PHYSICAL REVIEW A 86, 053815 (2012)

Localized modes in χ (2) media with PT -symmetric localized potential

F. C. Moreira,1,2 F. Kh. Abdullaev,3 V. V. Konotop,2 and A. V. Yulin2

1Universidade Federal de Alagoas, Campus A. C. Simões, Avenida Lourival Melo Mota, s/n, Cidade Universitária,
Maceió Alagoas 57072-900, Brazil

2Departamento de Fı́sica, Centro de Fı́sica Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa,
Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal

3Instituto de Fisica Teórica, Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 Barra Funda, São Paulo,
Código de Endereçamento Postal 01140-070, Brazil

(Received 14 September 2012; revised manuscript received 11 October 2012; published 15 November 2012)

We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized
parity-time- (PT )-symmetric modulation of the linear refractive index. Families of stable one- and two-hump
solitons are found. The properties of nonlinear modes bifurcating from a linear limit of small fundamental
harmonic fields are investigated. It is shown that the fundamental branch, bifurcating from the linear mode of the
fundamental harmonic is limited in power. The power maximum decreases with the strength of the imaginary
part of the refractive index. The modes bifurcating from the linear mode of the second harmonic can exist even
above the PT -symmetry-breaking threshold. We found that the fundamental branch bifurcating from the linear
limit can undergo a secondary bifurcation colliding with a branch of two-hump soliton solutions. The stability
intervals for different values of the propagation constant and gain or loss gradient are obtained. The examples of
dynamics and excitations of solitons obtained by numerical simulations are also given.

DOI: 10.1103/PhysRevA.86.053815 PACS number(s): 42.65.Tg, 42.65.Sf

I. INTRODUCTION

Non-Hermitian Hamiltonians satisfying the parity-time
(PT ) symmetry can have real eigenvalues [1]. Although these
ideas were originally developed in the quantum-mechanical
context, it soon became clear that new broad applications
could be found in optics. The first suggestions of the optical
analogs of the PT -symmetric Hamiltonians were proposed
in Ref. [2] and were based on a linear planar waveguide
structure. PT -symmetry effect predictions are confirmed in
experiments with light propagation in couplers with gains
and losses [3]. Later on, exploring nonlinear optical media
obeying PT symmetry was suggested, and, in particular, the
possibility for soliton propagation was shown in such media
[4]. The PT symmetry was modeled by the refractive index
having a symmetric real part and an antisymmetric imaginary
part. A particularly interesting realization of a PT -symmetric
modulation of the refractive index is when losses and gains are
localized in space, giving rise to a PT -symmetric localized
impurity. Such impurities allow for the existence of localized
(defect) modes. In the linear theory, in Refs. [5,6], such modes
were studied for exactly integrable models, and their nonlinear
extension was reported in Ref. [4]. Solitons supported by other
PT -symmetric defects were also reported for focusing [7] and
defocusing [8] media. Linear scattering by a PT -symmetric
inhomogeneity and the emergence of the related spectral
singularities were described in Ref. [9]. The effect of two
and various randomly distributed PT -symmetric impurities
on the lattice dynamics was addressed in Ref. [10]. Switching
of solitons in a unidirectional coupler using PT -symmetric
defects was suggested in Ref. [11]. Nonlinear modes in even
more sophisticated double-well PT -symmetric potentials
were studied recently [12].

All of the papers mentioned above and devoted to nonlinear
modes dealt with the PT -symmetric media possessing Kerr
nonlinearity (see the list of recent papers on solitons in

Ref. [13]). Furthermore, it is a natural step to address the
possibility of the existence of defect modes and their stability
in another class of widely used optical media, which is the
χ (2) materials. Solitons in quadratic nonlinear media with
conservative defects were investigated in Refs. [14,15] where
it was shown that solutions were dynamically stable in the
case of attractive impurities. In the present paper, we study the
existence of solitons in the media with quadratic nonlinearity
and localized PT -symmetric potentials.

The paper is organized as follows. In Sec. II, the model
and statement of the problem are formulated. The properties
of localized modes for different ratios between fundamental
and second harmonics are studied in Secs. III–V. The stability
and dynamics of localized solutions are considered in Sec. VI.

II. STATEMENT OF THE PROBLEM

We consider the system,

i
∂u1

∂ζ
+ ∂2u1

∂ξ 2
+ V

(
1

cosh2 ξ
+ iα

sinh ξ

cosh2 ξ

)
u1 + 2u1u2 = 0,

(1a)

i
∂u2

∂ζ
+ 1

2

∂2u2

∂ξ 2
+ 2

(
V

cosh2 ξ
+ q

)
u2 + u2

1 = 0,

(1b)

describing spatial second-harmonic generation in a χ (2) ma-
terial with localized modulation of the refractive index, u1

and u2 being the dimensionless fields of the first and second
harmonics, ξ and ζ are the dimensionless transverse and
propagation coordinates, scaled to the characteristic size of
the modulation of the refractive index, which is characterized
by the amplitude V . The mismatch in the propagation constants
of field components is described by q. We notice that,
experimentally, the introduced model can describe a medium

053815-11050-2947/2012/86(5)/053815(7) ©2012 American Physical Society

http://dx.doi.org/10.1103/PhysRevA.86.053815


MOREIRA, ABDULLAEV, KONOTOP, AND YULIN PHYSICAL REVIEW A 86, 053815 (2012)

with active dopants, typically having rather narrow spectral
resonances, i.e., affecting only a limited range of frequencies.
In particular, such impurities can induce gain and dissipation,
whose strengths are characterized by α only for one of the field
components, which, in our case, is the Fundamental Field (FF).

Before getting into the detailed study of the system (1), we
note that, in the standard way, solitonic solutions can be found
in the analytical form in the limit of large mismatch parameter
−q � 1 when u2 ≈ −u2

1/(2q), and the system (1) reduces to
the nonlinear Schrödinger equation with the PT -symmetric
potential for the fundamental harmonic u1,

i
∂u1

∂ζ
+ ∂2u1

∂ξ 2
+ V

(
1

cosh2 ξ
+ iα

sinh ξ

cosh2 ξ

)
− 1

q
|u1|2u1 = 0.

(2)

For q < 0, the bright soliton solution has the form Ref. [4]

u1 =
√

|q|Asech(ξ ) exp

[
i
αV

3
tan−1[sinh(ξ )] + iζ

]
,

(3)

A =
(

2 − V + α2V 2

9

)
.

We are interested in the localized solutions,

un(ξ,ζ ) = wn(ξ )einbζ , n = 1,2, (4)

where b is the propagation constant of the first harmonic and
w1,2 solves the system,

d2w1

dξ 2
+

[
V

(
1

cosh2 ξ
+ iα

sinh ξ

cosh2 ξ

)
− b

]
w1 + 2w1w2 = 0,

(5a)

1

2

d2w2

dξ 2
+ 2

(
V

1

cosh2 ξ
+ q − b

)
w2 + w2

1 = 0

(5b)

subject to the zero boundary conditions w1,2(ξ ) → 0 as |ξ | →
∞.

We restrict our consideration mainly to solutions bifur-
cating from the linear limit, which is understood as a limit
where, at least, one of the harmonics vanishes. It follows from
Eqs. (1), that the so-defined linear limit does not necessarily
imply that both amplitudes w1 and w2 are infinitely small.
Indeed, it is sufficient to require that only the amplitude of
the fundamental harmonic w1 is infinitely small to consider
Eqs. (1) in the linear limit. That is why one can distinguish
three different bifurcations of the fundamental soliton solution
from the linear limit:

(i) The amplitude of the second harmonic is on the order of
the squared amplitude of the first harmonic, i.e., is negligible
compared to the amplitude of the first harmonic,

w2 = O
(
w2

1

)
, w1 → 0. (6)

(ii) The second harmonic is finite in the limit of the
negligible first harmonic,

w2 = O(1), w1 → 0. (7)

(iii) Both harmonics are on the same order,

w2 = O(w1), w1 → 0. (8)

As follows from Eqs. (6) and (8), the maximal intensities
of both w1 and w2 go to zero at the bifurcation point for
cases (i) and (iii). However, in case (ii), the maximum of the
absolute value of field w2 goes to a finite value when the
solution approaches the bifurcation point. Which of the cases
is realized depends on the parameters of the system and, in
particular, on the mismatch q. Below, we consider these three
cases separately.

III. MODES WITH NEGLIGIBLE SECOND HARMONICS
IN THE LINEAR LIMIT

Let us start with the conditions necessary for Eq. (6) to
occur. In this limit, the nonlinear term in Eq. (5a) can be
neglected, and in the leading order, we have the eigenvalue
problem,

L1,αw1l = bw1l , (9a)

L1,α = d2

dξ 2
+ V

(
1

cosh2 ξ
+ iα

sinh ξ

cosh2 ξ

)
. (9b)

Equation (9) has been studied before. Therefore, below, we
only briefly outline the features necessary for our analysis,
referring to Refs. [5,6] for more details.

For V > 0, Eq. (9) possesses localized solutions. When α

is below the PT -symmetry-breaking threshold,

α < αcr = 1 + 1

4V
, (10)

the spectrum of Eq. (9) has discrete real eigenvalues given by
Ref. [16]

b1,n = (n − η1)2, n = 0,1, . . . < η1, (11)

where

η1 = 1
2 (

√
V (αcr − α) + √

V (α + αcr ) − 1). (12)

Above the symmetry-breaking point (α > αcr ), the eigenval-
ues of the bound states are complex valued. We notice here that
no fundamental branch, satisfying condition (6) with α > αcr ,
was found.

Therefore, from now on, we concentrate only on the
results for PT -symmetry preserving case (10). Moreover, our
consideration will be limited to nonlinear modes that bifurcate
from the ground state of the defect potential in Eq. (9), i.e.,
n = 0. The respective eigenstate of the linear problem (9)
reads [5,6]

w1l(ξ ) = W1sechη1 (ξ ) exp

[
i

2
� tan−1(sinh ξ )

]
, (13)

where W1 is a constant and

� =
√

V (αcr − α) −
√

V (α + αcr ). (14)

Passing to Eq. (5b), in the small amplitude limit, one
can look for a solution with w1 ≈ w1l , which plays the role
of the inhomogeneous term in the linear equation for w2.
However, to obtain the complete families of solutions, one
has to consider both Eqs. (1). We did this using the relaxation
Newton-Raphson method using the described linear solutions
as the initial ansatz.

053815-2



LOCALIZED MODES IN χ (2) MEDIA WITH . . . PHYSICAL REVIEW A 86, 053815 (2012)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

b

P

0 0.02 0.04
0

0.02

0.04

0.06

α=1.47

α=1.452

α=1.44

α=1.49 α=1.39

α=1.385

FIG. 1. (Color online) Families of the solutions bifurcating from
b1,0 = η2

1 in case (i) for several values of α. Thick (thin) lines represent
stable (unstable) solutions. The inset shows the branches for larger
values of α. The black circles represent solutions studied in the text.
The parameters of the structure are V = 1/2, q = 0, and αcr = 1.5.

Figure 1 shows the dependence of the total power,

P = P1 + P2, Pn = n

∫ ∞

−∞
|wn|2dξ, (15)

on the propagation constant with several values of α. We
observe that the fundamental branches shown in Fig. 1 have
a maximal in power PM, P � PM . The value of PM is
decreasing as α increases. We also observed that PM → 0 as
α → αcr , i.e., the fundamental branch disappears. For a given
α, the position of PM with respect to b approaches b = 0 as q

decreases as one can see in Fig. 2. The position of PM moves
to the right in the case of q > 0. In Fig. 2, one can also see that,
in this case, the branch ends in b = −q because the mismatch
shifts the position of the continuum spectrum of the linear
part of Eq. (5b) and w2 becomes delocalized. There is no low
amplitude linear limit for w1 in this case.

In Fig. 3, we show the typical distributions of the fields wn

and the real-valued currents jn defined as (n = 1,2)

jn(ξ ) = |wn|2 dθn

dξ
, θn(ξ ) = arg wn(ξ ). (16)

0 0.01 0.02 0.03 0.04 0.05
0

0.02

0.04

0.06

0.08

0.1

b

P

q=0.01

q=0
q=−0.01q=−0.05

q=−0.1

FIG. 2. (Color online) Families of the solutions bifurcating from
b1,0 = η2

1 and corresponding to case (i) for several values of q. Thick
(thin) lines represent stable (unstable) solutions. The parameters of
the structure are V = 1/2 and α = 1.44.

FIG. 3. (Color online) Upper panels: Spatial distributions of the
intensities |wn|2 and lower panels: the currents jn. The left panels
correspond to a stable solution with b = 0.024 as marked by a
black circle in Fig. 1, pertaining to the fundamental branch that
bifurcates from b1,0 = 0.034 where V = 1/2, q = 0, and α = 1.452.
The right panels correspond to a stable solution with b = 0.82,
pertaining to the fundamental branch that bifurcates from b1,0 = 0.92
where V = 2, q = 0, and α = 0.5. Shadowed domains show the
localized impurity (darker areas represent higher values of its real
part V sech2ξ .

By construction, |wn|2 and jn are even functions. The effective
width of the intensities of modes may significantly exceed the
size of the impurity, particularly, in the modes closer to the
edge of the continuous spectrum (the left panels of Fig. 3).

We note that, in the left panels of Fig. 3, the solution
has a relatively small amplitude. This is a peculiarity of the
chosen strength of the potential (it was V = 1/2, solitons with
larger amplitudes were be found to be unstable). Although this
potential (ensuring the existence of only one linear defect level
in the localized potential) is used below in the text, in the right
panels of Fig. 3, we show a higher amplitude soliton for the
potential well having the width V = 2 (and the parameters
α = 0.5 and q = 0).

IV. NONLINEAR MODES WITHOUT LINEAR LIMITS

In this section, we investigate the case when the second
harmonic remains finite at w1 → 0. Then, one can neglect the
nonlinear term w2

1 in Eq. (5b) reducing it to the well-known

−0.25 −0.2 −0.15 −0.1 −0.05 0
0

0.02

0.04

0.06

0.08

0.1

0.12

q

m
in

 |W
2|

0 20 40 60 80 100
−1

−0.5

0

0.5

1

Re(W
2
)

Im
(W

2)

−0.25 −0.2
0

2

x 10

0 0.5 1
−1

0

1

q
0

W
2
=0.072

FIG. 4. (Color online) Left panel: The eigenvalues W2 of Eq. (17).
The parameters of the structure are V = 1/2, α = 1.4, and q = 0.
The inset shows the first few eigenvalues in detail. Right panel: The
lowest |W2| of Eq. (17) as a function of the mismatch q of the lowest
P branch. The inset shows how the minimum value of W2 reaches
zero at q = q0.

053815-3



MOREIRA, ABDULLAEV, KONOTOP, AND YULIN PHYSICAL REVIEW A 86, 053815 (2012)

linear eigenvalue problem (see, e.g., Ref. [17]),

L2w2l = 2bw2l , (17a)

L2 = 1

2

d2

dξ 2
+ 2

(
V

1

cosh2 ξ
+ q

)
, (17b)

whose eigenvalues are

b2,n = (n − η2)2

4
+ q, n < 0,1, . . . , η2 =

√
1

4
+ 4V − 1

2
.

(18)

Here, we again consider the case where there is only one
localized mode. This leads to the requirement that η2 � 1 and,
consequently, V � 1/2. The corresponding eigenfunction of
b2,0 reads

w2,0(ξ ) = W2sechη2ξ, (19)

where W2 is some constant which must be determined. This
can be performed from the condition that the propagation
constants in Eqs. (5a) and (17) are the same. In the vicinity
of the bifurcation point, one can approximate w2 ≈ w2l , i.e.,
Eq. (5a) can be approximated by the following linear system:

L

(
w̄1

w1

)
= W2

(
w̄1

w1

)
, (20a)

L = −1

2
coshη2 (ξ )

(
0 L1 − b2,0

L̄1 − b2,0 0

)
. (20b)

Let us note that (20b) is an eigenvalue equation, and
so, the allowed values of W2 are simply the eigenvalues.
Next, we define bra and ket vectors: 〈ψ | ≡ (ψ(ξ ),ψ̄(ξ )) and
|ψ〉 ≡ (ψ̄(ξ ),ψ(ξ ))T (T stays for the transpose matrix), where
ψ(ξ ) → 0 at ξ → ±∞ and verify that the operator L is
Hermitian with respect to the weighted inner product,

〈ψ1|ψ2〉 =
∫ ∞

−∞
sechη2 (ξ )[ψ1(ξ )ψ̄2(ξ ) + ψ̄1(ξ )ψ2(ξ )]dξ.

(21)

0.2 0.3 0.4 0.5 0.6 0.7
0

0.5

1

1.5

2

2.5

b

P

0.2 0.3 0.4
0

0.5

1
α=1.5

 

 
P

1

P
2

P

α=1.5

α=2

α=2.5

α=0

FIG. 5. (Color online) The power diagrams of the fundamental
branches bifurcating from the linear mode b2,0 = 0.25 for several
values of α. Thick (thin) lines represent stable (unstable) solutions.
The inset shows, in detail, that P1 goes to zero in the vicinity of
b2,0, whereas, P2 remains finite. Filled circles represent two stable
solutions shown below in Fig. 6. The parameters of the structure are
V = 1/2 and q = 0.

The Hermiticity of L ensures the reality of the admissible W2.
We also note that, if (w̄1(ξ ),w1(ξ ))T is a solution of Eq. (17)
with eigenvalue W2, then −W2 is also an eigenvalue with
eigenfunction (−iw̄1(ξ ),iw1(ξ ))T . This allows us to restrict
the consideration to W2 > 0.

We investigated the system (17) numerically and found that
there is an infinite number of discrete eigenvalues W2. The
ones having the smallest absolute values are shown in Fig. 4.
The amplitude of the second harmonic W2 depends on b = b2,0

[see Eq. (20b)]. In Fig. 4, we show the dependence of W2 on
q corresponding to the lowest |W2| branch. The special case
when, for a certain value of q, the amplitude of the second
harmonic W2 → 0 happens when b1,0 = b2,0. This leads to

q = q0, q0 = η2
1 − 1

4η2
2, (22)

which is precisely the case, however, (8); we consider it in the
next section.

We numerically studied the existence of bifurcations satis-
fying (7) in Fig. 5. It is possible to see, in the inset of Fig. 5,
that, at the bifurcation point, the branches satisfy P1(b2,0) = 0
and, consequently, P (b2,0) = P2(b2,0). A simple integration
in (15) after the substitution w2 = w2l reveals that, for each
α when V = 1/2 and η2 = 1 (the case of Fig. 5), one has
P (b2,0) = 4W 2

2 .
It can be seen in Fig. 5 for the power diagrams of the

fundamental branches where stable solutions exist above
the PT -symmetry-breaking point. The existence of stable
nonlinear modes, even when the spectrum of the linear system
is not purely real, has been reported earlier in Ref. [18] (see
also recent paper [19]).

In Fig. 6, there is an example of a mode in case (ii).

V. BIFURCATION OF THE NONLINEAR MODES
FROM THE LINEAR SPECTRUM

Now, we consider case (iii) for which the relation (8) holds.
Previously, we have shown that, if the bifurcation point is, at the
same time, an eigenvalue of Eq. (9) and (17), i.e., b1,0 = b2,0,
then the mismatch must have the special value q = q0. We also
have seen that, in this case, W2 = 0. As a direct consequence,
(17) reduces to Eq. (9) and not only w2 → w2l , but also the
FFs satisfy w1 → w1l at the bifurcation point.

Figure 7 shows power diagrams of solutions satisfying (8)
for several values of α. It is possible to see that two bifurcations
occur at b = b1,0 = b2,0 (see the dashed line in the inset of
Fig. 7). The branch that goes to the right is a bifurcation of

FIG. 6. (Color online) An example of a stable solution of case
(ii) with b = 0.6 and α = 2 > αcr , marked with the filled circle in
Fig. 5. Shadowed domains show the localized impurity (darker areas
representing higher values of the real part of the localized potential).
The parameters of the structure are V = 1/2 and q = 0.

053815-4



LOCALIZED MODES IN χ (2) MEDIA WITH . . . PHYSICAL REVIEW A 86, 053815 (2012)

0 0.05 0.1 0.15 0.2

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

b

P

0 0.1 0.2
0

0.05

0.1
α=1.3

α=1.2

α=1.25

α=1.3

α=1.35

α=1.4

α=1.44
α=1.48

SHFF

FIG. 7. (Color online) Several fundamental branches for case (iii)
with different values of α. Note that the branch disappears when
α → αcr = 3/2. Stable (unstable) solutions are represented by thick
(thin) lines. The inset shows the regions of bifurcations from b1,0 (FF)
and b2,0 separated by a vertical dashed line. The parameters of the
structure are V = 1/2 and q = q0 (αcr = 1.5).

b2,0, and the branch that goes to the left is a bifurcation of b1,0.
Both branches have behaviors very similar to the branches of
cases (i) and (ii).

In the numerical simulations, we observed that there may be
a collision of the fundamental branch with a nonfundamental
branch with two peaked solutions. Figure 8 shows the corre-
sponding bifurcation diagrams, whereas, Fig. 9 illustrates the
distribution of the intensities and the currents in a two-hump
soliton solution. The intensities and the current j2 are largely
distributed far from the center of the potential, whereas, the
current j1 is localized at the defect.

In Fig. 10, one can see that, as b decreases, |w1(0)|2
decreases at the same time that the two emergent peaks become
increasingly separated. The intensity |w2(0)|2 (not shown)
decreases as well.

With respect to phase, we found all stable solutions that
bifurcate from b1,0 to satisfy wn(ξ ) = wn(−ξ ). This means
that the peaks of the two-hump solution shown in Fig. 10 are
out of phase.

0 0.01 0.02 0.03
0

0.035

0.07

0.105

0.14

b

P

0 0.005 0.01 0.015 0.02
0

0.02

0.04

0.06

0.08

b

P

Two peakedTwo peaked

FIG. 8. (Color online) Left panel: line: shows the fundamental
branch of case (iii) with α = 1.3 near b = 0 and the merged two-
peaked branch. Lines: thick lines are stable (unstable) solutions. Right
panel: Shows a fundamental branch and a two-peaked branch with
α = 1.44 near b = 0. Black circles are solutions represented in Figs. 9
and 14. The parameters of the structure are V = 1/2 and q = q0.

FIG. 9. (Color online) Stable double-peaked solution with b =
0.0057, corresponding to the black circle in the left panel of Fig. 8.
The left panel shows the intensities |wn|2; the right panel shows
the currents jn. The shadowed domain shows the localized impurity
V sech2(ξ ) (darker areas represent higher values of the real part of
the localized potential). The parameters of the structure are V =
1/2, α = 1.3, and q = q0.

VI. STABILITY AND DYNAMICS OF LOCALIZED
SOLUTIONS

The stability was studied by direct numerical simulations
of the system (1) and within the framework of eigenvalue
evaluation of the eigenvalue problem, obtained from perturba-
tions of the form

un(ξ,ζ ) = [wn(ξ ) + pn+(ξ )e−iλζ + pn−(ξ )eiλζ ]einbζ , (23)

with pn+(ξ ) and pn−(ξ ) being small perturbations. The
resulting eigenvalue problem is given by⎛

⎜⎜⎜⎝
L2 2w1 0 0

2w̄1 L1,α 0 2w2

0 0 −L̄2 −2w̄1

0 −2w̄2 −2w1 −L̄1,α

⎞
⎟⎟⎟⎠

⎛
⎜⎜⎜⎝

p2+
p1+
p2−
p1−

⎞
⎟⎟⎟⎠ = λ

⎛
⎜⎜⎜⎝

p2+
p1+
p2−
p1−

⎞
⎟⎟⎟⎠.

(24)
Whenever an eigenvalue λ with Im(λ) > 0 occurs, the solution
is unstable.

Let us start the stability analysis with case (i). Then, the
branches have two stable regions, one close to b1,0 and the other
close to b = 0 as shown in Fig. 1. It was found numerically

FIG. 10. The intensity profiles |w1|2 of solutions, pertaining to
the fundamental branch of case (iii) bifurcating from b1,0 = 0.063 at
different b’s illustrating the transition of a single-peaked profile into
a double-peaked one. The local minimum occurs exactly at ξ = 0.
The parameters of the structure are V = 1/2, α = 1.3, and q = q0.

053815-5



MOREIRA, ABDULLAEV, KONOTOP, AND YULIN PHYSICAL REVIEW A 86, 053815 (2012)

FIG. 11. (Color online) Upper left panel: the evolution of a
stable solution with b = 0.076 and upper right panel: an unstable
solution with b = 0.09 of the fundamental branch of case (i). The
corresponding eigenvalues of the linear stability analysis are given in
the lower panels. The parameters of the structure are V = 1/2, q = 0,
and α = 0.9.

that only low amplitude solutions are stable. The instability
is produced by pairs of purely imaginary eigenvalues λ, see
Fig. 11, showing the eigenvalues and the typical evolution of
the soliton, resulting in its rapid decay.

For case (ii), we investigated the stability and found that
all the solutions were stable if α � 0.9, for α � 0.9, some
parts of the bifurcation curve became unstable, see Fig. 12. In
Fig. 12, one can see that the unstable part of the bifurcation
curve becomes larger when α increases, but the stable solutions
survive even for α � αcr . The linear stability analysis shows
that the instability arises from quartets of complex eigenvalues
(see Fig. 13).

Finally, we discuss case (iii). We found that, with respect
to stability, the behavior is similar to cases (i) and (ii). For
values b > b1,0, i.e., bifurcations of b2,0, the stability behaves,
such as in case (ii) with the appearance of instability intervals
that increase in length as α increases. The region b < b1,0 has
solutions that bifurcate from b1,0. There is always a stable
region adjacent to b1,0 and another stable region close to b =

0.25 0.3 0.35 0.4
0

0.5

1

1.5

2

2.5

b

α

Stable

Unstable

FIG. 12. (Color online) The panel shows the maximal value of α

for which a fundamental branch solution is stable as a function of b.
The branch bifurcates from b2,0. The parameters of the structure are
V = 1/2 and q = 0.

FIG. 13. (Color online) Upper left panel: the evolution of a stable
solution with b = 0.6 and upper right panel: an unstable solution with
b = 0.27 of the fundamental branch of case (ii) that bifurcates from
b2,0. Both solutions were perturbed by 10% of amplitude random
noise. The corresponding eigenvalues of the linear stability matrix
are given in the lower panels. Both solutions are marked by black
circles in Fig. 5. The parameters of the structure are V = 1/2, q = 0,
and α = 2.

0. The instability, when observed, was due to a quartet of
complex eigenvalues of the stability matrix contained in the
region b > b1,0 and two purely imaginary eigenvalues in the
region contained in b < b1,0 (see the middle panel of Fig. 14).
We observed that there were stable solutions in the region

FIG. 14. (Color online) Left panels: Propagation of 10% of
amplitude perturbations of solutions marked by black circles in the
lower panel of Fig. 8, corresponding to case (iii). The upper left
panel has b = 0.0172, the middle left panel has b = 0.0072, and
the lower left panel has b = 0.0017. Right panels are the respective
eigenvalues of the linear problem. Parameters of the structure are
V = 1/2, α = 1.45, and q = q0.

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LOCALIZED MODES IN χ (2) MEDIA WITH . . . PHYSICAL REVIEW A 86, 053815 (2012)

where the fundamental branch, bifurcating from b1,0, merged
with a two-peaked branch.

VII. CONCLUSION

To summarize, we showed the existence of solitons in
quadratic nonlinear media with localized PT -symmetric
modulations of linear refractive index. The families of stable
one- and two-hump solitons were found. The properties of
nonlinear modes bifurcating from a linear limit of small
fundamental harmonic field were investigated. It was shown
that the fundamental branch had a maximum in a power. This
maximum was decreasing with the strength of the imaginary
part of the refractive index α. For the case when both harmonics
were on the same order, the scenarios of bifurcations of
different branches of solutions on the propagation constant
b were investigated. It was shown that modes bifurcating
from the linear mode of the second harmonic can exist, even
above the PT -symmetry-breaking threshold. We found that

the fundamental branch bifurcating from the linear limit can
undergo a secondary bifurcation colliding with a branch of
two-hump soliton solutions.

For nonlinear modes, having no linear limit, i.e., |u2| ∼
O(1), different branches of solutions, in dependence on the
parameters b and the phase mismatch q, had been investigated.
The stability intervals for different values of parameters b and
α were obtained. The examples of dynamics and excitations
of solitons by numerical simulations of the full χ (2) system of
equations with the PT -symmetric potential were confirmed
theoretical predictions.

ACKNOWLEDGMENTS

F.C.M. acknowledges support from the EU Alban pro-
gram. V.V.K. and A.V.Y. acknowledge support from the FCT
(Portugal) under Grants No. PTDC/FIS/112624/2009 and
No. PEst-OE/FIS/UI0618/2011. F.K.A. acknowledges support
from the FAPESP (Brazil).

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