech4(ηξ) 2sech2(ηξ) ,
dz dz cosh3(ηξ) − dv +∞ (x ξ)dx
(cid:18) (cid:19) =2bη W(x) −
(cid:2) (25) (cid:3) dz cosh2[η(x ξ)]
where c is substituted as per Eq. (17). Z−∞ −
tanh(ηξ)
Thus, the motion of the soliton interacting with the =4γ2η3 [2ηξtanh(ηξ) 1], (30)
cosh4(ηξ) −
localized potential is described, in the simplest ap-
PT
proximation,bythethird-ordersystemofcoupledODEs,
where we have inserted W(x) = 2γδ′(x), as per Eq. (2),
Eqs. (23) and (25). For the fast incident soliton, i.e.,
expression (29) for b, and Eq. (23) for dη/dz.
when dξ/dz(z ) = c is large, Eqs. (23) and (25)
→−∞ 0 Thisapproximationforthedynamicsofbrightsolitons
canbe solvedperturbatively,assuming,inthezero-order
is completely different from that derived in Ref. [23] for
approximation,
another localized -potential. Comparison of predic-
PT
ξ(z)=c z. (26) tions basedonEqs.(23)and (25) or(30) with numerical
0
findings will be presented in Section III. In particular,
However, the first-order collision-induced changes of the
thepost-adiabaticapproximation,whichmakesuseofEq.
soliton’samplitude andmomentum,∆η and∆(2ηc), ex-
(30), is accurateenoughforγ >0 andnegligiblysmallǫ,
actly vanish in this limit. Indeed, substituting approxi-
see Fig. 3(b,d) below.
mation(26)intotheexpressionsfollowingfromEqs. (23)
and (25),
+∞ dη +∞ d dξ C. Moving dark solitons
∆η = dz, ∆(2ηc)=2 η dz,
dz dz dz
Z−∞ Z−∞ (cid:18) (cid:19) Dark solitons are produced by the following modifica-
(27)
tion of Eqs. (1) and (2):
it is easy to check that both integrals are exactly equal
to zero. Thus, in the lowest-order approximation the
∂ψ 1∂2ψ
collisioniscompletelyelastic,whichisamanifestationof i = +[ǫδ(x)+iγδ′(x)]ψ+ ψ 2 µ ψ, (31)
∂z −2∂x2 | | −
the symmetry of the model.
PT (cid:0) (cid:1)
Numerical results displayed below [see Fig. 3(a)] where µ is the chemical potential (i.e., squared ampli-
demonstrate that the full approximation, based on Eqs. tude) of the continuous wave background maintaining
(23) and (25), is in agreement with simulations of the the dark-solitonsolution. Asymptotic theories forslowly
underlying equation (1) with g = +1 for 0 < γ < ǫ, i.e., movingdarksolitonshavebeendevelopedpreviously[31–
when the local defect is composed of the repulsive local 36]. Here, we aim to present a perturbation theory for a
potential and the dipole which is weaker than the movingshallow(light-gray)darksolitoninteractingwith
PT
potential. When ǫ<0, i.e., the local potential is attrac- the -symmetric dipole. Comparison of the analysis
tive, the disagreement is expected [see Fig. 3(c) below], withPnTumericalresultsisnotstraightforward,asthesim-
as the analysis does not take into regard the formation ulations, reported in the following section, demonstrate
ofthetrappedmode,whichinthelinearlimitisgivenby the generation of additional dark solitons, which is a
Eq. (14). clearly nonperturbative effect. Nevertheless, some quali-
For vanishing ǫ, the accelerationor decelerationof the tative comparison will be possible, and, in any case, the
solitoninteractingwiththedefectcanbeaccountedforif analysis may be of theoretical interest by itself.
the deviation of the phase of the perturbed soliton from We start by substituting into Eq. (31) the Madelung
the adiabatic approximation, corresponding to Eq. (16), form,
is taken into regard. Indeed, a well-known fact is that
the perturbed soliton, whose inverse width (alias ampli- ψ(x,z)=ρ(x,z)exp(iφ(x,z)), (32)
tude), η, varies in the course of the evolution, η = η(z),
generates an additional chirp term in the phase, hence replacing Eq. (31) by a system of real equations for the
ansatz (16) is replaced by amplitude and phase:
ψ(x,z)=η(z) sech[η(z)(x ξ(z)] ∂ρ 1 ∂2φ ∂ρ∂φ
− = ρ +ǫδ(x)ρ, (33)
exp ivx+ib(z)(x ξ(z)2)+iφ(z) , (28) ∂z −2 ∂x2 − ∂x∂x
× − ∂φ 1 ∂2ρ 1 ∂φ 2
where, as befor(cid:2)e, v = dξ/dz, and the express(cid:3)ion for the = ρ−1 γδ′(x) ρ2 µ . (34)
∂z 2 ∂x2 − 2 ∂x − − −
chirp coefficient is produced by the variational approxi- (cid:18) (cid:19)
(cid:0) (cid:1)
mation [30]:
As in the case of Eq. (30), we focus on the case when
dη only the imaginary potential is present, i.e., ǫ = 0 (the
b(z)= [2η(z)]−1 . (29) dynamics of dark solitons in the presence of various real
− dz
5
potentialswasstudiedindetailbefore[31–34]),whilethe where the prime stands for d/dx. Equation (43) is sup-
termγδ′(x)inEq.(34)maybetreatedasasmallpertur- plementedbythefollowingboundaryconditionsatx=0:
bation. Then, the standard approach to the description
of shallow dark solitons proceeds by setting [37] ∆bx=0 = 2γa(x=0), (44)
|
∆(a′) = 2ǫa(x=0), (45)
x=0
ρ=√µ(1+ερ ), (35) |
1
X 2√ε(x+√µz), Z √µε3/2z, (36) where ∆(...) stands for the jump of the respective func-
≡ ≡ tion at x=0. It is implied that functions a(x) and b(x)
where ε is a formal small parameter accounting for the in solution (42) are even and odd functions of x, respec-
shallowness of the gray soliton. tively, hence b(x = 0) = 0. The corresponding solutions
The result of the analysis in the case of γ = 0 is the to Eq. (43) are found in two different forms, depending
relation between φ and ρ1, ∂φ/∂X = ρ1/ 2√µ , and on the sign of ǫ, viz.,
−
the Korteweg-de Vries (KdV) equation for the evolution
of the amplitude perturbation: (cid:0) (cid:1) ψ(x)=√µ[cosθ+i sgn(x)sinθ]tanh[√µ(x +ξ)],
| |
(46)
∂ρ1 ∂ρ1 ∂3ρ1 for ǫ>0 (the repulsive dipole), and
6ρ + =0. (37)
∂Z − 1∂X ∂X3
ψ(x)=√µ[cosθ+i sgn(x)sinθ]coth[√µ(x +ξ)],
At the next order, via transformations (36), the pertur- | |
(47)
bation term γδ′(x) in Eq. (33) gives rise to the corre-
for ǫ < 0 (the attractive one). In fact, solution (47)
sponding perturbation dipole term in Eq. (37):
describes an antidark soliton pinned to the dipole.
PT
∂ρ ∂ρ ∂3ρ 4γ 2 The substitution of expressions (46) and (47) into Eqs.
1 6ρ 1 + 1 = δ′ X Z . (38) (44)and(45)yields aresultwhichis validforeither sign
∂Z − 1∂X ∂X3 ε3/2 − ε
(cid:18) (cid:19) of ǫ:
Equation (38) is, in turn, tantamount to the perturbed
1 4µ 2√µ
KdV equation studied in Ref. [38]. As shown in that ξ = ln +1+ , θ =arctanγ. (48)
work, solutions to Eq. (38) in the form of the soliton 2√µ rǫ2 |ǫ| !
interacting with the moving dipole can be looked for as
In the system with ǫ =0, which is simulated below, Eq.
2κ2 (48) yields ξ = , and the corresponding solutions (46)
ρ1 =−cosh2(κ(X 2Z/ε)+ζ(Z)), (39) and (47) degene∞rate into a constant-amplitude continu-
− ous wave (CW) with an embedded phase jump at x=0,
where the soliton’s amplitude, κ(Z), and position shift,
ζ(Z), evolve according to the following equations: ∆φ=2arctanγ. (49)
dκ 2γ κsinhζ The solutions given by Eqs. (46)-(48) are dark-soliton
= , (40)
dZ ε3/2 cosh3ζ counterparts of the exact stable solutions for pinned
bright solitons found in Ref. [25]), for ǫ < 0 (the at-
dζ 2 2γ 1
= κ 4κ2 + . (41) tractivedipole)andboththeself-focusinganddefocusing
dZ − ε ε3/2cosh2ζ
(cid:18) (cid:19) signsof the nonlinearity inEq.(1). In the limit ofǫ=0,
As showninRef. [38], dynamicalsystem (40), (41) gives the latter solution for the focusing nonlinearity amounts
rise to unbounded and trapped trajectories in the (ζ,κ) totheusualbrightsolitonwiththesameembeddedphase
plane, which, in terms of Eq. (31), correspond to solu- jump (49).
tions for freely moving dark solitons, and those trapped
by the dipole. Comparison of these results with nu-
mericalPsTimulation is possible in a qualitative form, as III. NUMERICAL RESULTS
shown in Section III.
To study the soliton scattering by the -symmetric
PT
dipole, we implemented the fourth-order Runge-Kutta
D. Exact solutions for pinned dark solitons method for integrating Eq. (1), with the Laplacian ap-
proximated by the three-point central discretization.
Stationary solutions to Eq. (31) for pinned dark soli- The simulations were carried out in spatial interval
tons can be looked for as ( L,L]withL 50,anddiscretestepsizes∆x=0.1and
− ≥
∆z =0.005 or smaller (it was checked that any decrease
ψ(x)=a(x)+ib(x), (42) of∆xand/or∆zdidnotproduceanyconspicuouseffect).
Following Ref. [25], the delta-function and its derivative
withψ(x) satisfyingthe stationaryversionofEq.(31)at
were approximated by
x=0,
6
s 2sx
− 21ψ′′+ |ψ|2−µ ψ =0, (43) δ(x)= π(x2+s2), δ′(x)=−π(x2+s2)2, (50)
(cid:0) (cid:1)
6
with s=0.1. This choice secured the inner width of the
25
regularized delta-functions to be much smaller than the 0.9
width of the incident soliton. 0.8
20
0.7
0.6
A. Scattering of Gaussian wave packets 15
0.5
z
0.4
First, we consider the passage of dispersive Gaussian 10 1.5
wave packets of width A−1/2 and velocity (spatial tilt) 1 0.3
v0 throughthelocalizeddefectinthelinearsystem,with 5 0.5 0.2
g = 0 in Eq. (1). To this end, the initial condition is 0 0.1
taken as 0 0 z 50 0
−50 0 50
ψ(x,0)=Ae−A(x−x0)2eiv0(x−x0), (51) x
(a)v0=1
where the amplitude A is fixed arbitrarily, as the model
is currently linear, and the initial position of the soliton
7
is x = 10. To provide a quantitative description of
0 − 0.9
the reflection and transmission, we compute the relative 6
0.8
powersofwavesthatremainbeforethedefect(atx<0),
5 0.7
and those which have been transmitted past the defect
0.6
(tox>0),P /P andP /P ,accordingtothefollowing 4
R I T I
definitions: z 0.5
3 1.5 0.4
0
P = ψ(x,z)2dx, 2 1 0.3
R
| |
Z−L 0.5 0.2
L 1
P = ψ(x,z)2dx, (52) 0 0.1
T Z0L | | −02 0 0 020 2 z4 406 0
P = ψ(x,0)2dx. x
I
| |
Z−L (b)v0=5
It will be then natural to compare their asymptotic val-
ues at z with the reflection and transmission co- FIG. 1: (Color online) The interaction of the incident Gaus-
efficients→for∞the plane waves, (R2+R2) and (T2+T2), sian wave packets with the localized defect, for ǫ = 0 and
1 2 1 2
as given by Eq. (8), with wavenumber q replaced by γ = 0.3. Shown is the distribution of |ψ(x,z)|. Solid blue
andredlinesin theinsetsdepict theevolutionoftherelative
incident velocity v .
0 powers defined as PR/PI and PT/PI, respectively, see Eq.
InFig.1wedisplaytheevolutionoftheincidentGaus-
(52). Theirasymptoticvaluesatz →∞arecompared tothe
sian wave packet impinging onto the defect with ǫ = 0,
reflection and transmission coefficients for the plane waves,
γ = 0.3, at two different values of v0. Shown is the top (R12+R22)and(T12+T22),givenbyEq. (8)(horizontaldashed
viewoftheabsolutevalueofthefield, ψ(x,z). Insetsto lines).
| |
thesamefigurepresentcoefficientsP /P andP /P ,as
R I T I
described above. One can observe that, naturally, larger
incoming velocity v makes the values of the coefficients
0 centered at x = 10, with initial velocity v > 0, and
atz closertoexactresultsfortheplanewavesgiven 0 − 0
→∞ η =1 (once ǫ=0 was set, η =1 may be always fixed by
byEq. (8),astheparameteraccountingforthedifference
rescaling).
oftheGaussianpulse(51)fromtheplanewaveis√A/v .
0 Shown in Fig. 2 are two pairs of examples of the in-
The case of γ < 0 is not shown here separately, as the
teraction of the soliton with the dipole. In panels (a,b),
respective results are quite similar to those presented in
the case of γ >0 is considered, which, according to Eqs.
Fig. 1.
(1), (2) and (50), implies that the incident soliton im-
pinges on the dipole from the side where the amplifying
(rather than attenuating) element is located. In panel
B. Dynamics of bright solitons
(a) of Fig. 2, the soliton gets trapped by the defect and
blows up, which happens when the initial velocity is suf-
In the model with the self-focusing nonlinearity, g =
ficiently small. On the other hand, when the velocity is
+1 in Eq. (1), we simulated collisions of the incident
sufficiently large, the incoming soliton, quite naturally,
bright soliton with the dipole, setting ǫ = 0 in Eq.
PT passes the defect, as seen in panel (b). These two exam-
(2). The initial conditions are taken as per expression
ples are typical for such outcomes of the collision.
(16), i.e.,
In panels (c,d) of Fig. 2, we display the evolution of
ψ(x,0)=η sech[η(x x ]exp(iv (x x )), (53) the soliton for γ = 0.5, when the the incident soliton
0 0 0
− − −
7
approaches the dipole from the side of the attenuating cumbersome form, due to the complexity of the respec-
element. Onthecontrarytopanel(a),wheretheblowup tive integrals in Eqs.(22), (24), and (30). Generally, the
was observed, in the present case the incident soliton is factthatthediscrepancybetweenthenumericalandana-
reflectedifitsvelocityissmallenough. Naturally,there- lyticalresultsinFig.3issmallerforγ >0isexplainedby
flected soliton has a smaller amplitude than the incident the factthat the largeramplitude ofthe pumped, rather
one, due to the action of the attenuation. On the other than attenuated, soliton in this case (see above) makes
hand, it is shown in panel (d) the the soliton passes the the local perturbation weaker in comparison with other
defectifthe velocityislargeenough,similartowhatwas terms in Eq. (1).
observed for γ > 0 in panel (b). In all the panels, the
insetsshowthe reflectedandtransmittedpowers,P /P
R I
and PT/PI, defined according to Eq. (52), as above. C. Dark solitons
Their asymptotic values at z are compared with
→ ∞
the exactresults,(R2+R2)and(T2+T2), forthe plane
1 2 1 2 To consider the interaction of dark solitons with the
waves, as given by Eq. (5).
dipole, we fix the CW-backgroundamplitude in Eq.
Obviously, an important characteristic of the interac- P(3T1) as µ = 1. In the absence of dark solitons, the CW
tion of the soliton with the dipole, which also in- background, ψ , is deformed by the potential [40, 41].
PT CW
cludes the attractive or repulsive local potential, as per Asshownabove,inthelimitofǫ=0andidealδ′function
Eq. (2), is the minimum (threshold) velocity necessary in Eq. (31), the deformationamounts to the phase jump
forthe solitontopassthisdefect. Weaimtoidentifythe (49) at x=0.
threshold velocity produced by the direct simulations of In Fig. 4(a), we plot the shape of the background ob-
Eq. (1), and compare it to predictions of the (semi-) an- tainedin the numericalform, with the δ′ function in Eq.
alyticalapproximationbasedonquasi-particleequations (31) replaced by regularization (50)], for γ = 0.3 and
(23), (25), and (30). Because not the entire power is ǫ = 0. Similarly to the previous works, we find that
reflected or transmitted as a result of the collision, we this groundstate,producedbythe stationarysolutionof
define the soliton as being transmitted past the defect Eq. (31), exists at γ < 0.49 (at γ exceeding this critical
when, at least,half ofits totalpower is transmitted, i.e., value, the system starts spontaneous generation of dark
in terms of the insets of Fig. 2, the transmission thresh- solitons [40, 41]). The difference of the backgroundfrom
oldcorrespondstothepointwherethesolidblueandred the above-mentioned analytical solution, which amounts
lines cross. to the phase jump (6) embedded into the constantback-
The threshold velocities produced by the direct simu- ground, is explained by the difference of approximation
lations are displayed in Fig. 3, as functions of the - (50)fromtheidealδ′ function. Additionally,wealsoplot
PT
dipole’s strength, γ,bycrosses. Predictionsproducedby in the same figure in panel (b) and (c) the profile of the
a numerical solution of Eqs. (23) and (25) are presented plane waves in the presence of nonzero ǫ.
too in this figure, by means of dashed lines. In addition Due to the presence of the non-uniform CW back-
to the approximation based on Eqs. (23) and (25), for ground(ψ ),wesimulatedthedynamicsofadarksoli-
CW
the caseofsmallǫ, suchasinpanels (b)and(d), we also ton in the framework of Eq. (1) with initial conditions
plot,bydashed-dottedlines,the predictiongeneratedby
the numerical solution of Eqs. (23) and (30), which was
ψ(x,0)=ψ 1 v2tanh 1 v2(x x ) +iv ,
derived for ǫ=0. CW − 0 − 0 − 0 0
(cid:20)q (cid:18)q (cid:19) (cid:21)
It is seen fromFig. 3(a) that, as mentioned in Section (54)
II.B, the quasi-particle adiabatic approximation, based where v0 and x0 determine the initial velocity and posi-
on Eqs. (23) and (25), is in good agreementwith the di- tion of the dark soliton.
rectsimulationsfor0<γ <ǫ. Ontheotherhand,panels In Fig. 5, we plot simulatedpictures ofthe interaction
(b) and (d) demonstrate that the post-adiabatic approx- of the dark soliton with the dipole for parameter
PT
imation, represented by Eqs. (23) and (30), which takes values indicated in the caption to the figure. Similar to
intoregardthethegenerationoftheintrinsicchirpinthe brightsolitonsconsideredabove,thedarksolitoniseither
soliton[seeEq. (29)],isrelevantfor ǫ γ .0.2(which transmitted or reflected. In panel (b) of Fig. 5, an addi-
| |≪
impliesthatγ ispositive). Thesameapproximationpro- tional weaker reflected dark soliton emerges, as a result
vides for a qualitative prediction of the threshold veloc- of the interaction, in addition to the main passing soli-
ity for γ <0 and ǫ γ too, even though in that case ton. Anotherparticularresultisseeninpanel(d),where
| |≪| |
thepredictionisquantitativelyinaccurate. Thereasonis the reflected feature, observed at x < 0, is not a soliton
that,ascanbeseenfromnumericaldata(notshownhere but a shelf, propagating with the speed determined by
in detail), the deformationof the solitonaroundx=0 is the background amplitude (the generation of shelves by
not small in the latter case, which cannot be taken into dark solitons was considered in Ref. [42]).
account by the perturbative treatment. In this sense, a The analytical approximation for the dark-soliton dy-
better agreement with the perturbation theory may be namics, based on equations (40)-(41) for variables κ(Z)
expected for a smoother shape ofthe dipole [see Eq. and ζ(Z), was derived in the framework of the adia-
PT
(50)], but in that case the analytical results take a more batic approach, which does not take into regard the
8
13 5
2.5
1.5
4.5 1.2
12 1
4
11 0.5 2 1
3.5
2 4 6 81012
10 z 1.5 3 0.8
z 9 z2.5
0.6
8 1 2 1.5
1.5 1 0.4
7
0.5 1 0.5
6 0.2
0.5 0
5 0 0 2 z 4 0
−20 −15 −10 −5 0 5 −20 −10 0 10 20
x x
(a)γ=0.3,v0=0.8 (b)γ=0.3,v0=5.0
60 1 30
1.2
50 25
0.8
1
40 20
0.6 0.8
z30 z15
0.6
1.5 0.4 1.5
20 10
1 1 0.4
10 0.5 0.2 5 0.5 0.2
00 20 z 40 60 00 10z 20 30
0 0
−20 −10 0 10 20 30 −20 0 20
x x
(c)γ=−0.5,v0=0.4 (d)γ=−0.5,v0=1.0
FIG. 2: (Color online) Examples of the trapping and blowup (a), and transmission (b), of the incident bright soliton by the
PT-symmetric dipole with γ = 0.3 and ǫ = 0. In panels (c) and (d), the incident soliton bounces back from the dipole, or
passes it, respectively, for γ =−0.5 and ǫ=0. Shown is thedistribution of |ψ(x,z)|. Similar to Fig. 1, theinsets present the
evolution of the scaled transmission and reflection powers, and compare their asymptotic values to the respective coefficients
for the plane waves.
generation of the additional dark soliton or non-soliton dipole.
shelf, hence this approximation cannot describe the ob-
served phenomenology well enough. Nevertheless, pre-
dictions of the analysis may qualitatively explain some IV. CONCLUSION
features of the dynamics revealed by numerical simula-
tions. For the sake of the comparison, obtaining coor-
We havestudied the dynamics ofbrightand darksoli-
dinate ζ from simulation results of Eq. (1) is straight-
tons in the model based on the focusing and defocusing
forward, while amplitude κ can be identified as κ(z) =
NLS equations with an embedded defect in the form of
sign(x0) (1 ψ(x=ζ,z)2)/2. Note also that our an- the -symmetric dipole, combined with a local repul-
alytical appro−xi|mation is d|erived under the assumption sivePorTattractive potential. The scattering problem for
p
v0 1. In that regard, our approximation should only plane waves and broad incident packets was considered
| | ∼
be compared with dynamics in panels (b) and (d). In too in the framework of the linear version of the model.
particular, in the case shown in Fig. 5(b), the approxi- The numerical study for the focusing nonlinearity has
mation correctly predicts that the incident dark soliton produced threshold values of the velocity of the incident
would pass through the dipole, although there is a bright soliton above which it passes the local defect.
PT
discrepancy in approximating the phase shift of the soli- For the defocusing nonlinearity, the interaction of dark
tonafter interaction– mostplausibly,causedby the fact solitons with the defect is studied in the numerical form
that the adiabatic approximation cannot take into ac- too. Parallel to the simulations, we have developed
count the generation of the additional reflected soliton, analytical approximations for both cases. For the bright
in this case. Nevertheless, the approximation correctly solitons, the adiabatic quasi-particle approximation
predictsthatthe solitonacceleratesinthe vicinityofthe yields accurate results in the case when the repulsive
9
1
1
0.8
0.8
0.6
n
0.6 mi
n
mi v
v 0.4
0.4
0.2
0.2
0 0
−0.4 −0.2 0 0.2 −0.4 −0.2 0 0.2
γ γ
(a)ǫ=0.2 (b)ǫ=0.02
1 1
0.8 0.8
0.6 0.6
n n
mi mi
v v
0.4 0.4
0.2 0.2
0 0
−0.4 −0.2 0 0.2 −0.4 −0.2 0 0.2
γ γ
(c)ǫ=−0.2 (d)ǫ=−0.02
FIG.3: (Color online) Theminimum velocity necessary for thetransmission of thesoliton past thePT dipole, which includes
the local potential, as per Eq. (2). The crosses and dashed lines represent, respectively, results of the direct simulations of
Eq.(1),and theapproximation produced bya numerical solution of Eqs. (23) and (25). For small ǫ in panels (b)and (d),the
approximation corresponding to Eqs. (23) and (30) is additionally plotted by thedashed-dotted line.
potential is stronger than the gain-and-loss component the exact solution for the dark soliton pinned by the
of the defect. For the negligibly weak local potential, -symmetric defect was found too.
PT
the analyticalconsiderationgoes beyondthe limit of the
adiabaticapproximation,taking into regardthe intrinsic Acknowledgement
chirp of the soliton. The respective semi-analytical
results predict the threshold velocity in a reasonably
accurateformtoo. Forthe darksolitons,the approxima- WH and HS thank the School of Mathematical Sci-
tion qualitatively explains the transmission,acceleration ences,UniversityofNottingham,foraSummerResearch
and deceleration of the incident soliton. In addition, Bursary (June–September 2013).
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