We consider bright solitons supported by a symmetric inhomogeneous defocusing
nonlinearity growing rapidly enough toward the periphery of the medium,
combined with an antisymmetric gain-loss profile. Despite the absence of any
symmetric modulation of the linear refractive index, which is usually required
to establish a PT-symmetry in the form of a purely real spectrum of modes, we
show that the PT-symmetry is never broken in the present system, and that the
system always supports stable bright solitons, fundamental and multi-pole ones.
Such phenomenon is connected to non-linearizability of the underlying evolution
equation. The increase of the gain-losses strength results, in lieu of the
PT-symmetry breaking, in merger of pairs of different soliton branches, such as
fundamental and dipole, or tripole and quadrupole ones. The fundamental and
dipole solitons remain stable for all values of the gain-loss coefficient.Comment: 4 pages, 4 figures, to appear in Optics Letter

Kartashov, Yaroslav V.

Malomed, Boris A.

Torner, Lluis

Optics Letters

English

arXiv

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT ) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multipole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain - loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain-loss coefficient.Peer ReviewedPostprint (published version

Torner Sabata, Lluís

UPCommons (Universitat Politècnica de Catalunya)

Unbreakable PT symmetry of solitons supported by inhomogeneous defocusing nonlinearity

Yaroslav V. Kartashov

Boris A. Malomed

Lluis Torner

Crossref

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain–loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multi-pole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain–loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain–loss coefficient.Peer ReviewedPostprint (published version

Kartashov, Yaroslav

UPCommons. Portal del coneixement obert de la UPC

Unbreakable PT symmetry of solitons supported byinhomogeneous defocusing nonlinearityYaroslav V. Kartashov,1,2,* Boris A. Malomed,3 and Lluis Torner11ICFO—Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya,Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain2Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region 142190, Russia3Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,Tel Aviv University, Tel Aviv 69978, Israel*Corresponding author: Yaroslav.Kartashov@icfo.euReceived July 28, 2014; accepted August 12, 2014;posted August 27, 2014 (Doc. ID 219921); published September 24, 2014We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidlyenough toward the periphery of the medium, combined with an antisymmetric gain–loss profile. Despite theabsence of any symmetric modulation of the linear refractive index, which is usually required to establish aparity-time (PT ) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is neverbroken in the present system, and that the system always supports stable bright solitons, i.e., fundamental andmulti-pole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of thegain–loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, suchas fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable atarbitrarily large values of the gain–loss coefficient. © 2014 Optical Society of AmericaOCIS codes: (190.5940) Self-action effects; (190.6135) Spatial solitons.http://dx.doi.org/10.1364/OL.39.005641The high current interest in parity-time (PT )-symmetricsystems with complex potentials is partially motivatedby the remarkable behavior of the corresponding linearspectrum, which remains purely real until the strengthof the imaginary part of the potential attains a certainPT -symmetry breaking threshold, above which it becomescomplex [1]. The effect has been experimentally observedin optics, where the similarity of the paraxial evolutionequation, governing the propagation of light beams in me-dia with an even spatial profile of the refractive-indexmodulation and an odd gain–loss profile, with the Schrö-dinger equation governing the evolution of the quantum-mechanical wave function in a complex potential, enablesvisualization of the PT symmetry and its breakup at acritical point [2]. While eigenmodes of linearPT -symmet-ric potentials are well-understood [1,3,4], the evolution ofnonlinear excitations in them remains a subject of activeresearch. In particular, the properties of solitons and dis-crete nonlinear modes have been studied in free-standingPT -symmetric waveguides [5], couplers [6–9], oligomers[10,11], and periodic lattices with [12] andwithout [13–17]transverse refractive-index gradients, and in truncated lat-tices [18], among other settings. An especially interestingsituation occurs when the underlying evolution equationscontain only nonlinear PT -symmetric terms [19,20], ormixed linear–nonlinear lattices [21–23].As mentioned previously, a generic property ofPT -symmetric structures is that they support stableexcitations only if the spectrum of the associated linearsystem is real, i.e., the symmetry is not broken. As aresult, the stability domain of nonlinear excitations, ifdefined in terms of the gain–loss strength, often coin-cides with the domain of the unbroken PT symmetryin the respective linear system. However, systems may beprepared to be nonlinearizable, i.e., the nonlinear termsin the underlying evolution equation cannot be omittedeven for the decaying tails of localized nonlinearexcitations. Under such conditions, no direct link canbe drawn between the spectra of the nonlinear systemand its linear counterpart.In this Letter, we address that case in a system with anodd gain–loss profile and defocusing nonlinearity, whoselocal strength grows toward the periphery. In theabsence of gain and loss, such a system supports brightsolitons in all three dimensions [24–28]. Existence ofbright solitons in spite of the self-defocusing nature ofthe nonlinearity is at first counterintuitive, but actuallyit is a consequence of the nonlinearizability of the respec-tive nonlinear Schrödinger equation on the soliton tails.In this Letter, we show that a system of this type, with aneven profile of the growing nonlinearity, supports stablebright solitons even in the presence of the odd (PT -symmetric) gain–loss profile, without any spatial modu-lation of the linear refractive index, which is required forsupporting the real spectrum in usual PT -symmetric sys-tems. Without the rapidly growing defocusing nonlinear-ity, the PT symmetry of the system considered in thisLetter is always broken; in the presence of the nonline-arity modulation the symmetry may be said to becomeunbreakable, as it holds at arbitrarily large strengths ofthe balanced gain and loss. Recently, unbreakable sym-metry was demonstrated for a dimer, but it was a veryspecial case of a PT -symmetric Hamiltonian system [29].We address the propagation of a laser beam along the ξaxis of a medium with a transverse modulation of thegain–loss and defocusing nonlinearity, obeying the para-xial nonlinear Schrödinger equation for scaled amplitudeq of the electromagnetic fieldi∂q∂ξ − 12∂2q∂η2 σηqjqj2  iRηq; (1)where the propagation distance ξ is normalized to thediffraction length kx20; the transverse coordinate η isOctober 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS 56410146-9592/14/195641-04$15.00/0 © 2014 Optical Society of Americanormalized to the characteristic transverse scale x0; thefunction ση > 0, which is assumed to be even, describesthe profile of a defocusing nonlinearity whose strengthgrows as η→ ∞; and the function Rη, assumed tobe odd, stands for the gain–loss profile. In accordancewith [24,25], we adopt the following nonlinearity andgain–loss profiles:σ  σ0  σ2η2 expαη2;R  βη exp−γη2: (2)Note that the necessary and sufficient condition for theexistence of bright solitons with power (or norm),U  R∞−∞ jqj2dη, sustained by a growing defocusing non-linearity, is weaker, namely, it is ση∕jηj →∞ at jηj → ∞[24]. Nevertheless, in this Letter we use the steep modu-lation profiles described by Eq. (2) because they createtightly localized solitons, which are convenient for thenumerical and analytical considerations alike. The oddprofile of Rη accounts for the mutually balanced attenu-ation of the field at η < 0, and amplification at η > 0. Non-linearity and gain landscapes are depicted in Fig. 1(j) forσ0  1, σ2  0, and α  γ  1∕2. The model gives rise tobright soliton solutions in the form qη; ξ  wrηiwiη expibξ. Despite the dissipative character ofthe system, such solitons form continuous families, likein conservative media, parameterized by the propagationconstant b < 0. The complex form of the solitons givesrise to the intrinsic currents, j  w2dϕ∕dη, wherew expiϕ  wr  iwi, with wr and wi being real andimaginary parts of the solution, respectively.We have found multiple soliton families, including fun-damental [Figs. 1(a) and 1(b)], dipole [Figs. 1(c) and1(d)], tripole [Figs. 1(e) and 1(f)], and quadrupole[Figs. 1(g) and 1(h)] soliton solutions, and even morecomplex states. Their multi-pole structure is readily vis-ible, mostly for small values of β, when soliton solutionsare close to their conservative counterparts. Increasing βleads to notable transformations of the field-amplitudedistributions, as shown in the top and bottom lines ofFig. 1. The intrinsic currents rapidly grow when βincreases, but the profile of jη remains bell-shaped,even for solitons with a large number of humps [Fig. 1(i)].Next, we vary the gain–loss strength, β, for a fixedpropagation constant, in order to study the impact ofthe imaginary potential in Eq. (1) on the soliton proper-ties. Without nonlinearity modulation, the PT symmetryin Eq. (1) is broken, as the real part of the linear potentialis zero. However, in what follows we show that the non-linear pseudo-potential [30], σηjqj2, does sustain the PTsymmetry. A key insight comes from the observation thatEqs. (1) and (2) with γ  0 admit the exact soliton solu-tion q  2σ2−1∕2α expibξ − iβη∕α − αη2∕2, with propa-gation constant b  −α σ0α2∕σ2  β∕α2∕2, whichcan be found for arbitrarily large values of the gain–losscoefficient β. Moreover, this solution can be completelystable (the stability was checked at σ2  α  β  1∕2),indicating the conservation of the PT symmetry in thepresence of pseudo-potential.The soliton families are characterized by the depend-ences of the total power on β, which are depicted inFigs. 2(a) and 2(b) for the four types of modes shownin Fig. 1. Here, subscripts f , d, t, and q refer to the familiesof fundamental, dipole, tripole, and quadrupole solitons,respectively. A first noteworthy result is that differentsoliton families, with completely different internal struc-ture when β  0, merge at different critical values of β,for a fixed propagation constant, b. Specifically, the fam-ily of fundamental solitons merges with the family of di-poles, the family of tripoles merges with the quadrupoles,and so on. We observed that the value, βupp, at whichmerging occurs increases with the order of the solitonfamilies, as seen by comparing the βupp values for the di-pole d and quadrupole q families in Fig. 2(c). Close to themerger point, the distributions of the absolute value ofthe field amplitude for solitons belonging to the mergingbranches are nearly indistinguishable, cf. Figs. 1(f) and1(h). The numerical calculations also reveal that thesoliton power decreases when the number of lobes inFig. 1. Profiles of (a) and (b) fundamental; (c) and (d) dipole; (e) and (f) tripole; (g) and (h) quadrupole solitons at b  −10. Panels(a) and (c) correspond to β  0.55; in (b) and (d) β  1.98; in (e) and (g) β  1.04; while in (f) and (h) β  3.47. Panel (i) showscurrents in fundamental solitons from (a) and (b). Panel (j) shows nonlinearity and gain–loss landscapes for β  3.47.5642 OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014the soliton profiles increases for given values of b and β,and that the largest gain–loss strength, βupp, at whichdifferent families merge, monotonously increases withjbj for all types of solutions considered in this Letter[Fig. 2(c)].Therefore, despite the absence of any linear refractive-index modulation profile in Eq. (1), the PT symmetry,supported by the nonlinear pseudo-potential, is indeednot broken in the present system, because the fundamen-tal and higher-order stable solitons are found for arbitrar-ily large β, at sufficiently large values of jbj. Belowwewillshow that such soliton solutions can also be stable atarbitrarily large β values, a clear indication of unbreak-able symmetry in the present setting.On physical grounds, self-localization around the mini-mum of ση is provided by the growth of the localstrength of the defocusing nonlinearity at η → ∞ atany rate faster than jηj, as mentioned previously. It is thusnot possible to drop the nonlinear term in Eq. (1) even forthe vanishing tails of the solution, due to the rapid growthof ση [24,25]. Accordingly, the nonlinear pseudo-potential σηjqj2 in Eq. (1) provides the balance withthe odd imaginary potential. This is in contrast with pre-viously considered PT -symmetric systems with uniformnonlinearity, where nonlinear terms may be droppedfar from the soliton center, and the conditions for main-taining the PT symmetry are determined by the linearterms in the evolution equation.To test the dynamic stability of the soliton families, westudied the evolution of the perturbed solutions in theform q  wr  iwi  u expδξ  iv expδξ expibξ,with ju; vj ≪ jwr;wij. Substitution of this field intoEq. (1) and linearization leads to the eigenvalue problemδu  − 12∂2v∂η2 Ru σ2uwiwr  v3w2i w2r  bv;δv   12∂2u∂η2 Rv − σ2vwiwr  u3w2r w2i  − bu; (3)that we solved numerically to obtain the growth rates δ ofthe perturbations. We found that the families of funda-mental and dipole solitons are fully stable for any valueof propagation constant b (even in small power limitU → 0) and gain/loss strength β, within the existence do-main of such states [Fig. 2(c)]. Instabilities emerge onlyfor some of the tripole and quadrupole solutions. Theunstable solutions are depicted in red in Fig. 2(b). In par-ticular, tripoles are stable for 0 < β < βcr, and unstablefor βcr < β < βupp. The corresponding critical gain–lossstrength βcr is shown, as a function of propagationconstant b, in Fig. 3(a), along with the βuppb curve.One observes that the stability region notably broadenswhen the propagation constant decreases. The shape ofthe stability area for quadrupole solitons is similar atsmall values of jbj, but additional instability windows ap-pear at large values of jbj. In general, tripole and quadru-pole solutions become unstable at different criticalvalues of the gain–loss strengths, but a situation is pos-sible when βcr nearly coincides for two families. Typicaldependences of the instability growth rates as a functionof β for tripole and quadrupole solutions families areshown in Fig. 3(b). Examples of the stable evolutionof the four soliton families considered in this Letterare displayed in Figs. 4(a)–4(c) and 4(e). Even in thepresence of strong perturbations, such solitons propa-gate for unlimited distances without visible distortions.In contrast, unstable tripoles and quadrupoles featureprogressively swinging amplitude oscillations inFigs. 4(d) and 4(f).In summary, we introduced the first example of anonlinear PT -symmetric system with an unbreakablesymmetry. Namely, for sufficiently large propagationconstants, soliton solutions of the system remain stablefor arbitrarily large values of the gain–loss strength. Inaddition, families of fundamental and dipole solitonsare completely stable. On physical grounds, in thisFig. 2. (a) Power U versus gain–loss strength β for fundamen-tal and dipole solitons and (b) for tripole and quadrupole sol-itons. Circles correspond to solutions shown in Fig. 1.The fundamental and dipole families merge at βf ;dupp  2.135,while the tripole and quadrupole ones merge at βt;qupp  3.565.Black and red lines correspond to stable and unstable subfami-lies, respectively; (c) largest β below which various soliton fam-ilies exist, as a function of propagation constant b (at β → 0, thecurves originate from b  0, which correspond to solitons withzero amplitude).Fig. 3. (a) Domains of stability (white) and instability (shaded)for tripole solitons in the plane of b; β. (b) Real part of theinstability growth rate versus β at b  −5 for tripoles and quad-rupoles.October 1, 2014 / Vol. 39, No. 19 / OPTICS LETTERS 5643system the spatially odd gain–loss distribution is bal-anced not by an even refractive-index profile, but bythe pseudo-potential induced by the spatially growingstrength of the defocusing nonlinearity.The second author appreciates the hospitality of theInstitut de Ciencies Fotoniques (ICFO).References1. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243(1998).2. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodou-lides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).3. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H.Musslimani, Opt. Lett. 32, 2632 (2007).4. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H.Musslimani, Phys. Rev. Lett. 100, 103904 (2008).5. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N.Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).6. H. Ramezani, T. Kottos, R. 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Mod.Phys. 83, 247 (2011).Fig. 4. Stable propagation: (a) fundamental and (b) dipole sol-itons at β  0.55; (c) tripole and (e) quadrupole at β  1.04. In-stability development: (d) a tripole at β  2.10; (f) a quadrupoleat β  2.30. In all the cases, b  −10.5644 OPTICS LETTERS / Vol. 39, No. 19 / October 1, 2014

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain–loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multi-pole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain–loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain–loss coefficient.Peer Reviewe

UPCommons

We consider bright solitons supported by a symmetric inhomogeneous defocusing nonlinearity growing rapidly enough toward the periphery of the medium, combined with an antisymmetric gain-loss profile. Despite the absence of any symmetric modulation of the linear refractive index, which is usually required to establish a parity-time (PT ) symmetry in the form of a purely real spectrum of modes, we show that the PT symmetry is never broken in the present system, and that the system always supports stable bright solitons, i.e., fundamental and multipole ones. This fact is connected to the nonlinearizability of the underlying evolution equation. The increase of the gain - loss strength results, in lieu of the PT symmetry breaking, in merger of pairs of different soliton branches, such as fundamental and dipole, or tripole and quadrupole ones. The fundamental and dipole solitons remain stable at arbitrarily large values of the gain-loss coefficient.Peer Reviewe

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Unbreakable PT-symmetry of solitons supported by inhomogeneous defocusing nonlinearity

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