Solitons in PT-symmetric nonlinear lattices
Fatkhulla Kh. Abdullaev, Yaroslav V. Kartashov, Vladimir V. Konotop, Dmitry A. Zezyulin
aa r X i v : . [ n li n . PS ] A p r Solitons in PT -symmetric nonlinear lattices Fatkhulla Kh. Abdullaev , Yaroslav V. Kartashov , Vladimir V. Konotop , and Dmitry A. Zezyulin Centro de F´ısica Te´orica e Computacional, Faculdade de Ciˆencias,Universidade de Lisboa, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya,Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain (Dated: October 29, 2018)Existence of localized modes supported by the PT -symmetric nonlinear lattices is reported. Thesystem considered reveals unusual properties: unlike other typical dissipative systems it possessesfamilies (branches) of solutions, which can be parametrized by the propagation constant; relativelynarrow localized modes appear to be stable, even when the conservative nonlinear lattice potentialis absent; finally, the system supports stable multipole solutions. PACS numbers: 42.65.Tg, 42.65.Sf
Since introduction of the concept of the PT -symmetricpotentials [1], this subject attracted a great deal of atten-tion [2]. While the primary interest was devoted to suchsystems in the context of non-Hermitian quantum me-chanics, recently new applications of the PT -symmetricpotentials have been found in optics in media with inho-mogeneous in space gain and damping, i.e. with properlydesigned imaginary part of the linear refractive index.The first experiments reporting the phenomenon are al-ready available [3]. As soon as the importance of the op-tical applications was realized, it became also clear thatthe phenomenon can be studied in the nonlinear context,from the point of view of existence of nonlinear local-ized modes in linear PT -symmetric potentials [4]. It isthen natural step to address the existence and stabilityof localized modes in nonlinear PT -symmetric potentialwhich in optics can be implemented by means of properspatial modulation of nonlinear gain and losses. As an ex-ample, such optical systems can be nonlinear waveguides,employing concantenated semiconductor optical ampli-fier and semiconductor doped two-photonic absorber sec-tions (notice that experimental implementation of the lin-ear PT symmetry breaking was reported in [5]).While it is now known that stable localized [6, 7] andmoving [8] solitons can exist in conservative purely non-linear lattices [6–8] (see also [9, 10] for review) the ex-istence of stable localized solitons in complex nonlinearlattices is still an open problem, since up to now onlyperiodic waves were found to be stable in such struc-tures [11]. The elucidation of stable localized solitonsin PT -symmetric nonlinear lattices is therefore a centralgoal of this Letter.We describe the propagation of laser radiation alongthe ξ -axis of the medium with periodic transverse mod-ulation of cubic nonlinearity and nonlinear gain with thecomplex nonlinear Schr¨odinger (NLS) equation for thedimensionless light field amplitude q : iq ξ = − q ηη − | q | q − [ V ( η ) + iW ( η )] | q | q (1)where η and ξ are the normalized transverse and lon- gitudinal coordinates, respectively. The functions V ( η )and W ( η ) describe transverse periodic modulations of theconservative and dissipative parts of the nonlinearity andare assumed to satisfy the PT symmetry relations. Wefurther assume that conservative and dissipative partsof nonlinearity have the same period π , i.e. V ( η ) = V ( − η ) = V ( η + π ) and W ( η ) = − W ( − η ) = W ( η + π ).These functions will be considered bounded with σ r and σ i being the maxima of V and W , respectively.We are interested in stationary localized solu-tions, which can be searched in the form q ( η, ξ ) = u ( η ) e iθ ( η )+ ibξ = [ w r ( η ) + iw i ( η )] e ibξ where u and θ arethe amplitude and phase of the mode. Eq. (1) can berewritten in the hydrodynamic form12 u ηη − bu +[1+ V ( η )] u − j u = 0 , j η = − W ( η ) u (2)where we have introduced the ”current density” given by j = u dθ/dη .Starting with general properties of stationary localizedsolutions, we notice that it follows from (2) that suchsolutions can exist only for b > η → ±∞ is given by u ∼ e ±√ bη and j ∼ e ± √ bη , i.e. the current density is localized muchstronger than the field. One easily finds that the field u can become zero only in the points where the current den-sity j is zero, as well. Furthermore, closely following theapproach described in [9], one obtains that b = O ( U )where U = U r + U i , with U r,i = R ∞−∞ w r,i ( η ) dη , is thetotal energy flow of the beam. This implies that in thelimit of small intensity ( U →
0) we have U ∼ √ b , andrespectively u max →
0. Following [9], one can obtainthe relation b ≤ u max (1 + σ i ). Thus the existence of thesolutions with u max → b → R ∞−∞ W ( η ) u ( η ) dη = 0. Since we considerodd functions W ( η ), this condition can be satisfied byany even function u ( η ). In other words, unlike this hap-pens in dissipative systems of a general kind [13], therequirement of balance between losses and gain in oursystem does not introduce a constraint selecting only onepossible mode (i.e. the propagation constant is not deter-mined by the balance between losses and gain). Thus, interms of the existence of branches of solutions the proper-ties of Eq. (1) resemble the properties of the conservativeNLS where the propagation constant b is determined bythe energy flow U and a continuous family of solutionsexist. This fact is illustrated in Fig. 1. Hereafter in allnumerical simulations we use V ( η ) = σ r cos(2 η ) and W ( η ) = − σ i sin(2 η ) , (3)where σ r and σ i are the modulation depths of the conser-vative and dissipative lattices. In Fig. 1 (a) we observethat increase of b results in monotonic growth of the soli-ton energy flow U and the contraction of light in a singlechannel of nonlinear lattice. Now, however the energyof the soliton is distributed between real and imaginarycomponents of the field, as it is shown in Fig. 1 (b). Theratio U i /U r of energy flows concentrated in imaginaryand real parts of the field takes on maximal value at in-termediate b values and diminishes at b → b → ∞ . U b (a) U i / U r b (b) FIG. 1: (Color online) (a) Energy flow versus propagationconstant for fundamental solitons at σ i = 0 (curve 1), 1 . . σ r = 1. (b) U i /U r versus propa-gation constant at σ i = 1 . σ r = 1. Let us now turn to more detailed study of the men-tioned limits of the propagation constant. First of all,Fig. 1 (a) supports the above estimate U ∼ b / , alsoillustrating that in the limit b → q ( η, ξ ) ≈ Q ( η, ξ ) + A ( η, ξ ) cos(2 η ) + iB ( η, ξ ) sin(2 η ) , where Q , A , and B arethe functions slowly varying on the scale π . Substitu-tion of this ansatz in Eq. (1) yields A = σ r | Q | Q/ B = − σ i | Q | Q , and cubic-quintic NLS equation for thefield Q : iQ ξ + 12 Q ηη + | Q | Q + 32 χ | Q | Q = 0 . (4) where χ = (3 σ r − σ i ) This equation does not containany imaginary part - the consequence of the oppositeparities of real and imaginary components of the nonlin-earity modulations. The solitonic solution of (4) whichexists at bχ > − / Q = 2 √ be ibξ h p χb cosh(2 √ bη ) i − / . (5)This solution is reduced to the conventional NLS solitonin the limit b →
0, revealing weak dependence of thesoliton on the parameter χ , what explains convergenceof all branches in Fig. 1 (a) at b → θ (0) = 0, it however can be changed due to the phaseinvariance of the complex NLS equation). The centers ofsuch solitons reside in the point where conservative partof nonlinearity takes on the maximal value, while dissi-pative part of nonlinearity is zero. Due to the fact thatleft wing of soliton resides in the domain with nonlin-ear losses, while its right wing is subjected to nonlineargain the solitons are characterized by the anti-symmetricimaginary parts of the field [Figs. 2(a) and (b)] indicat-ing on tilted phase fronts and the existence of internalcurrents directed into the domain with losses [Fig. 2 (d)].Returning to the simple approximation (5) we also ob-serve that at fixed b and σ r it suggests the existence ofthe upper limit, σ i ≤ σ uppi , of the strength of the dis-sipative term σ i , for which localized dissipative solitonsexist. This is indeed confirmed numerically in Fig. 3(a)[notice that the simple estimate for this upper limit σ ( upp ) i ≈ σ r + 3 / σ uppi ≈ .
22 while the numerical value is σ uppi ≈ . σ i results in the monotonic increase ofthe imaginary part of the field [c.f. also Figs. 2(a) and(b)] accompanied by a considerable increase of currentdensity [Fig. 2(d)]. The energy flow increases with σ i [Fig. 3(a), red curve] until the tangential line to U ( σ i )becomes vertical. Apparently, there exists another up-per branch of solutions joining with the lower branch inthe point σ i = σ uppi [Fig. 3(a), black curve] for whichthe energy flow is a monotonically decreasing function of σ i . The solitons belonging to this branch are character-ized by a double-hump field modulus profile [Fig. 2(c)].When σ i decreases the real part of the solutions decaysand only imaginary survives. The later is asymmetricand its maximum and minimum are located in a singleperiod of V ( η ) [this tendency is visible in Fig. 1(c)]. Thesolitons from upper branch in Fig. 3 (a) are unstable.Besides these simplest branches one can find a varietyof soliton families with more complicated internal phasedistributions, but we do not discuss them here becausethey are usually unstable.One of the most important results of this Letter is that -6 0 6-0.40.20.81.4 w i u , w r u (a) -6 0 6-1.0-0.10.81.7 w i u (b) w r u -6 0 6-1.90.01.9 uw i w r u (c) -2 0 2-4.0-2.4-0.80.8 j (d) FIG. 2: (Color online) The profiles of fundamental solitonsfrom lower branch at σ i = 0 . .
58 (b). (c) The profileof soliton from higher branch at σ i = 0 .
3. (d) The currentdensity for the fundamental solitons from lower branch at σ i = 0 . .
58 (curve 2). In all cases b = 1 and σ r = 0 . fundamental solitons can be stable despite the fact thatthe system (1) is characterized by the presence of do-mains where only losses or gain are acting. The outcomeof stability analysis is presented in Figs. 3 (b)-(d). Thefundamental solitons are stable for σ i below certain crit-ical value σ cri [see Fig. 3 (d) for a typical dependenceof the perturbation growth rate on σ i ]. Notice that for σ i > σ cri the growth rate δ r increases until one reachesthe border of existence domain σ i = σ uppi . For fixed σ r the stability domain on the plane ( b, σ i ) is rather complex[Fig. 3 (c)]. At σ r = 0 . σ cri and σ uppi increase as b → b values the domains of existence and stability monoton-ically expand with b . A similar situation is encounteredfor other values of σ r . The increase of the depth of modu-lation of conservative nonlinearity σ r at fixed b results inconsiderable expansion of existence domain on the plane( σ r , σ i ).However, especially interesting situation occurs at σ r = 0. In this case, there is no modulation of conser-vative nonlinearity at all, but our analysis still predictsstability of fundamental solitons between two red linesin Fig. 3(c) (for b > b cr ≈ .
05 the solitons are stablefor 0 < σ i < σ cri ). This fact is really remarkable tak-ing into account that now the symmetric conservative U i (a) cri i b (b) uppi cri uppi i b (c) ! r i (d) b =2.0 0.5 FIG. 3: (Color online) (a) The energy flow vs σ i for lower(red curve) and upper (black curve) branches of fundamentalsolitons at b = 1, σ r = 0 .
5. Circles correspond to solitonsshown in Figs. 2(a)–(c). Domains of existence and stabilityon the plane ( b, σ i ) for fundamental solitons at σ r = 0 . σ r = 0 (c). (d) The perturbation growth rate vs σ i at σ r = 0 . nonlinearity providing the restoring force in the case ofslight displacements of soliton center from the equilib-rium position η = 0, is absent. We observe that the lossof stability occurs at the soliton width, which is compa-rable with the characteristic scale of the lattice, i.e. tothe half-period π/
2. Thus, the modulation of conserva-tive nonlinearity is not a necessary ingredient for solitonstability, although it can change considerably stabilityproperties of low-power solitons with b → V . Therepresentative examples of such states that in the limit σ i → σ i →
0) changes its signbetween neighboring maxima of V . The current densityin such states is characterized by n ( n is the number ofpeaks in field modulus) negative spikes in the vicinityof maxima of V . Analogs of solitons with in-phase fieldpeaks were obtained too, but they all are unstable. Likefundamental solitons, multipole states are parameterizedby the propagation constant b . For a given σ r there exista cutoff on b below which multipole solitons do not exist,while increase of b results in growth of energy flow.Increase of gain-loss modulation σ i also causes increaseof U and fraction of power concentrated in imaginary partof the of multipole soliton [c.f. Figs. 4(a) and (b)], butsuch solitons can be found only at σ i < σ uppi [in Fig. 5(a)we show only the lower branch of dipole solitons although -4.5 1.5 7.5-1.80.01.8 w i u (a) w r u -4.5 1.5 7.5-2.30.02.3 uw r w i u (b) -9 0 9-1.70.11.9 uw r w i u (c) FIG. 4: (Color online) Dipole solitons at (a) σ i = 0 .
5, (b) σ i = 2 .
67, and tripole solitons at (c) σ i = 1. In all cases σ r = 1, b = 2 upper unstable branch can be found too]. Linear stabil-ity analysis predicts stability of the multipole solitons at σ i < σ cri as shown in Fig. 5(c). This domain of stabilitygradually broadens with increase of the depth of mod-ulation of conservative nonlinearity σ r [Fig. 5 (b)]. Incontrast to fundamental solitons multipole solitons canbe stable only if propagation constant is sufficiently large.This critical value of propagation constant increases withdecrease of σ r . This is because multipole solitons mayexist only if conservative nonlinearity is modulated andwhen this modulation is sufficient for compensation ofrepulsive forces acting between neighboring poles. In-crease of the number of poles in solitons does not resultin dramatic modifications of existence domain [0 , σ uppi ]but domain of stability [0 , σ cri ] shrinks with n .To conclude, we have reported a set of stable local-ized solutions supported by PT -symmetric nonlinear lat-tices. The system considered reveals a number of unusualproperties. First, although it is dissipative and the bal-ance between the gain and losses must be satisfied, itpossesses families (branches) of solutions, which can beparametrized by the propagation constant b , in contrastto other typical dissipative systems. Second, the modes,whose width is smaller than the lattice half-period appearto be remarkably stable, even when the conservative non-linear potential is absent. Finally, the system supportsstable multipole solutions.The work of FKA and VVK was supported by the U i (a) uppi cri i r (b) ! r i (c) b =2 1 FIG. 5: (Color online) (a) Energy flow vs nonlinear gain fordipole solitons at σ r = 1, b = 2. Only lower branch is shown.Circles correspond to profiles shown in Figs. 4(a) and (b). (b)Domains of existence and stability on the plane ( σ r , σ i ) fordipole solitons at b = 2. (c) Real part of perturbation growthrate vs the gain parameter at σ r . grant PIIF-GA-2009-236099 (NOMATOS). DAZ wassupprted by the grant SFRH/BPD/64835/2009. [1] C. M. Bender and S. Boettcher, Phys. Rev. Lett. ,5243 (1998).[2] see e.g. Special issue of J. Phys. A: Math. Gen. (2006); ibidem (2008).[3] C. E. Ruter et al. Nature Phys. , 192 (2010).[4] Z. H. Musslimani, et. al. Phys. Rev. Lett. , 030402(2008); K. G. Makris et. al. ibidem , 244019 (2008).[5] A. Guo et. al. , 093902 (2009)[6] H. Sakaguchi and B. A. Malomed, Phys. Rev. E ,046610 (2005).[7] Y. Sivan, G. Fibich, and M. I. Weinstein, Phys.Rev.Lett. , 193902 (2006).[8] F. Kh. Abdullaev and J. Garnier, Phys.Rev. A ,061605(R) (2005).[9] H. A. Cruz, et. al. Physica D , 1372 (2009)[10] Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev.Mod. Phys. (2011) (in press) [11] F. Kh. Abdullaev, et. al. Phys. Rev. E , 056606 (2010).[12] Kh. I. Pushkarov, D.I. Pushkarov, and I. V. Tomov,Opt. Electr.11