Gray solitons in parity-time symmetric potentialsHuagang Li,1 Zhiwei Shi,2,* Xiujuan Jiang,2 and Xing Zhu3

1Department of Physics, Guangdong University of Education, Guangzhou 510303, China2School of Information Engineering, Guangdong University of Technology, Guangzhou 510006, China

3Guangdong Engineering Research Center for Semiconductor Lighting, School of Science,South China University of Technology, Guangzhou 510640, China

*Corresponding author: szwstar@gdut.edu.cn

Received May 26, 2011; revised July 26, 2011; accepted July 28, 2011;posted August 1, 2011 (Doc. ID 148247); published August 15, 2011

We numerically study the gray solitons in parity-time (PT) symmetric potentials. Simulated results show thatthere are two kinds of gray solitons, the dip-shaped gray solitons and the hump-shaped solitons, and both of themcan be stable. Hump-shaped solitons can always exist, but the grayness of a stable dip-shaped gray soliton shouldexceed a threshold value. More interesting, it is discovered that when propagating in PT symmetric potentials, thegray solitons have no transverse deviation, and this is a phenomenon different from the usual gray solitons. © 2011Optical Society of AmericaOCIS codes: 190.3270, 190.6135.

Solitons caused by the interaction of nonlinear effect anddiffraction effect in the medium have been a hot topic innonlinear optics. They have been found in various media,such as Kerr media [1], nonlocal nonlinear media [2],photonic crystals, and Bose–Einstein condensates. Darksolitons are one kind of solution of the nonlinear Schrö-dinger (NLS) equation. They can be observed as localizedintensity dips on a cw background, which are usually ac-companied by a nontrivial phase change [3]. Spatial darksolitons exist in nonlinear planar waveguides [4,5]. Darksolitons are a particular type of gray solitons with a smal-ler and more gradual phase shift [6]. Gray solitons movein the transverse plane upon propagation and the gray-ness of solitons depends on the soliton velocity [2].The solitons in synthetic optical media with parity-time

(PT) symmetries have caught much attention in recentyears. Musslimani and his collaborators first discoveredthat a novel class of nonlinear self-trapped modes exist inoptical PT synthetic lattices [7], and that PT periodicstructures exhibit unique characteristics stemming fromthe nonorthogonality of the associated Floquet–Blochmodes [8]. The behavior of a PT optical coupled systemjudiciously involving a complex index potential was ob-served in an experiment in 2010 [9]. The analytical solu-tions to a class of NLS equations with PT-like potentials[10] and the stable dissipative defect modes in both fo-cusing and defocusing media with the periodic opticallattices imprinted in cubic nonlinear media with strongtwo-photon absorption [11] were also stated. However,thus far all studies focus on bright solitons in opticalPT symmetry media, and the dark and gray solitons inself-defocusing media are never reported.In this Letter, we investigate the gray solitons in PT

symmetric potentials. Simulated results show that thereare two kinds of gray solitons, the dip-shaped gray soli-tons and the hump-shaped solitons, and they can bestable in certain conditions. Compared with the gray so-litons in nonlocal nonlinear media [2], a more novelphenomenon is discovered that the gray solitons experi-ence no transverse deviation when propagating in PTsymmetric potentials.In a Kerr self-defocusing medium with PT symmetric

potential, the one-dimensional optical wave propagation

can be described by the normalized NLS-like equation forthe dimensionless light field amplitude q [7–11],

i∂q∂z

þ ∂2q

∂x2þ ½VðxÞ þ iWðxÞ�q − jqj2q ¼ 0; ð1Þ

where z is the propagation distance and VðxÞ and WðxÞare the real and the imaginary components of the com-plex PT symmetric potential, respectively. VðxÞ is aneven function andWðxÞ is odd [7]. We are going to searchfor a stationary soliton solution of Eq. (1) in the form ofqðx; zÞ ¼ uðxÞeibz, where u is a complex function and b isthe propagation constant of spatial solitons [12]. Thus,Eq. (1) can be changed into

∂2u

∂x2þ ½VðxÞ þ iWðxÞ�u − juj2u − bu ¼ 0: ð2Þ

The propagation constant b represents the asymptoticvalues of soliton intensity when VðxÞ ¼ WðxÞ ¼ 0, that is,juðx → �∞Þj2 ¼ −b. It sets the grayness of soliton definedas gd ¼ min jU j2, where jU j ¼ juj= ffiffiffiffiffiffijbjp

[2]. Here, we as-sume a Scarff II potential where VðxÞ ¼ V0sechðxÞ2 andWðxÞ ¼ W0sechðxÞ tanhðxÞ, with V0 andW0 being the am-plitudes of the real and imaginary parts. Although the PTsymmetric potential has crossed the phase transitionpoint, the solitons still exist because the amplitude ofthe refractive index distribution can be altered by thebeam itself through the optical nonlinearity [7]. The PTsymmetry will remain broken if it cannot be nonlinearlyrestored. By solving Eq. (2) using the finite differencemethod, we numerically obtain two kinds of gray solitonsat different values of V0 and W0.

First, when V0 ¼ −1 and W0 ¼ 0:3, the gray solitonsare dip-shaped. The real and imaginary components ofsuch a soliton are shown in Fig. 1(a). Since the valuesof such real and imaginary components are finite insteadof zero when x → �∞, ∂2q=∂x2 cannot be obtained withthe Fourier transform, and thus Eq. (2) cannot be solvedby using numerical methods such as the spectral re-normalization method [13]. The intensity distribution Iand the transverse phase distribution ϕ of the solitonare shown in Fig. 1(b). The errors of numerical solutions

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are shown in Fig. 1(c), and the solitons will not existwhen the error is greater than 10−10 [see Fig. 1(f)].In order to study the stability of the dip-shaped gray

solitons, we introduce the renormalized momentum Q ¼ði=2Þ R∞

−∞ðu∂xu� − u�∂xuÞð1þ b=juj2Þdx. In Fig. 1(d), the

solitons are stable because the condition dQ=dgd < 0 ismet [14], and the stable range is the same as that given bythe errors [Fig. 1(c)]. So, there exists a stability thresholdof gd, that is gd ≥ 0:18 (jbj ≥ 0:80). We have checked therobustness of the gray solitons by simulating the beampropagation with 1% random noise on both amplitudeand phase. The results shown in Figs. 1(g)–1(j) indicatethat the beams are stable, which supports the conclusionabout the stability of gray solitons given above.Second, we obtain the solitons under the conditions

V0 ¼ 1 and W0 ¼ 0:3. The real and imaginary compo-nents of such solitons are shown in Fig. 2(a), while I andϕ of the soliton are depicted in Fig. 2(b). As is seen, un-like the dip-shaped ones, these gray solitons are humpshaped. The grayness of the soliton is defined as gh ¼max jU j2. The errors of such numerical solutions areall less than 10−10, which indicates that the hump-shapedsoltions always exist. To check the stability of the soli-tons by the method of linear stability analysis, we assumef ðxÞ¼uðxÞeibzþϵðFðxÞeiδzþG�ðxÞeiδ�zÞeibz, where ϵ≪ 1,F and G are the perturbation eigenfunctions, and δ is thegrowth rate of the perturbation. By linearizing Eq. (2), wegain [7]

δF ¼�

∂2

∂x2þ VðxÞ þ iWðxÞ − 2juj2 − b

�F þ u2G; ð3Þ

δG ¼�−

∂2

∂x2− VðxÞ þ iWðxÞ þ 2juj2 þ b

�G − u�2F: ð4Þ

The hump-shaped gray soltions are linearly unstablewhen δ has an imaginary component; on the contrary,they are stable when δ is real. In Fig. 2(d), ImðδÞ, the ima-ginary component of δ, is zero, so the gray solitons arealways stable, and this confirms what results from theerrors of numerical solutions in Fig. 2(c). We simulate thestable propagation of beams under different conditions,as presented in Figs. 2(f)–2(j), which supports the aboveconclusion about the stability of hump-shaped soltionsas well.

We further study the influence ofW 0 on the solitons. Atb ¼ −10 and V0 ¼ −1, from Figs. 3(a) and 3(b), one cansee that the dip-shaped gray solitons can exist only whenW0 ≤ 1:34. Figure 3(c) verifies that the dip-shaped graysoliton cannot exist at W0 ¼ 1:37, while Fig. 3(d) verifiesthat the soliton can exist at W 0 ¼ 1:34. At b ¼ −1 andV0 ¼ 1, from Figs. 3(e) and 3(f), one can see that thehump-shaped gray solitons can exist only when W0 ≤

0:56. Figure 3(g) verifies that the hump-shaped gray so-liton cannot exist at W 0 ¼ 0:58, while Figs. 3(h) and 3(i)verify that the soliton can exist at W0 ¼ 0:56 and 0.48.

The real part of a PT complex potential must be aneven function of position, whereas the imaginary com-ponent should be odd. One of the most intriguing char-acteristics of such a PT symmetric Hamiltonian is theexistence of a critical threshold above which the systemundergoes a sudden phase transition because of spon-taneous PT symmetry breaking [7]. The nontrivial phasestructures of the nonlinear modes shown in Figs. 1(b)and 2(b) result in energy flow and those in Figs. 1(e)and 2(e) result in infinite domain. From Figs. 1(e) and2(e), one can find that the energy flow density S is posi-tive everywhere, which implies that the energy alwaysflows in one direction, i.e., from the gain region towardthe loss region [7].

A localized conservative potential can only determinethe position of the intensity dip of the dark solitons whenthe width of the potential is bigger [15], the dip will

Fig. 1. (Color online) (a) Real (dashed black curve) and ima-ginary (solid pink curve) components of the dip-shaped graysoliton; (b) I (solid pink curve) and ϕ (dashed black curve)of the soliton at b ¼ −1:46. The dash-dotted blue and dottedred curves in (a) and (b) correspond to the functions of VðxÞand WðxÞ, respectively. (c) Error of numerical solutions versusb, points marked with circles correspond to the cases shown in(f) and (g). (d) Q versus gd, point marked with circle corre-sponds to the case shown in (g). (e) Energy flow density of thesoliton at b ¼ −1:46. (f)–(j) Simulated propagation of the soli-tons with 1% random noise at b ¼ −0:78, −0:80, −1:00, −2:34,and −5:00, respectively.

Fig. 2. (Color online) (a) Real (dashed black curve) and ima-ginary (solid pink curve) components of the hump-shaped graysoliton; (b) I (solid pink curve) and ϕ (dashed black curve) ofthe soliton at b ¼ −0:74. The dash-dotted blue and dotted redcurves in (a) and (b) correspond to the functions of VðxÞ andWðxÞ, respectively. (c) Error of numerical solutions versus b.(d) ImðδÞ versus b. (e) Energy flow density of the soliton atb ¼ −0:74. (f)–(j) Simulated propagation of the solitons with1% random noise at b ¼ −0:05, −0:74, −1:49, −2:48, and −4:73,respectively.

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remain even in the absence of the conservative potential,and the solitons are the results of balance between non-linear effect and diffraction effect (V0 ¼ 0, W0 ¼ 0).Moreover, when the widths of a localized conservativepotential and an optical beam are almost the same, thecharacter of the potential will influence the type of solu-tion. This is just as in the PT complex potential (i.e., V0 ≠

0 and W0 ≠ 0) where the solitons are the results of theinteraction of nonlinear effect, diffraction effect, andconstraint force arising from VðxÞ. The solitons will bedip shaped when the total effects of the constraint forceand the diffraction are equal to the nonlinear effect, andon the other side, the solitons will be hump shaped whenthe constraint force is equal to the total effects of the dif-fraction and the nonlinearity.

In conclusion, we investigate two kinds of graysolitons in PT symmetric potentials, the dip-shaped graysolitons and the hump-shaped gray solitons. More impor-tant, we discover a phenomenon that the gray solitons inPT symmetric potentials have no transverse deviation intheir propagation.

This research is supported by the Natural ScienceFoundation of Guangdong Province, China (GrantNo. 10151063101000017), the Open Fund from KeyLaboratory for High Power Laser Physics of ChineseAcademy of Sciences (CAS) (Grant No. SG-001103),and the National Natural Science Foundation of China(NSFC) (Grant Nos. 10904041 and 10974061).

References

1. G. A. Swartzlander, Jr. and C. T. Law, Phys. Rev. Lett. 69,2503 (1992).

2. Y. V. Kartashov and L. Torner, Opt. Lett. 32, 946 (2007).3. Y. S. Kivshar, IEEE J. Quantum Electron. 29, 250 (1993).4. G. A. Swartzlander, Jr., D. R. Andersen, J. J. Regan, H. Yin,

and A. E. Kaplan, Phys. Rev. Lett. 66, 1583 (1991).5. B. Luther-Davies, R. Powles, and V. Tikhonenko, Opt. Lett.

19, 1816 (1994).6. Y. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81

(1998).7. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and

D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).8. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and

Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).9. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N.

Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192(2010).

10. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, andD. N. Christodoulides, J. Phys. A 41, 244019 (2008).

11. Y. V. Kartashov, V. V. Konotop, V. A. Vysloukh, andL. Torner, Opt. Lett. 35, 1638 (2010).

12. Z. Xu, Y. V. Kartashov, and L. Torner, Opt. Lett. 30, 3171(2005).

13. M. J. Ablowitz and Z. H. Musslimani, Opt. Lett. 30, 2140(2005).

14. Y. S. Kivshar andW. Królikowski, Opt. Lett. 20, 1527 (1995).15. D. E. Pelinovsky, D. J. Frantzeskakis, and P. G. Kevrekidis,

Phys. Rev. E 72, 016615 (2005).

Fig. 3. (Color online) (a) Error of numerical dip-shaped graysolutions versus W0; points marked with circles correspondto cases shown in (c) and (d). (b) Q versus gd of the dip-shapedgray solitons; point marked with circle corresponds to the caseshown in (d). (c) and (d) Simulated propagation of the gray so-litonwith 1%noise atW0 ¼ 1:37 and 1.34, respectively. (e)Errorsof numerical hump-shaped solutions versus W0; points markedwith circles correspond to the cases shown in (g) and (h).(f) ImðδÞ versus b of the hump-shaped gray solitons; pointmarked with circle corresponds to the case shown in (h). (g)–(i) Simulated propagation of the hump-shaped gray solitonswith1% noise at W0 ¼ 0:58, 0.56, and 0.48, respectively.

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