Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media
aa r X i v : . [ n li n . PS ] A p r Attraction centers and PT -symmetric delta-functional dipolesin critical and supercritical self-focusing media Li Wang , , Boris A. Malomed , and Zhenya Yan , , ∗ Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 59978, Israel
We introduce a model based on the one-dimensional nonlinear Schr¨odinger equation (NLSE) withthe critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family ofsolitons, which are unstable in this setting against the critical or supercritical collapse, is stabilizedby pinning to an attractive defect, that may also include a parity-time ( PT )-symmetric gain-losscomponent. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full familyof the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term,the solitons’ stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolovcriterion; in the presence of the PT -balanced gain and loss, the stability is explored by means ofnumerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical)medium if the gain-loss term is absent. A stability region for the pinned solitons persists in themodel with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss componentgives rise to intricate alternations of stability and instability in the system’s parameter plane. Thosesolitons which are unstable under the action of the supercritical self-attraction are destroyed by thecollapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss termis present, unstable solitons spontaneously transform into localized breathers, while the collapsedoes not occur. The same outcome may be caused by a combination of the critical nonlinearitywith the gain and loss. Instability of the solitons is also possible when the PT -symmetric gain-lossterm is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is brieflyconsidered too, producing completely stable families of pinned localized states. I. INTRODUCTION AND THE MODEL
It is well established that non-Hermitian Hamiltoni-ans, subject to the constraint of the parity-time ( PT )symmetry, may satisfy the fundamental condition of thereality of energy spectra, hence they may be physicallyrelevant objects [1]-[5]. A single-particle PT -symmetricHamiltonian usually contains a complex potential, U ( r ) ≡ V ( r ) + iW ( r ) , (1)whose real V ( r ) and imaginary W ( r ) parts are, respec-tively, even and odd functions of coordinates, i.e., U ∗ ( r ) = U ( − r ) , (2)where the asterisk stands for the complex conjugate.Usually, the energy spectrum generated by the PT -symmetric potential remains real below a threshold valueof the amplitude of its imaginary part, W ( r ) in Eq. (1),above which the PT symmetry breaks down [6, 7]. Nev-ertheless, examples of systems with unbreakable PT sym-metry are known too [8]. In fact, the linearized version ∗ Electronic address: [email protected] of the model introduced in the present work also avoidsthe breakdown, see Eqs. (18) and (19) below.While the PT symmetry was not experimentally im-plemented in quantum systems (and it was argued thatit does not hold in the framework of the quantum fieldtheory [11]), the possibility to realize the PT symmetryin classical photonic media with mutually balanced spa-tially separated gain and loss elements was elaboratedboth theoretically [12]-[44] and experimentally [45]-[50].In addition to that, the same concept can be realized inoptomechanics [51], acoustics [52, 53], magnetism [54],and Bose-Einstein condensates [55–57].Being a linear feature, the PT symmetry often oc-curs in combination with the Kerr nonlinearity of opti-cal media in which it is implemented. The respectivemodel amounts to the nonlinear Schr¨odinger equation(NLSE) with a complex potential which is subject to con-dition (2). The NLSE gives rise to PT -symmetric soli-tons, which were a subject of intensive theoretical work,see, e.g., original papers [14], [20]-[38], [43, 44, 55, 58],and reviews [59, 60]. The existence of PT -symmetricsolitons was also experimentally demonstrated in opticallattices [48]. Although PT -symmetric media are, as amatter of fact, dissipative ones, solitons in these mediaappear in continuous families, which is typical for conser-vative models [61, 62]. It is relevant to stress that PT -symmetric solitons lose their stability at a critical valueof the strength of the gain-loss terms which is smallerthan the above-mentioned threshold value, above whichthe PT symmetry breaks down in the given system. Be-tween these values, the solitons still exist, but they areunstable [20, 21, 23, 28].A specific form of the one-dimensional (1D) PT -symmetric potential (1) is represented by a delta-functional dipole , namely, U ( x ) = − [ εδ ( x ) + iγδ ′ ( x )] (3)[ δ ′ ( x ) stands for the first-order derivative of the delta-function], which was considered in various contexts (in-cluding, naturally, a regularized version of the delta-function) [56]-[67]. In particular, an advantage offeredby potential (3) is that it allows one to construct exactsolutions for solitons pinned to the PT -symmetric de-fect embedded in a spatially uniform nonlinear medium[56, 63], although the stability of such states, which isa crucially important issue in the context of PT sym-metry, was addressed by means of numerical methods.Actually, models of this type were previously elaboratedonly for embedding media with the cubic self-focusingor defocusing nonlinearity. While it indeed representsthe most common type of the self-interaction in pho-tonics and other settings which admit the realization ofthe PT symmetry, higher-order nonlinearities also occurin optics. In particular, it was experimentally demon-strated that combinations of cubic, quintic, and septimalterms in the optical response of colloidal waveguides canbe efficiently engineered by adjusting the size and den-sity of metallic nanoparticles in the colloidal suspension[68, 69]. This technique makes it possible to create anoptical medium with a nearly pure quintic or septimalnonlinearity of either sign. On the other hand, a 1DNLSE with the quintic self-attraction is an approximatemodel of the super-Tonks-Girardeau state, i.e., a quan-tum gas of strongly interacting bosons in a highly excitedstate [70]. The attraction center in the bosonic gas canbe implemented, as usual, by means of a tightly focusedred-detuned laser beam. The consideration of these pos-sibilities is relevant, in particular, because the quinticself-focusing in 1D gives rise to the critical collapse andone-dimensional solitons of the Townes type [71], suggest-ing one to look for possibilities to stabilize such solitons,which are unstable in uniform media [72, 73].The objective of the present work is to consider soli-tons pinned to the PT -symmetric defect (3), embeddedin the 1D uniform medium with general self-focusing non-linearity, of power 2 σ + 1 with σ > σ = 1 / ,
1, and2 correspond, respectively, to the quadratic, cubic, and critical quintic self-focusing nonlinearities), the respec-tive scaled NLSE for the wave amplitude ψ taking theform of iψ z = − ψ xx − | ψ | σ ψ − [ εδ ( x ) + iγδ ′ ( x )] ψ, (4)where z and x are the propagation distance and trans-verse coordinate, in terms of the underlying optical pla-nar waveguide, the nonlinearity coefficient is normalizedto be +1 (which implies self-focusing), ψ xx represents theparaxial diffraction, and ε > z axis), while its gain-losscomponent is accounted for by coefficient γ ≥
0. Thelatter ingredient of the model can be implemented, alsoby means of the doping technique, as parallel narrow am-plifying and absorbing stripes, separated by a small dis-tance, the respective transverse sizes being on the orderof one or few wavelengths of the light beam, while theuse of the paraxial propagation equation (4) implies thatthe transverse size of the beam is much larger than thewavelength, therefore the δ -functions and its derivativeoffer a natural approximation in this case. The above-mentioned model of the super-Tonks-Girardeau gas maybe approximated by Eq. (4) with z replaced by scaledtime, t , the quintic self-attraction ( σ = 2), and γ = 0.The remaining scaling invariance of Eq. (4) allows one tofix ε ≡
1, replacing the original variables by˜ ψ = ε /σ ψ, ˜ z = ε z, ˜ x ≡ εx. (5)This rescaling does not affect γ .The analysis reported below includes the case of theattraction center in the conservative medium, i.e., γ = 0,which was previously studied only for the linear and cubicembedding media ( σ = 0 or 1). The present model with γ = 0 and higher-order nonlinearity, σ >
1, includingthe above-mentioned critical (quintic) case, σ = 2, and supercritical one, σ >
2, makes it possible to constructa full family of exact solutions for solitons pinned to theattractive defect, and predict their stability in an ana-lytical form, by means of the Vakhitov-Kolokolov (VK)criterion, which applies to a broad class of conservativemodels with self-attraction [73, 75–78] (its generalizationfor solitons in self-repulsive media, in the form of the anti-VK criterion , is known too [79]). Analytical solutions inthe form of a family of pinned solitons are also obtainedfor the PT -symmetric defect, with γ >
0, although theirstability is investigated by means of a numerically im-plemented method, with the delta-function replaced, asusual, by its regularized version, see Eq. (39) below.The rest of the paper is arranged as follows. The gen-eral theory and analytical results, including the exact so-lutions for the delta-functional defect, application of theVK criterion for the stability analysis, and also a briefsummary of results for the model with the self-repulsivenonlinearity, are presented in Section II. Numerical find-ings, including solution of the stability problem throughthe computation of eigenvalues for small perturbations,and simulations of the evolution of unstable solitons, arereported in Section III. The paper is concluded by Sec-tion IV.
II. GENERAL THEORY AND ANALYTICALRESULTSA. Stationary states
Stationary localized states with propagation constant k > ψ ( x, z ) = e ikz φ ( x ) , (6)with complex function φ ( x ) satisfying the ordinary dif-ferential equation with the singular PT -symmetric po-tential, − kφ + 12 d φdx + | φ | σ φ + [ δ ( x ) + iγδ ′ ( x )] φ = 0 (7)[recall ε ≡ φ ( x ) in Eq. (7) can be rewritten as [30] φ ( x ) = ϕ ( x ) exp (cid:20) i Z x −∞ v ( x ′ ) dx ′ (cid:21) with real amplitude ϕ ( x ) and local wavenumber v ( x ) sat-isfying coupled nonlinear ordinary differential equations: ϕ ′′ xx + 2 ϕ σ +1 + (cid:2) δ ( x ) − v − k (cid:3) ϕ, ( ϕ v ) ′ x = − γδ ′ ϕ . (8)Stability of the stationary states is explored by consider-ing their perturbed version, ψ ( x, z ) = n φ ( x ) + η h F ( x )e iωz + G ∗ ( x )e − iω ∗ z io e ikz , (9)where η is an amplitude of the infinitesimal perturbationwith a (generally, complex) eigenvalue, ω , and compo-nents F ( x ) and G ( x ), which satisfy the matrix equation,produced by the substitution of ansatz (9) and lineariza-tion: ˆ L ˆ L − ˆ L ∗ − ˆ L ∗ ! (cid:18) F ( x ) G ( x ) (cid:19) = ω (cid:18) F ( x ) G ( x ) (cid:19) . (10) Here ∗ stands for the complex conjugate, and the con-stituent operators areˆ L = 12 ∂ x + δ ( x ) + iγδ ′ ( x ) + ( σ + 1) | φ | σ − k, ˆ L = σ | φ | σ − φ . (11)The generic stationary states are stable if all the corre-sponding eigenvalues ω have zero imaginary parts (theabove-mentioned Townes’ solitons, that are actually de-generate states, feature a specific subexponentially grow-ing instability, which is accounted for by additional zeroeigenvalues). B. Exact solutions and the Vakhitov-Kolokolovstability criterion
In the conservative version of the model, with γ = 0, anexact real solution to Eq. (7), which may be consideredas a soliton pinned to the attractive defect, can be readilyobtained: φ γ =0 ( x ) = np k ( σ + 1) sech h σ √ k ( | x | + ξ ) io /σ , (12)with real parameter ξ > (cid:16) σ √ k ξ (cid:17) = 1 √ k , (13)i.e., ξ = (cid:16) σ √ k (cid:17) − ln √ k + 1 √ k − ! . (14)The squared amplitude of the pinned soliton is A ( σ, k ) ≡ φ γ =0 ( x = 0) = [(1 + σ ) ( k − / /σ . (15)It follows from Eq. (15) that the solutions exist with thepropagation constant exceeding a cutoff value, k > k cutoff ≡ / , (16)as the amplitude vanishes at k → / PT -symmetric so-lutions to Eq. (7) can also be found in the presence ofthe gain-loss term, i.e., for γ >
0, similar to how it wasdone in Ref. [63] for the cubic nonlinearity ( σ = 1): φ γ> ( x ) = φ γ =0 ( x ) exp (cid:2) − i sgn( x ) tan − ( γ ) (cid:3) ≡ φ γ =0 ( x ) 1 − iγ sgn( x ) p γ , (17)where φ γ =0 ( x ) is the solution for γ = 0, as given byEqs. (12) and (13). The jump in the phase in expres-sion (17) is produced by the singular term − iγδ ′ ( x ) incomplex potential (3).Note that the exact localized solution for the linearizedsystem, with nonlinear term | φ | σ φ dropped in Eq. (7), isgiven by Eqs. (12) and (17) with ξ → ∞ . This solutionexists for the single value of the propagation constant, k = 1 / φ (linear) k =1 / ,γ> ( x ) = φ exp (cid:2) −| x | − i sgn( x ) tan − ( γ ) (cid:3) , (18)where φ is an arbitrary constant. In addition to thislocalized mode existing at the single positive value of k ,the linearized system supports a continuous spectrum ofdelocalized modes with k < φ (linear) k< ,γ> ( x ) = φ sin h √− k | x | − tan − (cid:16) √− k (cid:17)i × exp (cid:2) − i sgn( x ) tan − ( γ ) (cid:3) . (19)Solutions (18) and (19) exist at all values of the gain-lossstrength γ , i.e., unlike the above-mentioned generic situ-ation, in the present linearized model the PT symmetrydoes not suffer the breakdown with the increase of γ .A fundamental characteristic of the family of localizedstates is its norm (alias the integral power, in terms ofspatial solitons in the optical waveguide), which is givenby the same expression for the solutions with γ = 0 and γ > N ( σ, k ) ≡ Z + ∞−∞ | φ γ =0 ( x ) | dx = Z + ∞−∞ | φ γ> ( x ) | dx = √ σ + 1) /σ σ k /σ − / × Z ∞ " sech y + 12 ln √ k + 1 √ k − !! /σ dy, (20)In the limit of k → /
2, Eq. (20) demonstrates that thenorm vanishes as N ( σ, k ) ≈ h (1 + σ ) (cid:16) √ k − (cid:17)i /σ . (21)According to the VK criterion [73, 75–78], the neces-sary stability condition in the conservative model, with γ = 0, is ∂N/∂k >
0. In particular, in the case of theusual Kerr nonlinearity ( σ = 1), Eq. (20) takes a verysimple form [63], N ( σ = 1 , k ) = 2 (cid:16) √ k − (cid:17) , (22)which implies an evident result, that the pinned solitonsare VK-stable modes in the cubic medium. Similar tothis result, the norm of the pinned solitons diverges, at k → ∞ , for all subcritical nonlinearities, with σ <
2, as N ( σ, k ) ≈ r π σ + 1) /σ σ Γ (1 /σ )Γ (1 / /σ ) k /σ − / , (23) FIG. 1: (Color online). (a) Norm N ( σ, k ) versus parameter k at σ = 2. The solid green line represents the exact so-lution corresponding to Eq. (24), while the dotted red lineis produced by the numerical solution of Eq. (12) with thedelta-function regularized as per Eq. (39). The applicationof the VK criterion to the N ( σ = 2 , k ) dependence suggeststhat the entire family of the 1D Townes’ solitons is completelystabilized by pinning to the attractive defect. (b) The approx-imate dependence N ( α, k ) given by Eq. (26) with σ = 2 . where Γ is the Gamma-function.More interesting is the critical case of σ = 2 (the quin-tic nonlinearity), which was not considered previously inthe combination with the attractive defect. In this case,Eq. (20) yields N ( σ = 2 , k ) = r π − − s √ k + 1 √ k − , (24)see Fig. 1(a). The uniform medium [with ε = 0, in termsof Eq. (3)] formally corresponds to k → ∞ , which impliesthe degeneracy of the corresponding family of the 1DTownes’ solitons: their norm takes a single value, N Townes = p / π/ ≈ . , (25)which does not depend on the soliton’s propagationconstant. In terms of the VK criterion, the constantnorm formally corresponds to the neutral stability, with ∂N/∂k = 0. However, it is well known that the Townes’solitons are always unstable against the spontaneous on-set of the critical collapse , although their instability issubexponential. as mentioned above [73] (for this rea-son, it is not detected by the VK criterion).In the case of γ = 0 and σ = 2, Eq. (24) clearly demon-strates ∂N/∂k > k > (see Fig. 1),suggesting that the 1D Townes’ solitons, which are com-pletely unstable in the free space, are completely stabilized by the attractive center, irrespective of its strength. Thisconjecture was corroborated by the full stability analysisbased on the numerical solution of eigenvalue equation(10), as well as by direct simulations of the perturbedevolution of the pinned solitons.For the slightly supercritical case, with 2 < σ ≪ N ( σ, k ) dependence derived from Eq. (20) can beapproximated by N ( σ, k ) ≈ r
32 (2 k ) (2 − σ ) / π − − s √ k + 1 √ k − . (26)This dependence satisfies the VK criterion at12 < k < k cr ≈ π ( σ − , (27)and does not satisfy it at k > k cr . The approximatedependence (26) attains a maximum at k = k cr , which,in the lowest approximation, turns out to be N max = N Townes , see Eq. (25). Accordingly, the pinned solitonsare expected to be stable in this supercritical case (with γ = 0, i.e., in the absence of the gain and loss) in theregion of 1 / < k < k cr , and to suffer the onset of the supercritical collapse (spontaneous blowup) at k > k cr .In particular, for σ = 2 . σ = 2, butstill relatively close), the dependence given by Eq. (26)is plotted in Fig. 1(b), where k cr ≈ .
75, while approxi-mation (27) yields k cr ≈ .
07 for σ = 2 . ∂N max /∂σ slowly diverges at the border of the su-percritical case, i.e., at σ → ∂N max ∂σ ≈ − r π (cid:18) σ − (cid:19) . (28)This result agrees with the fact that N max = ∞ at σ < N ( σ, k ), numericallycalculated as per Eq. (20), for different critical and su-percritical values of the nonlinearity power, viz ., σ =2 , . , . , . ,
3. While, as said above, in the criticalcase of σ = 2 condition ∂N ( σ, k ) /∂k > k > , at each supercritical value, σ >
2, the N ( σ, k ) de-pendence indeed features the critical point, k cr , at whichthe slope, ∂N/∂k , changes its sign, and the norm attainsits largest value, N max = N ( k cr ) [note that, as followsfrom Eq. (20), N ( σ, k → ∞ ) ∼ (2 k ) (1 /σ − / →
0, forany σ > γ = 0, the pinned solitons are stable in theregion of 1 / < k < k cr , i.e., 0 < N < N max , is corrob-orated by the calculation of the stability eigenvalues viaEq. (10), as well as by direct simulations of perturbedevolutions of the solitons. In particular, the VK-unstablesolitons, existing at k > k cr , are indeed destructed by thespontaneous blowup, see Fig. 4(a) below. Thus, the at-tractive defect provides for the partial stabilization of thesolitons in the case of the supercritical nonlinearity, forall values of σ >
2, while all solitons are strongly unsta-ble in this case in the absence of the defect. Naturally,
FIG. 2: (Color online). (a) The dependence of norm N ( σ, k )on propagation constant k , numerically computed as perEq. (20), for the critical case, with σ = 2, and the supercriti-cal one, with σ = 2 . , . , . ,
3. (b) The maximum normof the pinned solitons, attained at the stability-boundary, k = k cr , at which ∂N/∂k = 0, versus σ (in the supercriticalinterval, 2 < σ ≤ γ = 0, the pinned solitonsare stable in the region of 1 / < k < k cr , which correspondsto 0 < N < N max . Fig. 2(a) demonstrates that the stability region (in bothforms of 1 / < k < k cr and 0 < N < N max ), maintainedby the interplay of the supercritical self-attractive non-linearity and attraction center, shrinks with the increaseof the nonlinearity power, σ . Nevertheless, the stabilityregion persists even at large values of σ . Indeed, if σ istreated as a large parameter, while k − as a small one,Eq. (20) amounts to N ( σ ≫ , k ) ≈ (2 k ) − / ( √ k − /σ (29)[cf. Eq. (21)], which yields k cr − ≈ /σ, lim σ →∞ N max = 1 . (30)The limit value of N max = 1, given by Eq. (30), agreeswith the numerical results displayed in Figs. 2(b) and3(d).Finally, to better understand the physical meaning ofthe PT -symmetric stationary state, φ γ> ( x ), given byEq. (17), it is relevant look at its power flux (the Poyntingvector), S ( x ) = Im (cid:0) φ ∗ ( x ) ddx φ ( x ) (cid:1) . The substitution ofEqs. (7) and (17) yields a singular expression, S ( x ) = − γφ γ =0 ( x ) δ ( x ) , (31)A regular characteristic of the transport in the PT -symmetric pattern is provided by the globally normalizedflux, ˆ S ≡ N − ( σ, k ) Z + ∞−∞ S ( x ) dx = − γA ( σ, k ) /N ( σ, k ) < , (32)where A ( σ, k ) is the amplitude of the pinned soliton,given by Eq. (15). An essential feature of this expres-sion is its sign, which, in comparison with Eq. (4), clearlydemonstrates that the power flows, as it should, from thesource (gain) towards the sink (loss). C. Exact solutions with the self-defocusingnonlinearity
It is relevant to briefly consider the model with the self-repulsive uniform nonlinearity, i.e., with Eq. (4) replacedby iψ z = − ψ xx + | ψ | σ ψ − [ εδ ( x ) + iγδ ′ ( x )] ψ. (33)In this case, the exact solution for γ = 0 is readily foundas φ (defoc) γ =0 ( x ) = np k ( σ + 1) / sinh h σ √ k ( | x | + ξ ) io /σ , (34)with ξ = 12 σ √ k ln √ k − √ k ! , (35)and the squared amplitude B ( σ, k ) ≡ h φ (defoc) γ =0 ( x = 0) i = [(1 + σ ) (1 / − k )] /σ . (36)As it follows from Eq. (36), the existence region for thelocalized modes pinned to the attractive defect embeddedin the defocusing medium is k < k cutoff ≡ /
2, which isexactly opposite to that in the case of self-focusing, cf.Eq. (16).In the presence of the gain-loss dipole, which intro-duces the PT symmetry, i.e., γ >
0, the exact solutionis generated from the one for γ = 0 by the same relation(17) which is relevant in the case of self-focusing. Asconcerns the norm, it takes a simple form in the cubicdefocusing medium ( σ = 1), N (defoc) ( σ = 1 , k ) = 2 (cid:16) − √ k (cid:17) (37)[cf. Eq. (22)]. On the other hand, in the limit case oflarge σ , the result is N (defoc) ( σ ≫ , k ) ≈ (2 k ) − / (cid:16) − √ k (cid:17) /σ , (38)cf. Eq. (29). It can be checked that, as well as it isobvious in Eqs. (37) and (38), the N ( σ, k ) dependencecorresponding to the defocusing nonlinearity satisfies theabove-mentioned anti-VK criterion, ∂N/∂k < σ . Accordingly, all the pinned modes arestable, both at γ = 0 and γ >
0, similar to the case ofthe self-defocusing cubic nonlinearity [63].
FIG. 3: (Color online). The norm of stable solitons (blue cir-cles) and unstable solitons (red stars) produced by the numer-ical solution of Eq. (12) with the regularized delta-function(39), for different values of the nonlinearity degree σ and gain-loss coefficient γ : (a) γ = 0, (b) γ = 0 .
1, (c) γ = 0 .
2. Thestability is identified through the numerical solution of eigen-value problem (10), using the same regularization. Note thedifference in the scales of the horizontal axes between panels(a) and (b,c). The top and bottom curves in panel (d) displaystability boundaries in the plane of ( σ , N ) for γ = 0 (dash-dotted blue line), 0 . . γ = 0 . γ = 0 . III. NUMERICAL SOLUTIONS AND THEIRSTABILITYA. Stationary states and small perturbations
While the model with the ideal delta-functional de-fect admits exact solutions for all values of σ and γ , asgiven by Eqs. (12), (14), and (17), their stability in the PT -symmetric model with γ > δ ( x ) → e δ ( x ) = 1 √ πa exp (cid:18) − x a (cid:19) , (39)with width a much smaller than a characteristic size ofthe pinned mode. This condition is secured by fixing a =0 .
03, which is adopted here. It was checked that takingsmaller a does not affect the results in any conspicuousway.Stationary equation (12) with δ ( x ) and δ ′ ( x ) replacedby e δ ( x ) and e δ ′ ( x ) can be efficiently solved by means ofthe Petviashvili iteration method [80], using the above-mentioned analytical solution, valid in the limit of theideal delta-function, as the initial guess. Then, Eqs.(10) and (11), regularized as per substitution (39), arenumerically solved to predict the stability boundary.Finally, perturbed evolution of the pinned modes wassimulated by means of fourth-order Runge-Kutta time-stepping scheme. A known necessary and sufficient con-dition for the stability of the direct simulations amountsto an inequality imposed on the time and space steps,∆ t/ (∆ x ) ≤ √ /π [78]. As a result, it is concludedthat the real numerically generated solutions for γ = 0,as well as real parts of the complex solutions for γ > σ = 2 and 2 . , . , . ,
3, in the absence of the gainand loss ( γ = 0). It is seen that the stability precisely fol-lows the VK criterion, and, in the exact agreement withthe analytical prediction, the solitons are completely sta-ble for σ = 2, while the stability region shrinks with theincrease of σ in the supercritical case. The respective ex-istence and stability boundary, N max ( σ ), which is shownby the top curve in Fig. 3(d), is identical to its analyti-cally predicted counterpart, cf. Fig. 2(b).As it may be expected, the stability pattern becomesmore complex in the presence of the gain-and-loss term,i.e., at γ >
0, see Figs. 3(b) and (c) for γ = 0 . . γ = 0 .
2, Fig. 3(c) demonstrates strong shrinkage of thestability region, in comparison with the case of γ = 0;in particular, the critical nonlinearity, with σ = 2, keepsthe pinned solitons stable only at N < N max ≈ . γ = 0, the same nonlinearity maintains thestability in the entire existence region of the solitons, upto N = N Townes , see Eq. (25). The numerically foundstability boundaries for γ = 0 . .
2, in the form of N max ( σ ), are displayed by two bottom curves (which arevirtually identical to each other) in Fig. 3(d).Figure 3(b) features an intricate structure of the sta-bility pattern, with multiple stability intervals , at smallervalues of the gain-loss coefficient, such as γ = 0 .
1. Asseen in Fig. 3(d), the lowest stability boundary for γ = 0 . γ = 0 .
2. However,the actual number of such boundaries observed in 3(b)is, at least, five. Probably, many more boundaries maybe revealed by extremely precise numerical data, and itseems plausible that the exact structure of the stabilityislands, embedded in the instability area, may be fractal ,as suggested, in particular, by the fractal alternation ofregions of elastic and inelastic collisions between solitonsin some nonintegrable conservative models [81–83]. Thefractal structure, if any, may strongly depend in values of γ and σ . A rigorous analysis of this challenging problemis beyond the scope of the present work.In the presence of the gain and loss, the instability oc-curs even in the subcritical case, i.e., at σ <
2. Whilein Figs. 3(b, c) the subcritical branches, correspondingto σ = 1 .
8, are completely stable in the displayed rangeof k , they develop weak instability at essentially largervalues of k , where the VK derivative, ∂N/∂k ∼ k /σ − / [see Eq. (23)] is very small, hence the the effect of thesubcriticality is insufficient to suppress the instability in-duced by the possibility of spontaneous breaking of the PT symmetry. In particular, for σ = 1 . γ = 0 .
2, thepinned soliton is unstable at k = 20, see Figs. 4(e2,e3)below. B. Direct simulations
The predictions for the (in)stability of the pinned soli-tons, produced by the application of the VK (or anti-VK)criterion to the analytical solutions in the case of γ = 0,and by the numerical solution of Eq. (10), subject toregularization (39), for the numerically constructed solu-tions in the case of γ ≥
0, were verified by direct simu-lations of perturbed evolutions of the pinned solitons. Ithas been concluded that all the modes which were pre-dicted to be stable are stable indeed (not shown here indetail, as simulations of the stable evolution do not revealnew features of the dynamics). The predicted instabilityis also corroborated by the direct simulations, as shownin Fig. 4. In particular, in the case of γ = 0 the insta-bility, displayed in panel (a2), leads to sudden blowup ofthe solution, which is a manifestation of the supercriticalcollapse. The same happens too, but faster, as a result ofthe interplay of the supercritical self-attraction and gain-loss ( γ = 0 .
2) term, see panel (b2). Panel (c2) demon-strates that, in the case of large k , when the instabilityrelated to the supercriticality is weak, as the derivative, | ∂N/∂k | , which determines the VK criterion, is small, seeFigs. 1(b) and 2(a), the interplay of the weak instabilityand gain-loss terms leads not to a blowup, but to the for-mation of a robust localized breather with an amplitudewhich is essentially smaller than in the original unstablesoliton.Recall that, in the absence of the gain-loss term, all thepinned solitons maintained by the critical nonlinearity,with σ = 2, are stable. The addition of the sufficientlystrong gain and loss [in particular, with γ = 0 .
2, as shownin Fig. 4(d2)] may destabilize the solitons in this case.It is observed in the figure that the initial stage of theinstability-driven evolution is chaotic, eventually relaxingto a robust breather.Finally, as it was mentioned above, the presence of thegain-loss term may induce instability in the subcriticalcase, σ <
2. Figure 4(e2) demonstrates an example ofthis instability for σ = 1 . γ = 0 .
2, provided thatthe soliton was created with a large value of the propaga-tion constant (in particular, k = 20 in this figure), where,as said above, the stabilizing effect of the subcritical in-stability is very weak. In this case too, the collapse doesnot occur (as there is nothing to drive it in the subcriticalcase), the instability leading to spontaneous transforma-tion of the unstable soliton into a robust breather withan essentially smaller amplitude.As seen from Fig. 3, the stability region of the PT -symmetric solitons shrinks with the increase of both thenonlinearity power, σ , and gain-loss coefficient, γ . Sys-tematically collecting numerical data, we conclude that,at a fixed value of σ , all the solitons become unstable at γ > γ max ( σ ), and, at a fixed value of γ , the same hap-pens at σ > σ max ( γ ). As concerns value N max of thenorm at which the stability ends, it essentially dependson σ and very weakly depends on γ , as shown by the sta-bility boundaries in the ( σ, N ) plane for γ = 0 . . γ . For a fixed value of the norm, N < N max ,the dependence of the stability border of γ is very weaktoo. IV. CONCLUSIONS AND DISCUSSIONS
The objective of this work is to introduce settings inwhich intrinsically unstable solitons in 1D media mod-elled by the NLSE with the critical ( σ = 2) and super-critical ( σ >
2) self-attractive nonlinearity may be stabi-lized by pinning to an attractive defect, including its PT -symmetric version with strength γ of the gain-loss com- ponent. The settings represent a planar nonlinear-opticalwaveguide, or a super-Tonks-Girardeau gas. A remark-able fact is that full families of the pinned solitons canbe found in the analytical form for the delta-functionaldefect, and, in the absence of the gain-loss term, theirstability too admits analytical investigation, by means ofthe VK criterion. In particular, a stability interval existsfor arbitrarily high values of the nonlinearity power σ .In the presence of the gain and loss, the stability of the PT -symmetric soliton families and the evolution of un-stable solitons were investigated by means of numericalmethods, revealing a nontrivial stability area in the ( σ, γ )plane. For relatively small values of γ , the structure ofthe stability area is intricate, featuring multiple stabilityboundaries (which may presumably form a fractal). Ifthe instability driven by the supercritical nonlinearity isstrong enough, unstable solitons are destroyed by the col-lapse, both in the absence and presence of the gain-lossterm. On the other hand, if the supercritical instabilityis weak, it does not lead, in the combination with thebalanced gain and loss ( γ > Acknowledgments
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(12), (14), and (17), and their counterparts obtainedfrom the numerical solution of Eq. (7), subject to regular-ization (39), labeled “n”, whose real and imaginary parts areshown by blue solid and red dashed lines, respectively. Asmall difference between the analytically predicted and nu-merically found imaginary parts of the solutions is a resultof the regularization, as the analytical version is discontinu-ous, see Eq. (17). The simulated evolution of the numericalsolutions from (a1), (b1), (c1), (d1), and (e1) is displayed,severally, in panels (a2), (b2), (c2), (d2), and (e2), and theirstability spectra are depicted in (a3), (b3), (c3), (d3), and(e3), respectively. The parameters are: γ = 0 , σ = 2 . , k = 8in (a1, a2, a3), γ = 0 . , σ = 2 . , k = 8 in (b1, b2, b3), γ =0 . , σ = 2 . , k = 20 in (c1, c2, c3), γ = 0 . , σ = 2 , k = 50 in(d1, d2, d3), and γ = 0 . , σ = 1 . , k, k