Romanian Reports in Physics, Vol. 67, No. 4, P. 1383–1400, 2015

Dedicated to International Year of Light 2015

LOCALIZED STRUCTURES IN NONLINEAR OPTICAL MEDIA: A SELECTION OF RECENT STUDIES

DUMITRU MIHALACHE

Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, RO-077125, Bucharest – Magurele, Romania

E-mail: Dumitru.Mihalache@nipne.ro

Received September 7, 2015

Abstract. We provide a brief overview of selected recent studies, which were performed in diverse relevant physical settings, on the creation and characterization of self-organized localized optical structures in either dissipative or conservative (lossless) nonlinear media.

Keywords: localized optical structures, optical solitons, pattern formation, optical rogue waves.

1. INTRODUCTION

Nonlinear waves are ubiquitous phenomena in many branches of physical and engineering sciences, such as hydrodynamics, plasma physics, acoustics, condensed matter physics, ultracold atomic gases (Bose-Einstein condensation and related phenomena), optics and photonics, etc. Such waves constitute fundamental physical objects of nature and have universal features. In this line of thought, optical solitons (more generally speaking, localized optical structures) are the key concepts of the fast growing field of nonlinear optics and photonics; for several relevant books and review articles published in this vast research area during the past two decades, see Refs. [1–39].

In the course of the last years a new level of understanding has been achieved about conditions for the existence, stability, excitation, and robustness of self-organized solitary localized structures (LSs) that are observed in spatially extended nonlinear systems. The LSs form in either dissipative or conservative (lossless) nonlinear media; see Refs. [40–93] for a series of representative works performed in this very broad area, by several research groups, over the past three decades. The solitary LSs in dissipative systems are now termed dissipative solitons [7] or autosolitons [16, 20]. Unique phenomena involving dissipative solitons have been observed in optics and photonics, biology, medicine, etc. [7].

1384 Dumitru Mihalache 2

In conservative (Hamiltonian) nonlinear media, the solitary waves are supported by a stable balance between diffraction and/or dispersion and nonlinearity. However, in the presence of dissipation, losses must be compensated either by gain or by feeding beams, in internally and externally driven systems, respectively, see e.g. [2, 7–9, 13, 14]. In the case of Hamiltonian systems (in completely integrable and nonintegrable models alike) solitons form continuous families with one or a few family parameters.

The condition of the balance between gain and loss in dissipative systems results in a zero-parameter family of soliton solutions. Thus a dissipative soliton has its field amplitude and velocity fixed by the coefficients of the governing nonlinear partial differential equation, i.e., the stationary soliton solutions are isolated ones, constituting attractors of the corresponding nonlinear system; see e.g. Ref. [4]. It is to be mentioned that a LS in a nonlinear dissipative medium requires a continuous energy flow into the medium, and into the LS itself in order to preserve its intrinsic characteristics such as shape, field amplitude, etc. Thus the balance between gain and loss (or between the input and output energies) is equally important as that between diffraction/dispersion and nonlinearity.

The dissipative nonlinear LSs occur in many areas of physics, chemistry, and biology; see the comprehensive book by Nicolis and Prigogine [94] for an earlier work on the morphogenesis of self-organized stable LSs allowing energy and matter flow between them and the environment in both animate and inanimate worlds. It is now commonly believed that the stable LSs forming in nonlinear optical media in either two-dimensional (2D) or three-dimensional (3D) physical settings could be used to store and process information in the forthcoming all-optical transmission and processing devices and systems.

The aim of this work is to provide an overview of selected recent studies on the creation and the characterization of localized optical structures in nonlinear media.

The paper is organized as follows. In Sec. 2 we briefly review the problem of morphogenesis (via spontaneous symmetry breaking) of two-dimensional self-organized localized structures in dissipative cubic-quintic nonlinear optical media. Recent results on existence, stability and generation of a variety of two-dimensional localized structures in parity-time symmetric photonic lattices are overviewed in Sec. 3. In Sec. 4 we discuss recent advances in the rapidly developing field of optical rogue waves, and in particular we consider the problem of vectorial rogue wave formation in the integrable Manakov system, i.e., the system of two coupled nonlinear Schrödinger (NLS) equations in the case of anomalous group-velocity dispersion. Finally, Sec. 5 concludes this paper.

3 Localized structures in nonlinear optical media: a selection of recent studies 1385

2. MORPHOGENESIS OF TWO-DIMENSIONAL LOCALIZED STRUCTURES VIA SPONTANEOUS SYMMETRY BREAKING

During the past decades symmetry breaking instabilities have been observed in a large variety of nonlinear systems in nature, ranging from biological systems [95] to various physical systems such as Bose-Einstein condensates [96], dual-core photonic lattices [97], optical cavities [98], dual-core nanolaser cavities [99] etc., just to name a few such relevant studies; see also the very interesting thoughts of Anderson [100] on “broken symmetry and the nature of the hierarchical structure of science”. Also, for a brief introduction to the topic of spontaneous symmetry breaking in both conservative and dissipative nonlinear systems, see the recent paper by Malomed [101]. The morphogenesis of self-organized LSs in nonlinear dissipative systems is driven by the supply of external energy and/or matter; see, for example, the book by Nicolis and Prigogine [94].

The complex Swift-Hohenberg and Ginzburg-Landau equations constitute the prototypical models that adequately describe the generation of dissipative LSs in systems ranging from optics and photonics, fluids, plasma, superfluidity and superconductivity, Bose-Einstein condensation, liquid crystals, quantum field theories, plant ecology and biological systems; see e.g., Refs. [2, 7–9, 13–18]. The above mentioned nonlinear evolution equations may be viewed as dissipative extensions to the prototype nonlinear Schrödinger equation; they can describe a broad range of phenomena suggested by the NLS dynamics, ranging from chaos and pattern formation to dissipative solitary structures; see, for example, the seminal papers published about two decades ago on spatial LSs in nonlinear optical systems, transverse pattern formation in lasers, spatial solitons in optical cavities, and on the possible application of cavity solitons as pixels in semiconductor cavities [102–115].

It is worth noting that the two generic nonlinear evolution equations (the complex Swift-Hohenberg and Ginzburg-Landau equations) have similar mathematical forms excepting the linear terms accounting for the spectral filtering phenomenon. In fact, in the case of the Swift-Hohenberg equation the spectral filtering term has the form (iβuxx + iγ2uxxxx), whereas in the case of the Ginzburg-Landau equation the spectral filtering term has a simpler form iβuxx, i.e., the fourth-order derivative with respect to the spatial/temporal coordinate is not taken into account.

The simplest one-dimensional (1D) complex Ginzburg-Landau equation with cubic (Kerr-type) nonlinearity includes the conservative diffraction/dispersion and Kerr-nonlinearity terms, the nonlinear (cubic) loss term that accounts for the phenomenon of two-photon absorption in the nonlinear optical medium, and the spatially uniform linear gain g that compensates the loss in the system:

( ) 211 | | 0,

2z xxiu i u iguu uε+ + + =− (1)

1386 Dumitru Mihalache 4 where u (x, z) is the complex amplitude of the wave, z is the propagation distance, and the positive parameter ε is the nonlinear (cubic) loss.

The exact solution of the 1D complex cubic Ginzburg-Landau equation is often called the Pereira-Stenflo dissipative soliton, which is a “chirped” nonlinear mode subject to instability due to the action of the constant (uniform) gain on the zero soliton background; see Ref. [116]. However, it is possible to achieve stabilization of such solitons in dual-core waveguides modeled by two linearly coupled equations, with the linear gain acting in one waveguide and the linear loss acting in the other one; see Refs. [117–119] and the recent comprehensive review by Malomed [37] on spatial solitons supported by localized gain applied at one or several hot spots. It is worth noting that the problem of creation of stable 1D and 2D LSs in lossy nonlinear optical media by employing a localized gain has attracted a lot of interest in the recent literature; see e.g. Refs. [120–124].

Stable solitons can be generated in dissipative systems described by the complex Ginzburg-Landau equation with the nonlinear (cubic) gain term and additional linear and quintic loss terms, the so-called complex cubic-quintic (CQ) Ginzburg-Landau equation. It is to be mentioned that the linear loss term accounts for the stabilization of the zero background solution of the corresponding CQ Ginzburg-Landau equation. The CQ Ginzburg-Landau equation in its 2D variant was first introduced, to the best of our knowledge, by Petviashvili and Sergeev [41]. The generic CQ Ginzburg-Landau model has then been studied by many authors; see e.g., Refs. [125–133], and it has been implemented in an optical setting by using a combination of linear amplification and saturable absorption, see, for example, Refs. [134–138].

In a recent work by Skarka et al. [139] a complex CQ Ginzburg-Landau equation with localized linear gain was put forward as a 2D generic model for the formation of different types of localized optical structures via spontaneous breaking of the axial symmetry. Thus the generic model introduced by Skarka et al. [139] was based on the (2 + 1)-dimensional complex cubic-quintic Ginzburg-Landau equation that governs the evolution of wave amplitude u (x, y, z) in a dissipative nonlinear medium with cubic-quintic nonlinearity:

( ) ( ) ( )2 41

1 | | | | ( ) ,2z xx yyiu u i u i u ig r uu u uε ν µ+ + + − − − = (2)

where the positive coefficients ν, ε, and µ account, respectively, for the saturation of the cubic nonlinearity, cubic gain, and quintic loss. In Eq. (2) the gain “iceberg” g(r) has the simple parabolic-type form:

2( ) .g r rγ= −Γ (3)

Here r is the radial coordinate, γ > 0 and Γ > 0, where γ is the gain amplitude and Γ is the gain curvature. The gain “iceberg” is protruding above the surface of the loss “sea”.

Note that in a previous work by Skarka et al. [69], it was investigated a different physical model with the submerged “iceberg”, where the above real-

5 Localized structures in nonlinear optical media: a selection of recent studies 1387 valued control parameter γ was negative. Starting from the steady-state solutions produced by the variational approximation developed by Skarka and Aleksić [55], extensive numerical simulations of the above Ginzburg-Landau equation generate a vast class of robust solitary-wave structures, such as axisymmetric vortex solitons, bell-shaped axisymmetric solitons without the central “crater” that is specific to a vortex soliton, rotating crescent-shaped solitons, vortex solitons with slanted “craters”, rotating elliptic vortex solitons, solitons consisting of two bell-shaped fragments, double-humped solitons, and four- to ten-pointed revolving “stars” with zero topological charge [139]. It was found in Ref. [139] that the four- and five-pointed “stars” feature a cyclic change of their structure in the course of the rotation, whereas six-, seven-, eight-, nine-, and ten-pointed “stars” steadily revolve, keeping constant their shapes and their angular momenta, having zero topological charges (vorticities).

In Fig. 1 we illustrate the formation of a four-pointed “star” with zero vorticity (zero topological charge). It is clearly seen from Fig. 1 that this four-pointed “star” features a periodic change of its structure during rotation. Figure 2 shows the evolutions of the revolving five- and six-pointed “stars”; notice that such star “patterns” are stable physical objects with nonzero angular momenta but with zero topological charges.

Fig. 1 – Formation of the four-pointed “stars”. The input ring-type structure evolves into a localized

vortex pattern (a vortex soliton), whose spontaneous symmetry breaking produces a four-pointed “star” (alias a “Celtic-cross” structure) that subsequently transmutes into other varieties of four-

pointed localized patterns (adapted from Ref. [139]).

Fig. 2 – The evolution of the revolving five-pointed “stars” [see panels (a), (b), (c), and (d)] and the

dynamics of the revolving six-pointed “stars” [see panels (e), (f), (g), and (h)] (adapted from Ref. [139]).

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Fig. 3 – Typical shapes of the revolving four-, five-, six-, seven-, eight-, nine-, and ten-pointed “stars”

with zero topological charges. The strongly asymmetric steadily rotating “cobra”-type localized structure is also shown as the last pattern on the right-hand side of this figure

(adapted from Ref. [139]).

The typical shapes of the rotating four-, five-, six-, seven-, eight-, nine-, and ten-pointed “stars” with zero topological charges are shown in Fig. 3. Also, the strongly asymmetric steadily rotating “cobra”-type localized structure is shown as the last pattern on the right-hand side of Fig. 3.

Finally we mention that very recently Aleksić et al. [140] by using the variational method and extensive numerical simulations of the 2D complex CQ Ginzburg-Landau equation have established the stability domains of parameters for dissipative vortex solitons with high topological charge ranging from S = 3 to S = 20. Also, a unique nesting property of low-vorticity solitons within those of higher vorticity was recently discovered by Aleksić et al. [140].

3. TWO-DIMENSIONAL LOCALIZED STRUCTURES IN PARITY-TIME SYMMETRIC PHOTONIC LATTICES

Bender and Boettcher have introduced in 1998 [141] the concept of parity-time (PT) symmetry in quantum physics for physical systems described by non-Hermitian Hamiltonians that contain complex-valued external potentials of the form V(x) = R(x) + iI(x), such that the real part R(x) is an even function of the spatial coordinate x, whereas the imaginary part I(x) is an odd function of the variable x; for a review of the interesting issue of making sense of non-Hermitian Hamiltonians; see Ref. [140]. In the two-dimensional case, the PT-symmetry condition is simply V(x, y) = V*(–x, –y), where the symbol “*” stands for the complex conjugate.

An interesting feature of non-Hermitian Hamiltonians with complex-valued PT-symmetric external potentials that is exploited in such PT-symmetric systems is the fact that the eigenvalue spectrum remains purely real below the so-called exceptional (critical) point. However, as the loss and gain of the medium are increased beyond some critical values, the eigenvalues move in the complex plane and become complex conjugates of each other; see e.g. the comprehensive review [142]. Actually, a sort of “phase transition” towards complex-valued eigenvalues connected with spontaneous symmetry breaking can be observed in such PT-symmetric systems. These physical systems are appealing from both theoretical

7 Localized structures in nonlinear optical media: a selection of recent studies 1389 and practical points of view, as is evident from the many papers published in the past decade in diverse areas, including quantum physics, optics and photonics, plasmonics and metamaterials; see Refs. [143–165]. In fact, a special class of dissipative physical systems was put forward and was practically implemented in optical settings, with exactly balanced lossy and amplifying elements that realize the above mentioned concept of PT symmetry, see Refs. [166–168].

In a seminal paper, Musslimani et al. [143] have theoretically studied the existence, stability, and robustness of optical solitons in PT-periodic potentials. A novel class of either 1D or 2D nonlinear self-trapped modes can exist in these PT synthetic optical lattices. These optical solitons were found to be stable over a wide range of parameters of the PT-symmetric external potential [143]. Driben and Malomed [145] have investigated the stability of optical solitons in PT-symmetric couplers. Families of analytical solutions were found for both symmetric and antisymmetric solitons in a dual-core system with cubic (Kerr) nonlinearity and PT-balanced gain and loss, and the stability of solitons of both types was investigated both analytically and numerically; see Ref. [145].

A detailed investigation of the stability of solitons in PT-symmetric nonlinear potentials was reported by Zezyulin et al. [146], by using both analytical and numerical methods. Alexeeva et al. [149] studied in detail the problem of existence and stability of spatial and temporal solitons in the PT-symmetric coupler with gain in one waveguide and loss in the other. The stability properties of the high- and low-frequency solitons were found to be completely determined by a single combination of the soliton’s amplitude and the gain-loss coefficient of the two coupled waveguide; see Ref. [149].

The existence and stability of lattice solitons in PT-symmetric mixed linear-nonlinear optical lattices in cubic (Kerr-type) nonlinear media were investigated by He et al. [151]. It was found that the combination of PT-symmetric linear and nonlinear lattices can stabilize such solitons and can provide unique soliton properties. It was revealed by He et al. [151] that the parameters of the linear lattice periodic potential play a significant role in controlling the extent of the stability domains and that the lattice solitons can stably propagate only in the low-power regime.

The existence, stability, and rich dynamics of dissipative latice solitons in optical media described by the complex cubic-quintic Ginzburg-Landau model with PT-symmetric periodic potentials were studied in detail by He and Mihalache [152]. Blow-up regimes in the PT-symmetric coupler and the actively coupled dimmer were investigated by Barashenkov et al. [155]. Defect solitons in 2D parity-time-symmetric photonic lattices were studied in detail by Xie et al. [158]. In the case of a positive defect, such solitons only exist in the semi-infinite gap and can propagate stably in their entire existence domain. In the case of a negative defect, the solitons can exist in both the semi-infinite gap and in the first gap. In the first gap, these solitons are stable only for moderate powers, whereas in the semi-

1390 Dumitru Mihalache 8 infinite gap these defect solitons were found to be unstable; see Ref. [158] for a comprehensive study of these issues.

In what follows we briefly discuss the problem of existence and stability of 2D localized structures in PT-symmetric triangular photonic lattices, see Ref. [169] for a detailed study of these issues. Fundamental-mode, dipole-mode (the two-peak mode), and multipole-modes (the four- and six-peak modes) solitons were studied in detail in Ref. [169]. Three-, four-, and six-peak vortex solitons in PT-symmetric triangular photonic lattices were also investigated in Ref. [169] and it was shown that such PT-symmetric optical lattices help to stabilize the dipole, multipole, and vortex nonlinear modes, in addition to the fundamental ones.

Fig. 4 – The refractive-index modulation in the PT-symmetric triangular photonic lattice:

(a) the real part; (b) the imaginary part (adapted from Ref. [169]).

Fig. 5 – Typical field profiles of fundamental solitons for low powers. The real part of the

corresponding optical field is shown in panel (a), whereas the imaginary part is displayed in panel (b) (adapted from Ref. [169]).

In Fig. 4 we show the complex-valued refractive-index modulation in the PT-symmetric triangular photonic lattice. The typical field profiles of fundamental solitons are displayed in Fig. 5 for low powers. We clearly see from Fig. 5 that for low powers the fundamental soliton features a main peak at its center, with six in-phase tails around it due to the action of the triangular optical lattice. However, by increasing the propagation constant and correspondingly, by increasing the

9 Localized structures in nonlinear optical media: a selection of recent studies 1391 soliton power, the fundamental soliton becomes narrower, featuring a single peak at its center. We show in Fig. 6 the typical field profiles of horizontal dipoles.

Fig. 6 – Typical field profiles of horizontal dipoles. The left panel shows the real part,

whereas the right panel displays the corresponding imaginary part (adapted from Ref. [169]).

Fig. 7 – Typical evolution of field profiles of stable three- and four- vortex solitons.

The corresponding phase structures at a large propagation distance are shown in the third column of this figure [panels (c) and (f)] (adapted from Ref. [169]).

Typical evolution of field profiles of stable three- and four-peak vortex solitons are displayed in Fig. 7, together with the corresponding phase structures of such lattice vortex solitons at a very large propagation distance.

It is worth noting that a realistic physical scheme to produce 1D and 2D weak-light solitons in atomic systems with PT symmetry was recently put forward by Hang and Huang [170]. The system under consideration was a cold three-level atomic gas with two species, which is driven by control and probe laser fields; see Ref. [170] for a detailed study of these issues.

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4. OPTICAL ROGUE WAVES

Originally, the terms “rogue wave” and “freak waves” were introduced in the scientific literature by Draper in 1964–1965; see Ref. [171] and a recent comprehensive book by Kharif et al. [172] on rogue waves in the deep ocean. This terminology refers to the gigantic ocean waves of high amplitudes that “appear from nowhere and disappear without a trace”; see the excellent work by Akhmediev et al. [173] on a parallel between the intrinsic mathematical structure of the high-amplitude waves in the deep ocean and the rational solutions of the focusing nonlinear Schrödinger equation.

During the past decade, more and more effort was devoted to the study of such extreme wave events in diverse areas such as optics and photonics [174, 175], Bose-Einstein condensates [176], plasma physics [177], capillary phenomena [178], financial markets [179]; see also a series of relevant recently published papers in this rapidly expanding research field [180–207].

The concept of optical rogue waves was introduced in 2007 by Solli et al. [174]. It was reported the observation of rogue waves in an optical system, i.e., the optical counterparts of infamous rare ocean waves with high amplitudes. The optical setting in the work by Solli et al. [174] was based on microstructured optical fibers, near the threshold of soliton-fission supercontinuum generation [208, 209], a nonlinear optical process in which a broadband optical field is generated from a narrow input.

In a recent work, Liu et al. [204] demonstrated that it is possible to trigger the onset of extreme events at the nanoscale in photonics “seas”. By using near-field imaging techniques, optical rogue waves characterized by a spatial localization of 206 nm and with an ultrashort duration of 163 fs at λ = 1.55 µm have been recorded; see Ref. [204].

Short-lived optical rogue wave events were also observed recently in incoherently pumped passive optical fiber ring cavities nearby the zero-dispersion wavelength of the fiber [205]. Note that Bragg grating rogue waves were also theoretically investigated in a recent work by Degasperis et al. [206]. In another optical setting, it was recently reported the experimental evidence of rogue-wave formation in a normal dispersion ytterbium fiber [207]. A large fraction of extremely high-energy pulses was observed; the amplitude of such pulses can be six times the significant wave height at the energy of about 50 nJ; see Ref. [207].

It is worth mentioning that in a recent work by Baronio et al. [202] it was shown that the baseband modulation instability is the origin of rogue wave formation. It was shown numerically that the rogues waves may only be excited from a weakly perturbed continuous wave whenever the baseband instability is

11 Localized structures in nonlinear optical media: a selection of recent studies 1393 present in the system under consideration; see Ref. [202] for a detailed study of this issue.

Very recently, Mulhern et al. [203] put forward a mechanism that might be responsible for the occurrence of extreme events in diverse physical settings, namely “the localization of energy at some few degrees of freedom”. In Ref. [203] it was studied a 1D chain of harmonically coupled units in an anharmonic potential and it was shown that due to the localization of energy, the dynamical systems exhibits extreme events. Thus it was demonstrated in Ref. [203] that the chain units show very large excitation amplitudes, a clear signature of occurrence of extreme events in the system under consideration.

In this Section we consider in much detail only a specific example of the formation of diverse rogue wave patterns in the vector nonlinear Schrödinger equations (also known as the completely integrable Manakov system). Peregrine-type, triple, quadruple, and sextuple vector rogue waves, either bright-dark or bright-bright in their respective components can form in the nonlinear Manakov system; see Ref. [210] for a detailed study of these issues.

Despite the diversity of such two-component (vectorial) optical rogue wave solutions to the Manakov nonlinear system, there exists a universal compossibility feature that different rogue wave states could coexist for the same set of parameters of the background of continuous waves.

The vector coupled nonlinear Schrödinger equations have the following dimensionless form

2 21

(| | | | 0,2

)t xxiu u vu u+ + =+

2 21

(| | | | 0,2

)t xxiv u vv v+ + =+ (4)

where u(t, x) and v(t, x) are the complex-valued envelopes of the two field components.

The above Manakov system of coupled nonlinear partial differential equations is of physical significance in optics for modeling the propagation of picosecond pulses in birefringent optical fibers, see e.g. Ref. [211], and in the study of spatial optical solitons in planar waveguides, see e.g., Ref. [212].

In Fig. 8 we show the typical field profiles of the first component |u| (the top panel) and of the second component |v| (the bottom panel) of the vectorial optical field for the fundamental (first-order) bright-black vector rogue waves that exist in the Manakov system.

Finally, in Figs. 9 and 10 we show the typical second-order bright-bright triplets and bright-bright quartets, respectively.

1394 Dumitru Mihalache 12

Fig. 8 – Typical field profiles |u| (top panel) and |v| (bottom panel) of fundamental (first-order)

bright-black vector rogue waves in the Manakov system (adapted from Ref. [210]).

Fig. 9 – Second-order rogue wave patterns in the form of bright-bright triplets; theinsetsinthe

upperrightcornersshowthecorrespondingcontourfielddistributions (adapted from Ref. [210]).

13 Localized structures in nonlinear optical media: a selection of recent studies 1395

Fig. 10 – Second-order rogue wave patterns in the form of bright-bright quartets; theinsetsintheupperrightcornersshowthecorrespondingcontourfielddistributions (adapted from Ref.

[210]).

5. CONCLUSIONS

In summary, in this paper we have attempted to provide the interested reader with a brief overview of selected recent studies on the creation and characterization of a diversity of localized structures that form in nonlinear optical media. The morphogenesis of such unique physical objects in several relevant physical settings was studied by considering both conservative (lossless) and dissipative nonlinear media.

We conclude with the hope that this brief overview on recent exciting developments in the field of creation and self-organization of localized optical structures will inspire further studies. We do believe that interesting times have arrived for this rapidly developing research area, and new exciting theoretical and experimental results will appear soon.

Acknowledgments. A part of the work overviewed in this paper has been carried out in

collaboration with B.N. Aleksić, N.B. Aleksić, S.H. Chen, Y.J. He, H. Leblond, M. Lekić, B.A. Malomed, and V. Skarka. I am deeply indebted to all of them for a fruitful collaboration and for many useful discussions.

D. M. acknowledges the support from the Romanian Ministry of Education and Research, Project PN-II-IDPCE-2011-3-0083.

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