Discrete solitons dynamics in PT -symmetric oligomers with complex-valued couplings
OOligomers with complex couplings as PT -symmetric systems O. B. Kirikchi
Department of Computing, Goldsmiths, University of London,New Cross, London SE14 6AD, United Kingdom andDepartment of Mathematical Sciences, University of Essex,Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom
N. Karjanto ∗ Department of Mathematics, University College, Sungkyunkwan University,Natural Science Campus, 2066 Seobu-ro, Jangan-gu, Suwon 16419, Gyeonggi-do, Republic of Korea
We consider a chain of dimers with a complex coupling between the arms as parity-time ( PT )symmetric systems. We study fundamental bright discrete solitons of the systems, their existence,and spectral stability. We employ a perturbation theory for small coupling between the arms andsmall gain-loss parameter to perform the analysis, which is then confirmed by numerical calcula-tions. We consider the fundamental onsite and intersite bright solitons. Each solution possessessymmetric, antisymmetric, and asymmetric configurations between the arms. The stability of thesolutions is then determined by solving the corresponding eigenvalue problem. We obtain that allthe solitons can be stable for small coupling, on the contrary to the reported continuum limit wherethe antisymmetric solutions are always unstable. The instability is either due to the internal modescrossing the origin or the appearance of a quartet of complex eigenvalues. In general, the gain-lossterm can be considered parasitic as it reduces the stability region of the onsite solitons. Addition-ally, we analyze the dynamic behavior of the onsite and intersite solitons when unstable, where notraveling solitons nor soliton blow-ups are observed. I. INTRODUCTION
Dissipative media featuring the parity-time ( PT )-symmetry has drawn a great deal of attention ever sincethe system was proposed by Carl Bender and his collabo-rators [1–4]. A system of equations is PT -symmetric if itis invariant with respect to the combined parity ( P ) andtime-reversal ( T ) transformations. The symmetry is fas-cinating as it forms a particular class of non-HermitianHamiltonians in quantum mechanics that may possess areal spectrum up to a critical value of the complex poten-tial parameter, above which the system is in the “broken PT -symmetry” phase [4–7].It is assumed that quantities in quantum physics thatwe observe are the eigenvalues of operators symbolizingthe dynamics of the quantities. Therefore, the eigenval-ues, which represent the energy spectra should be realand have lower bound so that the system has a stablelowest-energy state. To satisfy such demands, it wasspeculated that the operators must be Hermitian. Non-Hermitian Hamiltonians have been commonly related tocomplex eigenvalues and therefore decay of the quan-tities. However, it turned out that the Hermiticity isnot necessarily required by a Hamiltonian system to sat-isfy the Postulates of Quantum Mechanics [5]. A nec-essary condition for a Hamiltonian to be PT -symmetricis that its potential V ( x ) should satisfy the condition V ( x ) = V ∗ ( − x ) [8].The most basic configuration of a PT -symmetric sys-tem is a dimer, i.e., an oligomer system of two coupled ∗ [email protected] oscillators. One of them has damping loses and the otherone gains energy from external sources. Indeed, the ideaof PT -symmetry was realized experimentally for the firsttime on dimers consisting of two coupled optical waveg-uides [9, 10]. Optical analogs using two coupled waveg-uides with gain and loss were investigated in [11–13],where such couplers have been already considered pre-viously in the 1990s [14–16]. PT -symmetric analogs in coupled oscillators have alsobeen proposed theoretically and experimentally [17–20].A system of coupled oscillators with gain and loss havealready been studied [21]. PT -symmetric system withperiodically changing-in-time gain and loss modeled bytwo coupled Schr¨odinger equations (dimer) is studiedin [22, 23], where a comparison between analytical studyand numerical approach were presented to investigate anapproximate threshold for PT -broken symmetry phasecorresponding to the disappearance of bounded solutions.A continuum limit of a chain of coupled PT -symmetricdimers has been covered in [24], where the amplitudesystem remains conservative and the small-amplitudebreathers are stable for a finite time scale. Fascinatingclass of optical and other systems in which the communi-cation or coupling makes the systems PT -symmetric wasconsidered in [25], where it discussed the comparison be-tween the dynamical behaviors with that of the usual PT -symmetric systems with intrinsic loss-gain terms.In particular, we are interested in the nonlinear dynam-ics of PT -symmetric chain of dimers that can be mod-eled by the discrete nonlinear Schr¨odinger (DNLS) typeof equations due to its abundance applications in nonlin-ear optics and Bose-Einstein condensates (BEC) [26–28].Transport on dimers with PT -symmetric potentials are a r X i v : . [ n li n . PS ] M a y modeled by the coupled DNLS equations with gain andloss, which was relevant among others to experiments inoptical couplers and proposals on BEC in PT -symmetricdouble-well potentials [29]. This proposed model is inte-grable and its integrability is further utilized to buildup the phase portrait of the system. The existence andstability of localized mode solutions to nonlinear dynam-ical lattices of the DNLS type of equations with two-component settings have been considered and a generalframework has been provided in [30]. PT -symmetric systems have also been considered byseveral authors in various contexts. We present somerecent and relevant examples. A system modeling a PT -symmetric coupler composed by a chain of dimers witha cubic-quintic nonlinearity exhibits a snaking behaviorin the bifurcation diagrams for the existence of standinglocalized solutions [31]. A dual-core nonlinear waveguidewith the PT -symmetry has been expanded by includinga periodic sinusoidal variation of the loss-gain coefficientsand synchronous variation of the inter-core coupling con-stant [32]. The system lead to multiple-collision inter-actions among stable solitons. A study of the nonlinearnonreciprocal dimer in an anti-Hermitian lattice with cu-bic nonlinearity has been explored recently [33].In this paper, we consider the coupled discrete linearand nonlinear Schr¨odinger equations on oligomers withcomplex couplings as systems of PT -symmetric poten-tials. The model arises as nonlinear optical waveguidecouplers on BEC in PT -double well symmetric poten-tials. The phase portrait of the system and the behaviorof the solutions are discussed through analytical and nu-merical approaches.The manuscript is outlined as follows. In Section II,we present the equations of motion as the correspond-ing governing equation. We use perturbation theory forsmall coupling to analyze the existence of fundamentallocalized solutions. Such analysis is based on the con-cept of the so-called anticontinuum limit approach. Thestability of the solitons is then considered analytically inSection III by solving a corresponding eigenvalue prob-lem. In addition to small coupling, the expansion is alsoperformed under the assumption of the small coefficientof the gain-loss term due to the non-simple expression ofthe eigenvectors of the linearized operator. The findingsobtained from the analytical calculations are then com-pared with the numerical counterparts in Section IV. Weproduce stability regions for the fundamental onsite soli-tons numerically and present the typical dynamics of soli-tons in the unstable parameter ranges by direct numericalintegrations of the governing equation. We present theconclusion in Section V. II. MATHEMATICAL MODEL
The governing equations describing PT -symmetricchains of dimers are of the following form:˙ u n = i | u n | u n + i(cid:15) ∆ u n + γv n + iv n , ˙ v n = i | v n | v n + i(cid:15) ∆ v n − γu n + iu n , (1)where the dots represent the derivative with respect tothe evolution variable, which is the physical time t forBEC and the propagation direction z in the case of non-linear optics. Both u n = u n ( t ) and v n = v n ( t ) arecomplex-valued wave function at site n ∈ Z , 0 < (cid:15) << u n =( u n +1 − u n + u n − ) and ∆ v n = ( v n +1 − v n + v n − )are the discrete Laplacian terms in one spatial dimen-sion, the gain and loss acting from the complex couplingare represented by the coefficient γ , which without lossof generality can be taken to be γ >
0. We considerlocalized solutions satisfying the localization conditions u n , v n → n → ±∞ .In the uncoupled limit, i.e. when (cid:15) = 0, the chain (1)becomes the equations for the dimer with complex cou-plings. This type of PT -symmetric system with the com-plex coupling has been studied recently in [25]. A simi-lar setup was studied in [34] in the presence of gain-lossterms, Stokes variable dynamics of the dimer with gain-loss terms were developed as a subcase of a general dimermodel. The dimer itself may be considered for the firsttime in [35, 36], where the integrability was shown.The focusing system has static solutions that can beobtained from substituting u n = A n e iωt , v n = B n e iωt , (2)into (1) to yield the static equations ωA n = | A n | A n + (cid:15) ( A n +1 − A n + A n − ) − iγB n + B n ,ωB n = | B n | B n + (cid:15) ( B n +1 − B n + B n − ) + iγA n + A n , (3)where A n , B n are complex-valued quantities and thepropagation constant ω ∈ R .The static equations (3) for (cid:15) = 0 has been analyzed indetails in [25, 35, 36]. When (cid:15) is nonzero, but sufficientlysmall, the existence of solutions emanating from the un-coupled limit can be shown using the Implicit FunctionTheorem (see, e.g., The existence analysis of [29], whichcan be adopted here rather straightforwardly). However,below we will not state the theorem and instead derivethe asymptotic series of the solutions.Using perturbation expansion, solutions of the cou-pler (3) for small coupling constant (cid:15) can be expressedanalytically as A n = A (0) n + (cid:15)A (1) n + (cid:15) A (2) n + . . . ,B n = B (0) n + (cid:15)B (1) n + (cid:15) B (2) n + . . . . (4)By substituting the above expansions into equations (3)and collecting the terms in successive powers of (cid:15) , oneobtains the following equations at O (1) and O ( (cid:15) ), re-spectively A (0) n (1 + iγ ) = B (0) n ( ω − B (0) n B ∗ (0) n ) ,B (0) n (1 − iγ ) = A (0) n ( ω − A (0) n A ∗ (0) n ) . (5)and A (1) n (1 + iγ ) = B (1) n ( ω − B (0) n B ∗ (0) n ) − B (0) n B ∗ (1) n − ∆ B (0) n ,B (1) n (1 − iγ ) = A (1) n ( ω − A (0) n A ∗ (0) n ) − A (0) n A ∗ (1) n − ∆ A (0) n . (6)It is well-known two natural fundamental solutions arerepresenting bright discrete solitons that may exist forany (cid:15) , from the anticontinuum to the continuum limit, i.e.an intersite (two-excited-site) and onsite (one-excited-site) bright discrete mode. Here, we will limit our studyto these two fundamental modes. A. Dimers
In the uncoupled limit (cid:15) = 0, the time-independentsolution of (3), i.e. (5), can be written as A (0) n = ˜ a e iφa and B (0) n = ˜ b e iφ b , where both amplitudes are positivereal valued, i.e. ˜ a > b >
0. Solving the resultingpolynomial equations for ˜ a and ˜ b will yield [25]˜ a = ˜ b = 0 , (7)˜ a = ˜ b = (cid:113) ω − (cid:112) γ , (8)˜ a = − ˜ b = (cid:113) ω + (cid:112) γ , (9)˜ a = 1 √ (cid:113) ω + (cid:112) ω − γ ) , ˜ b = 12 (cid:113) ω + (cid:112) ω − γ ) (cid:104) ω − (cid:112) ω − γ ) (cid:105)(cid:112) γ ) , (10)and the phase φ b − φ a = arctan γ . The parameter φ a canbe taken as 0, due to the gauge phase invariance of thegoverning equation (1) and henceforth φ b = arctan( γ ).Solutions (8), (9), and (10) are referred to as the sym-metric, antisymmetric, and asymmetric solutions, respec-tively. The asymmetric solution (10) emanates from apitchfork bifurcation from the symmetric solution (8) at ω = 2 (cid:112) γ .Another variant of interesting dimers where the cou-pling between the oscillators provide a gain to the systemwas considered in [37–39]. Such a system may model thepropagation of electromagnetic waves in coupled waveg-uides embedded in an active medium. The dimer con-sidered herein when (cid:15) → B. Intersite solitons
The mode structure of the intersite solitons in the an-ticontinuum limit is given by A (0) n = (cid:26) ˜ a n = 0 , , ,B (0) n = (cid:26) ˜ b e iφ b n = 0 , , . (11)For the first-order correction due to the weak coupling,writing A (1) n = ˜ a , B (1) n = ˜ b e iφ b , and substituting theseinto equations (6) will yield˜ a = ˜ b ( ω − b ) + ˜ b (cid:112) γ , ˜ b = ˜ a ( ω − a ) + ˜ a (cid:112) γ , (12)Equations (11) and (12) are the asymptotic expansions ofthe intersite solitons. One can continue the same calcu-lation to obtain higher-order corrections, which we willomit here as considering the first two terms is alreadysufficient for our analysis. C. Onsite solitons
For the onsite soliton, i.e., a one-excited-site discretemode, one can perform the same computations to obtainthe mode structure of the form A (0) n = (cid:26) ˜ a n = 0 , ,B (0) n = (cid:26) ˜ b e iφ b n = 0 , , (13)and the first-order correction from (6)˜ a = ˜ b ( ω − b ) + 2 ˜ b (cid:112) γ , ˜ b = ˜ a ( ω − a ) + 2 ˜ a (cid:112) γ . (14)The asymptotic expansions of the onsite solitons are thusgiven by equations (13) and (14). Likewise, higher-ordercorrections can be obtained using a similar calculation. III. STABILITY ANALYSIS
In the following, we consider six configurations, whichare combinations of the intersite and onsite discrete soli-tons with the three solutions of the dimers (8)–(10). Wewill denote them by subscripts i and o for intersite andonsite solitons, and s , at , and as for the symmetric, an-tisymmetric, and asymmetric solutions, respectively.After we find discrete solitons, their linear stabilityis then determined by solving the corresponding lin- ear eigenvalue problem. To do so, we introduce thelinearisation ansatz u n = ( A n + (cid:101) (cid:15) ( K n + iL n ) e λt ) e iωt , v n = ( B n + (cid:101) (cid:15) ( P n + iQ n ) e λt ) e iωt , | (cid:101) (cid:15) | (cid:28)
1, and substi-tute this into Eq. (1) to obtain the linearised equationsat O ( (cid:101) (cid:15) ) λK n = − ( A n − ω ) L n − (cid:15) ( L n +1 − L n + L n − ) + γP n − Q n ,λL n = (3 A n − ω ) K n + (cid:15) ( K n +1 − K n + K n − ) + γQ n + P n ,λP n = − (cid:2) Re ( B n ) + 3 Im ( B n ) − ω (cid:3) Q n − (cid:15) ( Q n +1 − Q n + Q n − ) − B n ) Im( B n ) P n − γK n − L n ,λQ n = (3 Re ( B n ) + Im ( B n ) − ω ) P n + (cid:15) ( P n +1 − P n + P n − ) + 2 Re( B n ) Im( B n ) Q n − γL n + K n , (15)which have to be solved for the eigenvalue λ and the cor-responding eigenvector [ { K n } , { L n } , { P n } , { Q n } ] T . Thesolution u n is said to be (linearly) stable when Re( λ ) ≤ λ and unstable otherwise. However,as the spectra will come in pairs, a solution is thereforestable when Re( λ ) = 0 for all λ ∈ R . A. Continuous spectrum
The spectrum of (15) will consist of continuous anddiscrete spectra (eigenvalues). To investigate the former,we consider the limit n → ±∞ , introduce the plane-waveansatz K n = ˆ ke ikn , L n = ˆ le ikn , P n = ˆ pe ikn , Q n = ˆ qe ikn , k ∈ R , and substitute the ansatz into (15) to obtain λ ˆ k ˆ l ˆ p ˆ q = ξ γ − − ξ γ − γ − ξ − γ − ξ ˆ k ˆ l ˆ p ˆ q (16)where ξ = ω − (cid:15) (cos k − λ = − (1 + γ ) − ξ ± | ξ | (cid:112) − γ . (17)The continuous spectrum is therefore given by λ ∈± [ λ − , λ − ] and λ ∈ ± [ λ , λ ] with the spectrumboundaries λ ± = ± i (cid:113) γ + ω ∓ | ω | (cid:112) γ , (18) λ ± = ± i (cid:113) γ + ( ω + 4 (cid:15) ) ∓ | ω + 4 (cid:15) | (cid:112) γ , (19)obtained from (17) by setting k = 0 and k = π in theequation. B. Discrete spectrum
Following the weak-coupling analysis as in Section II,we will as well use similar asymptotic expansions to solvethe eigenvalue problem (15) analytically, i.e., we write X = X (0) + √ (cid:15)X (1) + (cid:15)X (2) + . . . , (20)with X = λ, K n , L n , P n , Q n . We then substitute the ex-pansions into the eigenvalue problem (15).At order O (1), one will obtain the stability equation forthe dimer, which has been discussed for a general valueof γ in [25]. The expression of the eigenvalues is simple,but the expression of the corresponding eigenvectors isnot, which makes the result rather impractical to use.Therefore, here we limit ourselves to the case of small | γ | and expand (20) further as X ( j ) = X ( j, + γX ( j, + γ X ( j, + . . . ,j = 0 , , , . . . . Hence, we have two small parameters, i.e. (cid:15) and γ , that are independent of each other. The stepsof finding the eigenvalues λ ( j,k ) , j, k = 0 , , , . . . havebeen outlined in details in [23] Here, we will present theresults.
1. Intersite soliton
We have three types of intersite solitons (i.e., symmet-ric, antisymmetric, and asymmetric ones). All of themhave in general one pair of eigenvalues that bifurcate fromthe origin for small (cid:15) and two pairs of nonzero eigenval-ues. They are asymptotically given by λ i,s = √ (cid:15) (cid:0) √ ω − − γ / (2 √ ω −
1) + . . . (cid:1) + O ( (cid:15) ) , (21) λ i,at = √ (cid:15) (cid:0) √ ω + 1 + γ / (2 √ ω + 1) + . . . (cid:1) + O ( (cid:15) ) , (22) λ i,as = √ (cid:15) (cid:0) √ ω + . . . (cid:1) + O ( (cid:15) ) , (23)for the eigenvalues bifurcating from the origin and λ i,s = (cid:16) √ ω − γ ω − √ ω − + . . . (cid:17) + (cid:15) (cid:16) √ ω − − γ ω √ ω − + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (cid:16) √ ω − γ ω − √ ω − + . . . (cid:17) + (cid:15) (cid:16) √ ω − + γ ω ω − / + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (24) λ i,at = (cid:16) i √ ω + 2 + γ i ( ω +4)2 √ ω +2 + . . . (cid:17) − (cid:15) (cid:16) i √ ω + 2 + γ i ( ω +5 ω +4)8( ω +2) / + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (cid:16) i √ ω + 2 + γ i ( ω +4)2 √ ω +2 + . . . (cid:17) + (cid:15) (cid:16) i √ ω +2 + γ i (5 ω +21 ω +12)8( ω +2) / + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (25) λ i,as = (cid:16) √ − ω − γ i √ ω − + . . . (cid:17) + (cid:15) (cid:16) iω √ ω − + γ iω ( ω − / + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (cid:16) √ − ω − γ i √ ω − + . . . (cid:17) + (cid:15) (cid:16) iω √ ω − + γ iω ( ω − / + . . . (cid:17) + O (cid:0) (cid:15) / (cid:1) , (26)for the nonzero eigenvalues.
2. Onsite soliton
Similarly, we also have three types of onsite solitonswith each one generally has only one nonzero eigenvalue for small (cid:15) given asymptotically by λ o,s = (cid:18) √ ω − γ ( ω − √ ω − . . . (cid:19) + (cid:15) (cid:18) √ ω − γ ω ω − / + . . . (cid:19) + O (cid:16) (cid:15) / (cid:17) , (27) λ o,at = (cid:18) i √ ω + 2 + γ i ( ω + 4)2 √ ω + 2 + . . . (cid:19) + (cid:15) (cid:18) i √ ω + 2 + γ iω ω + 2) / + . . . (cid:19) + O (cid:16) (cid:15) / (cid:17) , (28) λ o,as = (cid:18) i (cid:112) ω − − γ i √ ω − . . . (cid:19) + (cid:15) (cid:18) iω √ ω − γ iω ( ω − / + . . . (cid:19) + O (cid:16) (cid:15) / (cid:17) . (29) IV. NUMERICAL RESULTS
We have solved the steady-state equation (3) numeri-cally using a Newton-Raphson method and analyzed thestability of the numerical solution by solving the eigen-value problem (15). Below we will compare the analyticalcalculations obtained above with the numerical results.First, we consider the discrete intersite symmetric soli-ton. We show in the top panels of Figure 1 the spectrumof the soliton as a function of the coupling constant (cid:15) for ω = 2 and γ = 0 .
5. The dynamics of the non-zero eigen-values as a function of the coupling constant are shownin the right panels of the figure, where one can see thatfirstly there is only one eigenvalue and as the coupling in-creases, one of the nonzero eigenvalues that was initiallyon the imaginary axis becomes real, too.In the bottom panels of the same figure, we plot theeigenvalues for ω large enough. Here, in the uncoupledlimit, all the three pairs of eigenvalues are on the realaxis. As the coupling increases, two pairs go back to-ward the origin, while one pair remains on the real axis (not shown here). In the continuum limit (cid:15) → ∞ , we,therefore, obtain an unstable soliton (i.e., an unstablesymmetric soliton). In both figures, we also plot the ap-proximate eigenvalues in solid (blue) curves, where goodagreement is obtained for small (cid:15) .Next, we consider antisymmetric intersite solitons.Figure 2 shows a typical distribution of the spectra inthe complex plane of the discrete solitons for one partic-ular value of ω . There is an eigenvalue bifurcating fromthe origin. For the selected value of ω we choose here, wehave the condition that the nonzero eigenvalues λ satisfy λ < λ − in the anticontinuum limit (cid:15) →
0. The collisionbetween the eigenvalues and the continuous spectrum asthe coupling increases creates complex eigenvalues. Ad-ditionally, in the continuum limit the value of ω as wellas other values of the parameter that we computed forthis type of discrete solitons yield unstable solutions.The final case for intersite solitons is the asymmetricone. Figure 3 displays a common spectrum distributionin the complex plane for a particular choice of param-eters ω and γ . Although the complex eigenvalues are -1 -0.5 0 0.5 1 Re( ) -10-50510 I m () R e () -5 0 5 Re( ) -10-50510 I m () R e () FIG. 1. The spectra of intersite symmetric soliton with ω = 2, γ = 0 . ω = 5, γ = 0 . (cid:15) = 1. Right panels present the eigenvalues as a function of the coupling constant.The solid blue curves are the asymptotic approximations presented in Subsubsection III B 1 while the dots are obtained froma numerical calculation. -4 -2 0 2 4 Re( ) -10-50510 I m () (a) R e () (b) (c) FIG. 2. The spectra of antisymmetric intersite soliton with ω = 2 and γ = 0 .
5. Panel (a) displays the spectra in the complexplane for (cid:15) = 1. Panels (b) and (c) present the eigenvalues λ as a function of the coupling constant (cid:15) . The solid blue and dottedcurves are attained from the asymptotic approximation and numerical calculation, respectively. not visible, the asymmetric intersite solitons yield un-stable solutions for the set of calculated parameters inthe continuum limit. In the anticontinuum limit, the po-sition of the discrete spectrum for the previous case ofthe antisymmetric intersite is above all the continuousspectrum, viz. Figure 2. The main interesting part isthat the unstable eigenvalues bifurcate into the complexplane, i.e., the emergence of eigenvalues with non-zeroimaginary part. For the asymmetric intersite case, theposition of the discrete spectrum is in between the con- tinuous one and the imaginary part remains zero.We also study onsite solitons shown in Figures 4–6.Unlike intersite discrete solitons that are always unsta-ble, onsite discrete solitons may be stable. In Figure 4(a),we show the spectrum as a function of the coupling. Thechoice of ω , in this case, corresponds to stable discretesolitons. However, there are regions of instability for dif-ferent parameter values of ω that may depend on γ and (cid:15) .We present the (in)stability region of the discrete solitonsin the ( (cid:15), ω )-plane for three values of the gain-loss param-eter γ in Figure 4(b). Onsite symmetric discrete solitons -5 0 5 Re( ) -10-50510 I m () (a) R e () (b) (c) FIG. 3. The spectra of asymmetric intersite soliton with ω = 5 and γ = 0 .
5. Panel (a) displays the spectra in the real plane for (cid:15) = 1. Panels (b) and (c) present the eigenvalues λ as a function of the coupling constant (cid:15) . The solid blue and dotted curvesare attained from the asymptotic approximation and numerical calculation, respectively. (a) =0=0.1=0.5 (b) FIG. 4. (a) Eigenvalues as a function of the coupling andits approximation of symmetric onsite soliton with ω = 1 . γ = 0 .
5. (b) The stability region of the onsite soliton in the( (cid:15), ω )-plane for several values of γ . The solutions are unstableabove the curves. are unstable above the curves. In general, we obtain thatthe gain-loss term in the coupling can be beneficial as itincreases the stability region of the discrete solitons.Figure 5 shows that the antisymmetric solitons are gen- erally unstable due to a quartet of complex eigenvalues,as shown in the left panels of the figure. As the instabil-ity is due to the collision of an eigenvalue with the con-tinuous spectrum, stability regions may present prior tothe collision. Panel (c) shows the region, where antisym-metric solitons are unstable between the curves. Thesesolitons are unstable in the continuum limit. Figure 6shows asymmetric solitons that are stable in the regionof their existence. Note that this soliton bifurcates fromsymmetric ones.Finally, we present in Figures 7–10 the time dynamicsof the unstable solutions shown in Figures 1–5. What weobtain is that typically there is only one dynamics, i.e. inthe form of discrete soliton destructions. One may obtainoscillating solitons or asymmetric solutions between thearms. V. CONCLUSION
We have presented a systematic method to determinethe stability of discrete solitons in a PT -symmetric cou-pler by computing the eigenvalues of the correspondinglinear eigenvalue problem using asymptotic expansions.We have compared the analytical results that we ob-tained with numerical computations, where good agree-ment is obtained. From the numerics, we have also estab-lished the mechanism of instability as well as the stabil-ity region of the discrete solitons. The application of themethod in higher dimensional PT -symmetric couplers isa natural extension of the problem that is addressed forfuture work. ACKNOWLEDGEMENT
We are grateful to Professor Hadi Susanto, Department ofMathematical Sciences, University of Essex, UK and Depart-ment of Mathematics, Khalifa University, Abu Dhabi, TheUnited Arab Emirates for his assistance and valuable com-ments in improving this paper significantly. -0.15 -0.1 -0.05 0 0.05 0.1 0.15
Re( ) -10-50510 I m () (a) (b) =0=0.1=0.5 (c) FIG. 5. Panels (a) and (b) display eigenvalues of antisymmetric onsite soliton for ω = 2, γ = 0 .
5, and (cid:15) = 1. (c) The stabilitydiagram of the discrete solitons for several values of γ . Antisymmetric solitons are unstable between the curves. -1 0 1 2 3 4 5 6 7 Re( ) -8 -15-10-5051015 I m () FIG. 6. Eigenvalues of asymmetric onsite soliton for ω = 5, γ = 0 .
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