Acta Polytechnica
doi:10.14311/AP.2014.54.0079
Acta Polytechnica 54(2):79–84, 2014 © Czech Technical University in Prague, 2014
available online at http://ojs.cvut.cz/ojs/index.php/ap
TRACKING DOWN LOCALIZED MODES IN PT-SYMMETRIC
HAMILTONIANS UNDER THE INFLUENCE OF
A COMPETING NONLINEARITY
Bijan Bagchia, ∗, Subhrajit Modakb, Prasanta K. Panigrahib
a Department of Applied Mathematics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata-700
009, India
b Department of Physical Science, Indian Institute of Science, Education and Research (Kolkata), Mohanpur,
West Bengal 741 252, India
∗ corresponding author: bbagchi123@rediffmail.com
Abstract. The relevance of parity and time reversal (PT)-symmetric structures in optical systems
has been known for some time with the correspondence existing between the Schrödinger equation and
the paraxial equation of diffraction, where the time parameter represents the propagating distance
and the refractive index acts as the complex potential. In this paper, we systematically analyze a
normalized form of the nonlinear Schrödinger system with two new families of PT-symmetric potentials
in the presence of competing nonlinearities. We generate a class of localized eigenmodes and carry out
a linear stability analysis on the solutions. In particular, we find an interesting feature of bifurcation
characterized by the parameter of perturbative growth rate passing through zero, where a transition
to imaginary eigenvalues occurs.
Keywords: nonlinear Schrödinger equation, PT-symmetry, competing nonlinearity.
1. Introduction
Following Bender and Boettcher’s seminal paper [1],
in which they offered the first coherent explanation of
a special class of non-Hermitian but parity and time-
reversal (PT)-symmetric Hamiltonians to possess a
real bound-state spectrum, the last decade has wit-
nessed extensive theoretical work [2–4] being devoted
to this growing field of research. The interplay be-
tween the parametric regions where PT is unbroken
and the regions in which PT is broken as signaled
by the appearance of conjugate-complex eigenvalues
(see, for example, [5–7]) has for some time found re-
peated experimental support [8–17] as evidenced by
the observations of a phase transition that clearly
marks out the separation of these regions. It is useful
to bear in mind that the analytical studies in this
regard have mostly been carried out for the linear
domain. Of late, the relevance of PT-structure has
been noticed in various optical systems and inter-
esting features have been seen such as, for example,
power oscillations [9], unidirectional invisibilty [13],
coherent perfect absorber [12, 18], giant wave amplifi-
cation [19] and realiztion through electromagnetically
induced transparency [20]. In optical systems, PT-
symmetry has the implication that the index guid-
ing part nR(x) and the gain/loss profile nI(x) of the
complex refractive index n(x) = nR(x) + inI(x) obey
the symmetric nR(x) = nR(−x) and antisymmetric
nI(x) = −nI(−x) combinations (see, for example,
[21–23]). Balancing gain and loss [24–27] is an inter-
esting curiosity towards experimental realization of
PT-symmetric Hamiltonians.
Against the background of the experimental find-
ings, Musslimani et al [24, 25] have reported optical
solitons in PT-periodic potentials which are stable
over a wide range of potential parameters. Specifi-
cally they have considered optical wave propagation
with the beam evolution being controlled by a normal-
ized nonlinear Schrödinger (NLS) equation defined in
terms of an electric field envelop and a scaled prop-
agation distance. Indeed, the generalized NLS that
they consider, in the presence of a PT-symmetric
potential, is given by
iψz + ψxx +
[
V (x) + iW(x)
]
ψ + g|ψ|2ψ = 0, (1)
with the PT-symmetric potential possessing the usual
properties [28] V (−x) = V (x) and W(−x) = −W(x).
In (1), ψ represents the electric field envelope, z is
a scaled propagation distance and g = 1 or −1 cor-
responds to a self-focussing or a defocussing nonlin-
earity. Further, the ψxx term describes the optical
diffraction, V (x) is the index guiding and W(x) rep-
resents the gain/loss distribution of the optical poten-
tial. Musslimani et al [24, 25] studied nonlinear sta-
tionary solutions of the form ψ(x,z) = φ(x) exp(iλz),
λ being a real propagation constant and φ is the
signature of the nonlinear eigenmode. In the con-
text of nonlinear optics, localised modes are either
temporal or spatial depending on whether the con-
finement of light occurs in time or space during wave
79
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http://ojs.cvut.cz/ojs/index.php/ap
B. Bagchi, S. Modak, P. K. Panigrahi Acta Polytechnica
propagation.In particular, spatial modes represent
propagating transverse self-guided beams orthogonal
to the direction of movement. Because of a balance
between the diffraction and the Kerr effect, a spa-
tial mode does not change with propagation. These
modes are termed as spatial solitons. Both the types
of solitons emerge from a nonlinear change in the re-
fractive index of an optical material induced by the
light intensity. This phenomenon is referred to as
the optical Kerr effect. The intensity dependence of
the refractive index leads to spatial self-focussing (or
self-defocussing) and temporal self-phase modulation,
the two major nonlinear effects that are responsible
for the formation of optical solitary modes or optical
solitons [29].
In this article we report on some new localized solu-
tions of the NLS and study the distribution of eigen-
modes on the real and complex plane by incorporat-
ing the effects of higher degree nonlinear effects over
and above the minimal cubic term. By parametriz-
ing the coupling strength of the latter and arbitrar-
ily specifying the order of additional nonlinearity on
a Rosen-Morse potential we observe numerically for
one class of solutions the existence of a threshold
value of the growth rate parameter beyond which
suitably chosen pairs of discrete eigenmodes on the
real axis merge and subsequently appear in conju-
gate imaginary pairs exhibiting the qualitative char-
acter of bifurcation. In this connection it needs to be
pointed out that our model differs significantly from
those advanced so far to search for solitonic solutions
[30, 31]. For instance, the potentials of our interest
are markedly different from the Rosen-Morse type
considered in [31] because of the presence of an addi-
tional nonlinear term in our case. The PT-symmetric
potentials addressed in [30] are basically nonlinear
extensions of the Scarf II. Also, the above aspect of
bifurcation did not arise in the models considered in
[30, 31].
2. Mathematical Model and
Formulation
We consider here an optical wave propagation in the
presence of a PT -symmetric potential. In this case
the beam dynamics is governed by a generalized non-
linear Schrödinger model with competing nonlineari-
ties, i.e.,
i
∂Ψ
∂z
+
∂2Ψ
∂x2
+
[
V (x) + iW(x)
]
Ψ
+ g1|Ψ|2Ψ + g2|Ψ|2κΨ = 0, (2)
where κ is an arbitrary real number, Ψ(x,z) is a
complex electric field envelope, g1 and g2 control re-
spectively the strength of the cubic and the arbitrary
nonlinear term. It is clear that Eq. (2) admits sta-
tionary solutions Ψ(x,z) = φ(x)eiλz, where λ is a
real propagation constant and the complex function
Φ(x) obeys the eigenvalue equation
∂2Φ
∂x2
+
[
V (x) + iW(x)
]
Φ
+ g1|Φ|2Φ + g2|Φ|2κΦ = λΦ. (3)
We now show that this model supports two different
soliton solutions marked by Class I and Class II cases
provided we do not alter the imaginary part of the
potential but only choose the real part appropriately.
2.1. Class I solutions
We focus on a PT -symmetric Rosen-Morse potential
−a(a + 1) sech2 x + 2ib tanh x (a, b ∈ R) being sub-
jected to an additional term −V1 sech2κ x (V1, κ ∈
R), i.e.,
V (x) = −a(a + 1) sech2 x−V1 sech2κ x,
W(x) = 2b tanh x. (4)
It may be noted that a PT-symmetric Rosen-Morse
potential has been well studied in the literature (see
[32] and references therein). A noteworthy feature of
this potential is that while its real component van-
ishes asymptotically, it’s imaginary part remains non-
zero finite.
Corresponding to (4) there always exists for (3) a
typical solution
Φ(x) = Φ0 sech(x) eiµx, (5)
provided that the amplitude Φ0 and the phase factor
µ are related to the potential parameters through
Φ2κ0 =
V1
g2
, Φ20 =
a2 + a + 2
g1
,
b = µ, λ = 1 −µ2. (6)
Note that the imaginary strength of the potential
contributes only through the phase factor whereas
for the real part both the parameters a and V1 along
with the nonlinearity parameters g1 and g2 appear
with odd powers and thus their sign is not of relevance.
For this solution the transverse power flow defined
by
S =
i
2
(φφ∗x −φ
∗φx) (7)
turns out to be S = bΦ20 sech
2(x) implying that the
transmission depends upon the strength of the imag-
inary part of the potential.
2.2. Class II solutions
On the other hand, if we choose the extended Rosen-
Morse potential to have the form
V (x) = −a(a + 1) sech2 x−V1 sech
2
κ x,
W(x) = 2b tanh x, κ ∈ R\{0} (8)
then Eq. (3) enjoys a solution
Φ(x) = Φ0 sech
1
κ x eiµx (9)
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vol. 54 no. 2/2014 Tracking Down Localized Modes in PT-symmetric Hamiltonians
if the amplitude and phase factor are constrained by
the relations
Φ20 =
V1
g1
, b =
µ
κ
, λ =
1
κ2
−µ2,
Φ2κ0 =
1
g2
[
a(a + 1) +
(1
κ
+
1
κ2
)]
. (10)
Note that the solution (10) is valid irrespective of the
signs of g1 and g2. The transverse power flow, defined
by Eq. (5), turns out to be S = bΦ20κ sech
2
κ x, which
in this case is influenced by both b and κ including
of course the effects of their signs.
3. Numerical computations and
Eigenmode distribution
Solitary waves associated with the non-Kerr nonlin-
ear media retain their shape but their stability is not
guaranteed because of the nonintegrable nature of
the underlying extended NLS equation that we have
at hand. In fact, their stability against small pertur-
bation is an important issue because only stable (or
weakly unstable) self-trapped beams can be observed
experimentally. Let us impose a small perturbation
to determine whether it is stable or unstable against
this slight disturbance. More specifically we consider
a perturbation of the form [30]
Ψ(x,z) = φ(x)eiλz +
{[
v(x) + ω(x)
]
eηz
+
[
v∗(x) −ω∗(x)
]
eη
∗z
}
eiλz, (11)
where v(x) and ω(x) are infinitesimal perturbed eigen-
functions such that |v|, |ω| � |φ| and η indicates
the perturbed growth rate. Linearization of Eq. (1)
around Φ(x) yields the following eigenvalue problem(
0 L̂0
L̂1 0
)(
v
ω
)
= −iη
(
v
ω
)
, (12)
where L̂0 = ∂xx−λ+ (V +iW) +g1|φ|2 +g2|Φ|2κ and
L̂1 = ∂xx −λ + (V + iW) + 3g1|φ|2 + g2(1 + 2κ)|Φ|2κ
and η is the associated eigenvalue corresponding to
the growth rate parameter. The η-spectrum is called
the linear-stability spectrum for the localized modes.
It is easy to see if η is an eigenvalue then so are η∗,
-η, and -η∗ indicating that these eigenvalues always
appear in pairs or quadruples.
The continuous spectrum of Eq. (12) can be read-
ily recovered in the large-distance limit of |x| → ∞.
Under this limit, L̂0 and L̂1 move over to a simple
differential operator with a constant coefficient. In
order for η to be in the continuous spectrum, the
corresponding eigenfunction at large distance must
be a Fourier mode. If we observe the orientation
of the eigenmodes in the entire spectrum, we run
into three different kinds of modes. The appearance
of nonzero discrete eigenvalues in the linearization
spectrum of solitary waves is a signature of nonin-
tegrable character of the equation. If the spectrum
contains a real positive eigenvalue, the correspond-
ing eigenmode in the perturbed solution will grow
exponentially with time; hence the solitary wave is
linearly unstable. Generally if the spectrum contains
any eigenvalue with a real positive part then such
eigenvalues are unstable. Secondly if the spectrum
admits of a pair of conjugate-complex eigenvalues
(internal modes) the perturbed solution will exhibit
oscillations leading eventually to shape fluctuations
that would be smothered with time. Thirdly one can
encounter zero eigenvalues which are the so-called
Goldstone modes (see for a discussion on this point
[30]). The behavior of the eigenvalues η can be as-
certained by solving Eq. (12) numerically. Here we
adopt the Fourier collocation method [33, 34] to track
the tendencies of the eigenvalues. We now turn to
some discussions of our results.
4. Results and Discussion
Figures 1–4 give a graphical display of our numerical
results on the eigenmode distribution. The interplay
between the cubic and competing nonlinearity on the
soliton dynamics is best understood in terms of the
parameters Φ0, |g1|, |g2|. We also look for stability
around some specific value of κ, as mentioned in the
Figure captions. It should be borne in mind that
Eq. (6) and Eq. (10) constrain these parameters in
terms of the amplitude Φ0 and the phase factor µ as
well as the coupling constants of the potential. The
evolution of a Class I solution for different choices
of g1 and g2 is laid down in Fig.1. We start with a
sample choice of a, b and g2 (for specific values of
these parameters see the corresponding Figure cap-
tions) when the discrete modes lie on the real axis.
Of course a continuous change of the parameter a will
ultimately result in the eigenvalues appearing with a
non-vanishing imaginary part (discussed below).
Normalizing Φ0 to unity without any loss of infor-
mation, g1 gets automatically fixed while b, λ and
µ acquire their values from the consistency condi-
tions (6). In this manner of parametrization g1 and
g2 differ in sign while the magnitude of g1 turns out
to be weaker than g2. In Fig.2 the plots are sequen-
tially arranged according to the varying strengths of
the couplings corresponding to the cubic nonlinear
term and holding g2 fixed. We note that g1 changes
according to the potential parameter a. A new type
of solution develops at this point due to the sensitive
dependence of the perturbative growth rate parame-
ter η on a: around a = 0.03 we see that as parameter
a is varied the discrete modes initially lying on the
real axis mutually approach towards the zero-mode
eigenvalue. However, further change in a causes a
pair of imaginary eigenvalues to develop revealing a
typical feature of bifurcation. We next carry out com-
putations for equal and opposite values of |g1| and
|g2| couplings. In Fig.3 we see that in such a case
the real eigenmodes lead to the solitonic solutions be-
coming unstable. A similar situation exists in Fig.4
81
B. Bagchi, S. Modak, P. K. Panigrahi Acta Polytechnica
−0.2 −0.1 0 0.1 0.2 0.3
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re η
Im
η
−0.2 −0.1 0 0.1 0.2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re η
Im
η
−0.2 −0.1 0 0.1 0.2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re η
Im
η
−0.2 −0.1 0 0.1 0.2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re η
Im
η
Figure 2. Sequentially computed eigenmode behavior for b=.003, V1 = g2 = −4 and κ = 3. In this case the
coupling parameters corresponding to cubic nonlinearity are varied against the potential parameter a continuously
from a =.03 to .09 . Four different values of a as a=.03, a=.04, a=.05, a= .09 are considered and the Figures are
arranged in such a way that the lowest value of a corresponds to the figure at the left and the highest to the figure
at the right.
−3 −2 −1 0 1 2 3
−60
−40
−20
0
20
40
60
Re η
Im
η
Figure 1. Numerically computed eigenmode distri-
bution. In this case we have considered a = .01, b = .3,
V1 = g2 = −4, g1 = 2.01 and κ = 3. Specification of
g1 is done by choosing the potential parameter a.
−0.2 −0.1 0 0.1 0.2
−20
−15
−10
−5
0
5
10
15
20
Re η
Im
η
Figure 3. The figure shows the unstable modes for
the coupling parameters |g1| and |g2| with equal and
opposite strengths. Distribution of eigenmodes for
a=1, b=.003, g1=4, V1 = g2 = −4 and κ = 3.
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vol. 54 no. 2/2014 Tracking Down Localized Modes in PT-symmetric Hamiltonians
−0.2 −0.1 0 0.1 0.2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re η
Im
η
−0.2 −0.1 0 0.1 0.2 0.3
−4
−3
−2
−1
0
1
2
3
4
Re η
Im
η
Figure 4. Figure at the left showing the unstable modes corresponding to equal-strength couplings for the choice
g1 = 4, g2 = 4 obtained from (6) and κ = 3. The potential parameters are taken to be a = 1 and b = .003. The
figure at the right corresponds to the evaluation of Class II solution under the same parametrizations. The coupling
strengths for Class II are g1 = 4 and g2 = 2.44 obtained from (10).
where oscillatory instability along the imaginary axis
is caused by the equal-strength coupling parameters
from the two nonlinear terms.
Finally, let us point out that Class II solutions
can be evaluated under various parametric conditions.
Here we inevitably run into the unstable character of
the soliton solutions (in Figure 4 one such situation is
described to compare with the counterpart of Class
I solution). It should be emphasized that for the
various runs of the parameters we were unable to
track down any feature of bifurcation characterized by
the growth rate parameter crossing the zero-threshold
value and transiting to the complex plane.
Acknowledgements
We thank Dr. Abhijit Banerjee for useful discussions and
the referee for making a number of constructive sugges-
tions.
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Acta Polytechnica 54(2):79–84, 2014
1 Introduction
2 Mathematical Model and Formulation
2.1 Class I solutions
2.2 Class II solutions
3 Numerical computations and Eigenmode distribution
4 Results and Discussion
Acknowledgements
References