PT -symmetric couplers with competing cubic-quintic nonlinearities

Transcription

1 PT -symmetric couplers with competing cubic-quintic nonlinearities Gennadiy Burlak, Salomon Garcia-Paredes, and Boris A. Malomed, 3 ) Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Av. Universidad, Cuernavaca, Morelos 6, México. ) Department of Physical Electronics, School of Electric Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel. 3) Laboratory of Nonlinear-Optical Informatics, ITMO University, St.Petersburg 97, Russia. We introduce a one-dimensional model of the parity-time (PT )-symmetric coupler, with mutually balanced linear gain and loss acting in the two cores, and nonlinearity represented by the combination of self-focusing cubic and defocusing quintic terms in each core. The system may be realied in optical waveguides, in the spatial and temporal domains alike. Stationary solutions for PT -symmetric solitons in the systems are tantamount to their counterparts in the ordinary coupler with the cubic-quintic nonlinearity, where the spontaneous symmetry breaking of solitons is accounted for by bifurcation loops. A novel problem is stability of the PT -symmetric solitons, which is affected by the competition of the PT symmetry, linear coupling, cubic self-focusing, and quintic defocusing. As a result, the solitons become unstable against symmetry breaking with the increase of the energy (alias integral power, in terms of the spatial-domain realiation), and they retrieve the stability at still larger energies. Above a certain value of the strength of the quintic selfdefocusing, the PT symmetry of the solitons becomes unbreakable. In the same system, PT -antisymmetric solitons are entirely unstable. We identify basic scenarios of the evolution of unstable solitons, which may lead to generation of additional ones, while stronger instability creates expanding quasi-turbulent patterns with limited amplitudes. Collisions between stable solitons are demonstrated to be quasi-elastic. Losses are ubiquitous in physical media. In most cases, losses are considered as a detrimental factor, which must be compensated by a properly introduced gain or feeding field, in internally and externally driven systems, respectively. However, losses may play a positive role too, helping to create and/or stabilie modes which otherwise would not exist, or would be unstable. Examples are provided by nonlinear couplers (dual-core waveguides) in optics, where the dominant loss in one core secures the stability of two-component solitons supported by the linear gain applied to the mate core. More recently, it has been demonstrated that the special case of exactly balanced gain and loss in two cores of the coupler provides a realiation of the general concept of the paritytime (PT ) symmetry. This symmetry defines a class of systems with spatially separated and mutually symmetric gain and loss elements with equal strengths; although the respective Hamiltonian is evidently non-hermitian, it gives rise to a purely real (physically meaningful) spectrum, provided that the gain-loss strength does not exceed a certain critical value, beyond which complex eigenvalues emerge, signaling spontaneous breakup of the PT symmetry. Furthermore, the interplay of the PT symmetry, which is usually represented by linear terms in the underlying models, with the generic Kerr (self-focusing cubic) nonlinearity readily gives rise to families of PT -symmetric solitons. The consideration of the PT -symmetric nonlinear coupler makes it possible to find stable PT -symmetric solitons and their stability boundaries in exact analytical forms. The present work makes a next step in the studies of this topic, by considering the interplay of the PT -symmetry in the coupler with the intra-core nonlinearity combining self-focusing cubic and defocusing quintic terms. This setting can be implemented in available optical materials. We find that the defocusing term provides restoration of the broken PT symmetry above a certain level of the soliton s energy (or integral power, depending on the physical interpretation of the system). Eventually, if the strength of the defocusing quintic term is large enough, the PT -symmetry of the solitons becomes unbreakable, i.e., all the symmetric solitons are stable. On the other hand, PT -antisymmetric solitons are always unstable in this system. Basic scenarios of the evolution of unstable solitons are investigated too, by means of systematic simulations. I. INTRODUCTION Dynamical models based on nonlinear partial differential equations, which give rise to solitary modes, are naturally divided into two vast classes, conservative and dissipative. Accordingly, they give rise to solutions of different types, namely, solitary waves (which are often named solitons in physics literature, even if the underlying conservative model is not integrable), and dissipative pulses (which are often called dissipative solitons ). These classes feature the principal difference in the structure of the solution sets: conservative systems give rise to continuous families of solitons, parameteried by their

2 intrinsic frequency or propagation constant, which may be completely or partly stable, while stable dissipative pulses normally exist as isolated attractors, which are selected by the balance between gain and losses (unstable dissipative solitons exist as isolated solutions too, playing the role of separatrices, i.e., borders between different dynamical scenarios) 3. A specific class of systems was identified at the interface between conservative and dissipative ones, with spatially separated and mutually balanced gain and loss elements. Such systems realie the parity-time (PT ) symmetry, which was originally introduced in the quantum theory for non-hermitian Hamiltonians 4. PT -symmetric Hamiltonians produce purely real spectra up to a certain critical value of the gain-loss strength. Above the critical point, the spectrum becomes complex, signaling spontaneous breakup of the PT symmetry, and onset of instability in the system. Experimental realiation of the PT symmetry was suggested by the fact that the propagation equation for optical beams in the paraxial approximation has essentially the same form as the quantum-mechanical Schrödinger equation, which makes it possible to emulate the evolution of the wave function of a quantum particle by the transmission of an optical beam. The implementation of the PT symmetry in optics was proposed in Ref. 5 and experimentally demonstrated in Ref. 6, using symmetrically placed elements carrying equal amounts of optical gain and loss. The ubiquitous presence of the Kerr effect in optical media suggests to consider the interplay of the PT - symmetry with the cubic self-focusing nonlinearity, which naturally leads to the prediction of PT -symmetric solitons. They were theoretically studied in detail in terms of optics 7 and exciton-polariton condensates 8, and experimentally demonstrated in a nonlinear photonic lattice. Although systems which gives rise to the PT -symmetric solitons are, strictly speaking, dissipative ones, these solitons exist in continuous families, resembling solitons in conservative models, rather than as isolated solutions, which is typical to dissipative solitons. Recent reviews of the topic of PT -symmetric solitons are provided by Ref. 9. The most important issue in studies of such solitons is their stability, as the exact balance between the amplification and dissipation is fragile. A natural setting for the implementation of the PT symmetry in optics is offered by dual-core waveguides (couplers), with the linear gain and loss carried by different cores. If the Kerr nonlinearity is present in the cores, this system can readily give rise to solitons. Originally, dissipative solitons were studied in the dual-core waveguides with imbalanced gain and loss (normally, the loss must be stronger, to guarantee the stability of the ero background ), using a variety of models based on coupled systems of nonlinear Schrödinger (NLS) equations which include the gain and loss terms, see a review in Ref. 3. Recently, a similar scheme was elaborated for the application of gain and stabiliation to solitons in plasmonics 4, as well as for the creation of stable twodimensional dissipative solitons and vortices in dual laser cavities 5. PT -symmetric couplers with the balanced gain and loss in their Kerr-nonlinear cores are modeled by the following coupled NLS equations 6-9 (see also extensions of the model elaborated in Refs. -bilayer): iu + (/)u tt + u u + v = iγu, iv + (/)v tt + v v + u = iγu, () where u and v are amplitudes of the electromagnetic waves in the two cores, is the propagation distance, t is the local time or transverse coordinate in the temporal and spatial renditions of the coupler (the cores are, respectively, fibers or planar waveguides in these cases), and γ > is the gain-loss parameter. Further, coefficients in front of the second-derivative terms, which represent the temporal dispersion or paraxial diffraction, the nonlinearity factor, and the coupling constant are all scaled here to be. PT -symmetric solitons and their stability boundaries, generated by Eq. (), have been found in an exact analytical form 6,9. In fact, any exact solution, W (, t), of the usual NLS equation, iw + (/)W tt + W W =, gives rise to exact PT - symmetric (+) and antisymmetric ( ) solutions of system () with γ, v (, t) = (iγ ± ) γ u (, t) ( = exp ±i ) γ W (, t). () In the limit of γ =, solutions () coalesce into a single one, v = iu = V (, t) [taking γ > leads to instability of the ero solution of Eq. (), which makes the model irrelevant]. In particular, symmetric (+) solitons with propagation constant K (and respective amplitude K), V = ( Kt ) K exp (ik) sech, are stable against the spontaneous symmetry breaking at K K (/3) γ, (3) while the antisymmetric solitons, with sign in Eq. (), are completely unstable 6,9 (in the conservative system with γ =, the stability limit for the symmetric solitons, K = /3, was found earlier 3 ). At K > K, unstable solitons blow up in the PT -symmetric system (while in the case of γ =, they are replaced by stable asymmetric solitons 4, at γ > asymmetric solitons do not exist, as they cannot maintain the balance between the gain and loss). The destabiliation of the symmetric solitons with the increase of K is caused by the competition of the strengthening Kerr self-focusing, which tends to concentrate the energy in a single core, and the linear coupling, which drives the coupler towards even distribution of the energy between the cores. The stability region is additionally reduced by the presence of the gain and loss,

3 3 as is clearly shown by Eq. (3). On the other hand, many optical materials feature competing nonlinearities typically, a combination of the cubic self-focusing and quintic defocusing terms 5. In particular, it was recently demonstrated that the cubic-quintic (CQ) nonlinearity with virtually any set of the cubic and quintic coefficients can be engineered in colloids of suspended metallic nanoparticles 6. The conservative couplers with the CQ nonlinearity in their cores were considered, in the one- and two-dimensional (D and D) geometries, in Refs. 7 and 8, respectively, where it was found that the competition of the linear coupling and quintic defocusing with the cubic self-focusing leads to stability restoration for symmetric solitons (and for symmetric solitary vortices in D), which is accounted for by bifurcation loops combining the symmetry breaking and restoration. It is relevant to mention that the vortex modes considered in Ref. 8 represent spatiotemporal vortices (experimental creation of such modes in 3D was recently reported in Ref. 9 ). These results suggest to consider the PT -symmetric coupler with the CQ nonlinearity, which can be realied experimentally in optics, providing for the integration of several physically interesting factors: linear coupling, balanced gain and loss in two cores, self-focusing, and self-defocusing. The stability of D PT -symmetric and antisymmetric solitons in this system is the subject of the present work. In the D geometry, the destabiliation of the corresponding symmetric spatiotemporal solitons and collisions between moving stable ones were considered in Ref. 3. However, the most interesting issue of the symmetry restoration and formation of the bifurcation loops was not addressed previously (in fact, the quintic defocusing was introduced in Ref. 3 not for that purpose, but rather for the suppression of the collapse driven by the cubic self-focusing in D). In this work, we focus on the destabiliation and restabiliation of PT -symmetric solitons in D. The model, its solutions, and the framework for the analysis of the stability of the solitons in it are introduced in Section II. In that section, we also consider a related but simpler problem of the stability of PT -symmetric continuous waves (CWs) in the same system. Numerical results, which explicitly demonstrate the breakup and restoration of the stability of the symmetric solitons, including identification of an area of the unbreakable PT symmetry, are summaried in Section III. In the same section, we identify basic scenarios of the evolution of unstable solitons and, on the other hand, we demonstrated that collisions between stable solitons are quasi-elastic. The paper is concluded by Section IV. II. THE MODEL AND SOLUTIONS The PT -symmetric coupler with the CQ nonlinearity in its cores is based on the following system of coupled NLS equations, cf. Eq. () and the system with γ = introduced in Ref. 7 : iu + (/)u tt + u u g u 4 u iγu + v =, iv + (/)v tt + v v g v 4 v + iγv + u =, (4) where g > is the strength of the quintic self-defocusing. Thus, the model is controlled by two irreducible positive parameters, g and γ. For solution families, the propagation constant, K, will play the role of the third control parameter. The total energy of the fields governed by Eq. (4) (or the total power, in terms of the spatial-domain system) is E = + [ u (, t) + v (, t) ] dt E u + E v. (5) An obvious corollary of Eq. equation, (4) is the energy-balance de d = γ (E u E v ), (6) which demonstrates that solely symmetric or antisymmetric solitons, with E u = E v, may represent stationary modes. Any solution of the standard NLS equation with the CQ nonlinearity, iw + (/)W tt + W W g W 4 W =, (7) gives rise to PT -symmetric and antisymmetric solutions of Eq. (4) via Eq. (). Of practical interest is the wellknown soliton solution of Eq. (7) 3, K W sol = + (6/3)gK cosh ( ) e ik, Kt (8) where the propagation constant takes values in the interval of < K < K max 3/ (6g). (9) In the limit of K = K max, soliton solution (8) carries over into the CW one, with a constant amplitude: W CW (K = K max ) = (/) 3/g exp (ik max ). () The total energy (5) of the two-component soliton, built as per Eqs. (8) and (), is ( ) gk E sol = g ln. () 3 4 gk The divergence of E sol at K K max is explained by the divergence of the soliton s width at the constant amplitude, which is identical to W CW (K = K max ) in Eq. ().

4 4 The main objective of the present work is to identify stability limits of the PT -symmetric and antisymmetric solitons within the framework of system (4). This is done by means of the lineariation of the equations with respect to small perturbations, δu (, t) exp (ik) δu (, t) and δv (, t) exp (ik) δv (, t), and computing the respective stability eigenvalues in a numerical form 3. In the conservative system, the critical role in the destabiliation of the symmetric solitons, with u (, t) = v (, t), is played by antisymmetric perturbations, with δu = δv. In the case of γ =, the instability sets in when the eigenvalue of the antisymmetric mode passes ero 3. Similarly, in the system with γ > the destabiliation of the PT -symmetric soliton, with v = (iγ + ) γ u, is accounted for by the PT - antisymmetric perturbation mode, with δv = (iγ ) γ δu, () see Eq. (). The instability threshold corresponds to the respective eigenvalue passing ero. Then, setting δv = iγ γ δw and δu = δw/ iγ γ, which is compatible with Eq. (), the critical real eigenmode δw (t), associated with the ero eigenvalue, is determined by the single linear equation, which can be derived from the two linearied ones in the same way as it was done for the system with the cubic nonlinearity in Refs. 6 and 9 : [ d K + γ dt 3 W (t) + 5g W (t) 4 ] δw =, (3) where W (t) is given by Eq. (8). Rigorous justification of the reduction of the twocomponent linear-stability problem to one based on the single equation (3) can be developed in essentially the same way as it was done for the model of the PT - symmetric coupler with the cubic-only nonlinearity in Ref. 9. The consideration makes it possible to decompose infinitesimal perturbations into a component lying in the manifold of Eq. (3) and the transversal component, the former one playing the critical role for the destabiliation of the PT -symmetric solitons. Accordingly, the stability may be lost via a pair of pure imaginary eigenvalues generated by Eq. (3) (not through the birth of a complex quadruplet). Note that Eq. (3) is different from the traditional Sturm-Liouville problem for eigenvalue K, due to the fact that K also appears in the expression for W (t), according to Eq. (8). In Refs. 6 and 9, Eq. (3) was solved exactly for the cubic model, with g =, which produced result (3). In the present case, an analytical solution to Eq. (3) with g > is not available, hence the equation should be solved numerically. It is relevant to mention that the same analysis can be applied, in a simpler form, to the stability of PT - symmetric CW states, which are generated by Eq. () from the commonly known modulationally stable CW solution of Eq. (7): ( W CW = (g) + ) 4gK exp (ik), (4) with propagation constant K taking values in the semiinfinite interval, < K / (4g) K largest. (5) At K = 3/ (6g), see Eq. (9), solution (CW) is identical to the one given by Eq. (). The largest CW amplitude is attained at K = K largest : ( W CW ) largest = / (g). (6) There is also another branch of the CW solutions, corresponding to the opposite sign in front of 4gK in Eq. (4), which exists in a finite interval, < K < / (4g); however, this branch is modulationally unstable in the framework of Eq. (7). The analysis of the stability of the PT -symmetric CW state against the symmetry breaking reduces to the t- independent version of Eq. (3), which implies a simple algebraic relation, tantamount to vanishing of the expression in the square brackets, K + γ 3 W CW + 5g W CW 4 =. (7) Finally, the substitution of the CW amplitude from Eq. (4) in Eq. (7) readily shows that Eq. (7) is never satisfied, except for the limit case of γ =, K = K largest, see Eq. (5). Thus, the PT -symmetric CW states are never subject to the instability. III. NUMERICAL RESULTS A. The stability area for the PT -symmetric solitons To identify stability boundaries for the symmetric solitons, we solved Eq. (3) for eigenvalue K by means of the relaxation (Newton s) method with the use of a finite-difference scheme on a numerical grid that spans the domain of T/ t T/ with ero boundary conditions 33. This approach allows one to find, simultaneously, both eigenvalues (there are two of them, see below) and the corresponding localied eigenfunctions. The target relative accuracy of the numerical solutions was 6. To reach it, the numerical domain was initially covered by a grid of N = points, and N was then increased iteratively until the convergence with the necessary accuracy was reached. This actually happened at N 5. Figure shows the so found eigenvalues K as functions of the gain-loss parameter γ for different values of the quintic coefficient, g. It is seen that there are two solutions, K, (denoted by red and blue colors in Fig. ), the

5 Unstable solitons K K.8.7 Bifurcation boundary γ c (g) K, K.5 B.9. γ c Stable solitons A γ. g c g FIG.. Eigenvalues K and K (red and blue lines, respectively) produced by the numerical solution of Eq. (3) as functions of γ, at different values of the quintic coefficient, g, which are indicated by arrows.in the interval of g < g c.65, branches K and K are completely separated (the branch corresponding to g = g c is labeled by symbol B). At g > g c, the branches partly merge. As a result, they originate from a junction point at a finite value of the gain-loss parameter, γ = γ c. The dashed curve (labeled A) shows the exact analytical solution given by Eq. (3) for the cubic system with g =. The branch of K practically overlaps with A at g <.35. PT -symmetric solitons being unstable in the interval of K < K < K, and stable at K < K and K > K. The upper stability region, which does not exist in the cubic system, cf. Eq. (3), is a specific feature of the CQ system, in which the quintic self-defocusing term provides for the restoration of the stability at K > K. In particular, in the absence of the gain and loss (γ = ), the stability and instability intervals on the vertical axis in Fig. exactly correspond to those for the symmetric solitons in the conservative coupler with the CQ nonlinearity, which were found in Ref. 7. On the other hand, in the limit of γ the stability region for solitons at K > K must vanish, which implies that K (γ = ) = 3/ (6g), see Eq. (9). This conjecture is fully corroborated by numerically generated data. In the limit of vanishingly small g, branch K (γ) in the CQ model naturally becomes very close to the exact one in the cubic system, which is given by Eq. (3), as can be clearly seen in Fig.. In the same limit, branch K is expelled to a region of very large values /g, cf. Eq. (9). Branches K and K remain completely separated in the entire interval of γ < if the quintic coefficient is not too large, vi., at g < g c.65. At g > g c, the branches partly merge and thus disappear at small γ < γ c (g), hence all the solitons are stable at γ < γ c. In this case, branches K and K originate from point FIG.. The value of the gain-loss coefficient, γ c, at the critical point from which branches K, originate at g > g c.65 (see Fig. ), versus the quintic-defocusing coefficient, g. The critical value attains γ c = at g = g max.9, which implies that all the PT -symmetric solitons are stable at g > g max. γ = γ c, thus reducing the instability region to γ > γ c (g). Note that all values of K presented in Fig. really exist, in the sense that they satisfy restriction (9), K < K max, and, as said above, it can be checked that the value of K = K max 3/(6g) is attained precisely at γ =. The merger of the K and K branches at γ = γ c for g > g c is, clearly, a bifurcation phenomenon. The respective bifurcation diagram, showing the value of the gain-loss coefficient at the merger point, γ c, as a function of g is plotted in Fig.. It can be clearly identified as a supercritical bifurcation (in other words, it is the phase transition of the second kind) 34. Accordingly, in the plane of the two control parameters of Eq. (4), g and γ, all the PT -symmetric solitons are stable on the right-hand side of the γ c (g) curve. On its left-hand side, the solitons are unstable in interval K < K < K, and stable outside of it, as stated above. It is worthy to mention that the presently found value of g c.65 fully agrees, up to the numerical accuracy, with the critical point found in Ref. 7 in the CQ system with γ =, at which the bifurcation loop shrinks to ilch and disappears. A simple analytical approximation for the critical value of the quintic coefficient, g c.5, was obtained in Ref. 3. A remarkable feature revealed by Figs. and is that γ c (g) attains the limit of γ = (up to the numerical accuracy) at g = g max.9, all the PT -symmetric solitons being stable at g > g max, with the corresponding K max. given by Eq. (9). Note that g max is, approximately, eight times larger than value g c, defined above, at which the symmetric solitons become completely stable in the system with γ =. Thus, if the quintic self-defocusing is strong enough, it makes the PT symmetry of the solitons unbreakable. It is relevant to mention that another

6 6 (a) (b) t FIG. 3. The perturbed evolution of the v component of the stable PT -symmetric soliton at g = gc =.65, γ =., and K =.9, which falls into the lower stability area, K < K, in Fig.. The evolution of the u component is similar. model, in which the self-defocusing nonlinearity gives rise to D solitons with the unbreakable PT symmetry, was elaborated in Ref.35. This conclusion is consistent with the above-mentioned stability of the PT -symmetric CW states (4). B. The evolution of stable and unstable solitons The above predictions for the stability and instability of the PT -symmetric solitons were verified by direct simulations of the perturbed evolution of the solitons. To this end, Eq. (4) was simulated with initial conditions u(, t) = v(, t) = W ( =, t), (8) where W ( =, t) is exact solution (8) for the CQ soliton. Although this input is different from the exact PT symmetric solution given by Eq. (), as the phase shift between the two components is missing, the input, with small perturbations added to it for testing the stability, readily evolves into a stable soliton, provided that parameter K in Eq. (8) belongs to either of the two predicted stability areas, K < K or K > K (see Fig. ). Figure 3 presents an example of the perturbed evolution for a soliton which is expected to be stable, according to the above analysis, and indeed remains stable, at least, up to, which roughly corresponds to 5 characteristic dispersion/diffraction lengths of the soliton. A characteristic example of the evolution of a soliton belonging to the instability area is displayed in Fig. 4. The domination of the self-defocusing quintic term at large amplitudes prevents the blowup of unstable solitons, unlike what happens in the cubic model6,9. Instead, the unbalanced pump of the energy into the un- FIG. 4. The perturbed evolution of an unstable soliton with g =., γ =.45, and K =.9, which belongs to the instability region, K < K < K, pursuant to Fig.. Panels (a) and (b) display top views of the evolution of fields u (, t) and v (, t), respectively. The dissociation of one unstable soliton into three oscillating pulses is observed. stable configuration creates a spatially symmetric pair of two additional tilted (moving) solitons, on top of generation of a considerable amount of radiation. The newly formed solitons seem as oscillating breathers, and in this connection it is relevant to mention that dissociation of an unstable soliton into a pair of breathers in the system with the cubic nonlinearity, based on Eq. (), was observed in Ref.9. As concerns PT -antisymmetric solitons, which correspond to the bottom sign in Eq. (), they all were found to be unstable. An example of a relatively weak instability is displayed in Fig. 5. In this case, the instability sets it within a single dispersion length, 5. The further evolution transforms the unstable soliton into a stable PT -symmetric one, which is accompanied by the emission of radiation. Stronger instability of the antisymmetric solitons causes a different outcome. As shown in Fig. 6, the intensive pump of the energy into the u component gives rise to a quasi-turbulent pattern with a limited amplitude [the magnitude of which is roughly approximated by Eq. (6)], symmetrically expanding along the t axis, after an initial spontaneous shift of the unstable soliton (eventually, the expanding pattern hits edges of the simulation domain). The fronts which separate the domains with u (, t) = and u(, t) = may be related to the exact front solution of Eq. (7), that was found in Ref.36 : v ) ( u 3 3i u ( )] exp Wfr = t [. g + exp ± 3/ (g)t 6g (9) While this solution produces a quiescent front, moving

7 7 (a) 5 (a) (b) (b) V U t 5 5 t t FIG. 5. The same as in Fig. 4, but for the evolution of an unstable PT -antisymmetric soliton, with the same parameters as chosen for the stable symmetric soliton in Fig. 3. FIG. 7. The soliton-soliton collision for K =.4, γ =., g = gc =.65, and velocity v =.7 in Eq. (), shown for the U and V components in panels (a) and (b), respectively. solitons in the course of the collision does not cause instability, the outcome must be exactly the same as for solitons in the NLS equation (7) with the CQ nonlinearity, where the collisions are known to be quasi-elastic37. To verify this expectation, simulations of Eq. (4) were run for pairs of quiescent and moving solitons, originally separated by distance t. The corresponding initial conditions for Eqs. (4) were taken as u(, t) = U (, t) + U (, t t)eiv t, v(, t) = U (, t) + U (, t t)eiv t, FIG. 6. The outcome of the evolution of a strongly unstable PT -antisymmetric soliton with parameters g =.5, γ =., and K =.9. ones, such as the fronts observed in Fig. 6, can be generated by the Galilean transformation, as the presence of the gain and loss terms in Eq. (4) does not break the system s Galilean invariance. IV. COLLISIONS OF SOLITONS As said above, the presence of the gain and loss does not break the Galilean invariance of the NLS equations, therefore the consideration of moving solitons makes sense in the present model. If PT -symmetric solitons are stable, one may expect that collisions between moving ones are elastic6. Indeed, if the overlapping of the () cf. Eq. (8), where v is the velocity of the moving soliton. In the simulations we varied v. Figure 7 shows a typical result of the collision, in terms of both components U and V. It is observed that the collision indeed looks quasi-elastic, with a shift of each soliton, which is always generated by elastic collisions. This outcome of the interaction implies, as mentioned above, that the overlap of the interacting solitons does not excite any instability breaking the PT symmetry. V. CONCLUSION We have introduced the D model the of PT -(paritytime ) symmetric coupler with the mutually balanced linear gain and loss applied to its two cores, and the CQ (cubic-quintic) nonlinearity acting in each one, which represents the competition between the cubic selffocusing and quintic defocusing. Such models may be realied in dual-core optical waveguides, in the spatial and temporal domains alike. Stationary solutions for PT symmetric and antisymmetric solitons in these systems amount to their counterparts in the coupler without gain and loss. The stability of the solitons is the most essential problem. The symmetric solitons become unstable

8 8 against the breaking of the PT symmetry with the increase of the energy (or integral power, in terms of the spatial-domain system), and retrieve the stability at still larger energies. Numerical results for the stability boundaries of the solitons were obtained by means of the numerical solution of the linearied equation for the critical mode of small perturbations, and verified by direct simulations. Naturally, the increase of the strength, g, of the repulsive quintic term leads to expansion of the stability domain. At g > g c, the instability sets in only at finite (i.e., not too small) values of the gain-loss coefficient, and at g > g max, which is roughly eight times larger than g c, the PT symmetry of the solitons becomes unbreakable. On the other hand, PT -antisymmetric solitons are always unstable. Basic scenarios of the spontaneous evolution of unstable solitons were identified. These include the transformation of weakly unstable solitons into stable ones and the generation of additional solitons, while stronger instability gives rise to an expanding area filled by a quasi-turbulent state. Collisions between stable solitons are found to be quasi-elastic. As an extension of the analysis, it may be relevant to develop it in the full form for fundamental and vortex solitons in the D version of the system. VI. ACKNOWLEDGMENTS This work was partially supported by CONACyT (México) through grant No , and by grant No. 566 from the joint program in physics between the National Science Foundation (US) and BSF. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press: San Diego, 3); T. Dauxois and M. Peyrard, Physics of Solitons (Cambridge University Press: Cambridge, UK, 6). N. Akhmediev and A. Ankiewic, eds., Dissipative Solitons, Vol. 66 of Lecture Notes in Physics (Springer, 5); N. N. Rosanov, Dissipative Optical Solitons: from Micro to Nano and Atto (Fimatlit, ) (in Russian); L. Lugiato, F. Prati, and M. Brambilla, Nonlinear Optical Systems (Cambridge University Press: Cambridge, UK, 5). 3 B. A. Malomed, Physica D 9, 55 (987); S. Fauve and O. Thual, Phys. Rev. Lett. 64, 8 (99); I. V. Barashenkov, N. V. Alexeeva, and E. V. Zemlyanaya, Phys. Rev. Lett. 89, 4 (). 4 C. M. Bender and S. Boettcher, Phys. Rev. Lett. 8, 543 (998); C. M. Bender, D. C. Brody, and H. F. Jones, ibid. 89, 74 (); C. M. Bender, Rep. Prog. Phys. 7, 947 (7). 5 A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 38, L7 (5); R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 3, 63 (7); M. V. Berry, J. Phys. A: Math. Theor. 4, 447 (8); S. Klaiman, U. Günther, and N. Moiseyev, Phys. Rev. Lett., 84 (8); S. Longhi, ibid. 3, 36 (9); K. Li, and P. G. Kevrekidis, Phys. Rev. E 83, 6668 (); H. Rameani, D. N. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Phys. Rev. Lett. 9, 339 (). 6 A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aime, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 3, 939 (9); C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nature Phys. 6, 9 (); A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 67 (). 7 Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett., 34 (8); F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zeyulin, Phys. Rev. A 83, 485(R) (); X. Zhu, H. Wang, L.-X. Zheng, H. Li, and Y.-J. He, Opt. Lett. 36, 68 (); S. V. Suchkov, B. A. Malomed, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. E 84, 4669 (); J. Zeng and Y. Lan, Phys. Rev. E 85, 476 (); M.-A. Miri, A. B. Aceves, T. Kottos, V. Kovanis, and D. N. Christodoulides, Phys. Rev. A 86, 338 (); Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, Opt. Commun. 85, 33 (); C. Li, H. Liu, and L. Dong, Opt. Exp., 683 (); A. Khare, S. M. Al-Maroug, and H. Bahlouli, Phys. Lett. A 376, 88 (); Yu. V. Bludov, R. Driben, V. V. Konotop, and B. A. Malomed, J. Opt. 5, 64 (3); X. Zhang, J. Chai, J. Huang, Z. Chen, Y. Li, and B. A. Malomed, Opt. Exp., 397 (4); Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 39, 564 (4). 8 J.-Y. Lien, Y.-N. Chen, N. Ishida, H.-B. Chen, C.-C. Hwang, and F. Nori, Phys. Rev. B 9, 45 (5). 9 S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, Laser Photonics Rev., 77 (6); V. V. Konotop, J. Yang, and D. A. Zeyulin, Rev. Mod. Phys. 88, 35 (6). D. A. Zeyulin, Y. V. Kartashov, and V. V. Konotop, EPL 96, 643 (); S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 38 (). M. Wimmer, A. Regensburger, M.-A. Miri, C. Bersch, D. N. Christodoulides, and U. Peschel, Nature Communications 6, 778 (5). B. A. Malomed and H. G. Winful, Phys. Rev. E 53, 5365 (996); J. Atai and B. A. Malomed, Phys. Rev. E 54, 437 (996). 3 B. A. Malomed, Chaos 7, 377 (7). 4 A. Marini, D. V. Skryabin, and B. A. Malomed, Opt. Exp. 9, 666 (); C. Milián, D. E. Ceballos-Herrera, D. V. Skryabin, and A. Ferrando, Opt. Lett. 37, 4 (); Y. Xue, F. Ye, D. Mihalache, N. C. Panoiu, and X. Chen, Laser Phot. Rev. 8, L5 (4). 5 P. V. Paulau, D. Gomila, P. Colet, N. A. Loiko, N. N. Rosanov, T. Ackemann, and W. J. Firth, Opt. Exp. 8, 8859 (); P. V. Paulau, D. Gomila, P. Colet, B. A. Malomed, and W. J. Firth, Phys. Rev. E 84, 363 (). 6 R. Driben and B. A. Malomed, Opt. Lett. 36, 433 (). 7 F. Kh. Abdullaev, V. V. Konotop, M. Øgren, and M. P. Sørensen, Opt. Lett. 36, 4566 (). 8 R. Driben and B. A. Malomed, EPL 96, 5 (). 9 N. V. Alexeeva, I. V. Barashenkov, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. A 85, (). R. Driben and B. A. Malomed, EPL 99, 54 (); I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 5389 (). P. Li, L. Li, and B. A. Malomed, Phys. Rev. E 89, 696 (4). J. Čtyroký, V. Kumiak, and S. Eyderman, Opt. Exp. 8, 585 (); S. Savoia,G. Castaldi, V. Galdi, A. Alú, and N. Engheta, Phys. Rev. B 89, 855 (4); N. V. Alexeeva, I. V. Barashenkov, K. Rayanov, and S. Flach, Phys. Rev. A 89, 3848 (4); B. Dana, A. Bahabad, and B. A. Malomed, ibid. 9, 4388 (5); J. Cuevas-Maraver, B. Malomed, and P. Kevrekidis, Symmetry 8, 39 (6). 3 E. M. Wright, G. I. Stegeman, and S. Wabnit, Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers, Phys. Rev. A 4, 4455 (989). 4 C. Paré and M. F lorjańcyk, Phys. Rev. A 4, 687 (99); A. I. Maimistov, Kvantovaya Elektron. (Moscow) 8, 758 (99) [English translation: Sov. J. Quantum Electr., 687 (99); J. M. Soto-Crespo and N. Akhmediev, Phys. Rev. E 48, 47 (993); P. L. Chu, B. A. Malomed, and G. D. Peng, J. Opt. Soc. Am. B, 379 (993); B. A. Malomed, I. Skinner, P. L. Chu, and G. D. Peng, Phys. Rev. E 53, 484 (996).

9 9 5 F. Smektala, C. Quemard, V. Couderc, and A. Barthélémy, J. Non-Cryst. Solids 74, 3 (); G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, F. Sanche, Opt. Commun. 9 (3) 47 (3); C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, J. Opt. Soc. Amer. B 9, 369 (); R. A. Ganeev, M. Baba, M. Morita, A. I. Ryasnyansky, M. Suuki, M. Turu, and H. Kuroda, J. Opt. A: Pure Appl. Opt. 6, 8 (4); E.L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues Jr., J. Opt. Soc. Am. B 4, 948 (7). 6 A. S. Reyna and C. B. de Araújo, Phys. Rev. A 89, 6383; A. S. Reyna, K. C. Jorge, and C. B. de Araújo, ibid. 9, (4); A. S. Reyna and C. B. de Araújo, Opt. Express, 456 (4). 7 L. Albuch and B. A. Malomed, Mathematics and Computers in Simulation 74, 3 (7). 8 N. Dror and B. A. Malomed, Physica D 4, 56 (). 9 N. Jhajj, I. Larkin, E. W. Rosenthal, S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, Phys. Rev. X 6, 337 (6). 3 G. Burlak and B. A. Malomed, Phys. Rev. E 88, 694 (). 3 Kh. I. Pushkarov, D. I. Pushkarov, and I. V. Tomov, Opt. Quant. Electr., 47 (979); S. Cowan, R. H. Enns, S. S. Rangnekar, and S. S. Sanghera Can. J. Phys. 64, 3 (986). 3 J. Yang, Nonlinear Waves in Integrable and Non-integrable Systems (SIAM: Philadelphia, ). 33 W. H. Press, S. A. Teukovsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C++ (Cambridge University Press: Cambridge, ). 34 G. Iooss, D. D. Joseph, Elementary Stability Bifurcation Theory (Springer, New York, 98). 35 Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 39, 564 (4). 36 Z. Birnbaum and B. A. Malomed, Physica D 37, 35 (8). 37 A. S. B. Sombra, Opt. Commun. 94, 9 (99).

10 K K K, K.5 B A γ.9

11 .9 Unstable solitons.8.7 Bifurcation boundary γ c (g).6 γ c.5 Stable solitons.4.3. g c g

12

13 5 (a) (b) t - -

14 5 (a) (b) t - -

15

16 (a) (b) U.5 5 t 4 5 V.5 5 t 4 5