Stability of solitons in PT -symmetric couplers Rodislav Driben, 1,2 and Boris A. Malomed 2 1 Jerusalem College of Engineering - Ramat Beit HaKerem, POB 3566, Jerusalem, 91035, Israel 2 Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel Compiled September 28, 2011 Families of analytical solutions are found for symmetric and antisymmetric solitons in the dual-core system with the Kerr nonlinearity and PT -balanced gain and loss. The crucial issue is stability of the solitons. A stability region is obtained in an analytical form, and verified by simulations, for the PT -symmetric solitons. For the antisymmetric ones, the stability border is found in a numerical form. Moving solitons of both types collide elastically. The two soliton species merge into one in the “supersymmetric”case, with equal coefficients of the gain, loss and inter-core coupling. These solitons feature a subexponential in- stability, which may be suppressed by periodic switching (“management”). c  2011 Optical Society of America OCIS codes: 060.5530, 190.6135, 190.5940, 230.4320 Dissipative media featuring the parity-time (PT ) sym- metry have recently drawn a great deal of attention. The introduction of this symmetry in optics followed works extending the canonical quantum theory to non- Hermitian Hamiltonians that may exhibit a real spec- trum [1]. The Hamiltonian is PT -symmetric if it in- cludes a complex potential V (x) which satisfies con- straint V (x)= V * (-x). Such potentials were proposed [2]- [7] and realized [8, 9] in optics, by juxtaposing spa- tially symmetric patterns of the refractive index and ap- propriately placed gain and loss elements, see Ref. [7] for a review. Nonlinear PT systems [10]- [15] and respective solitons [16] were introduced too. A medium which is akin to PT systems is a dual- core waveguide with gain and loss acting separately in two cores, which are linearly coupled by the tunneling of light [17]. This system predicts stable 1D solitons in optical [17]- [19] and plasmonic [20] waveguides with the Kerr nonlinearity, as well as 2D dissipative solitons and vortices [21, 22]. The system is made PT symmetric by adopting equal strengths of the gain and loss in the cores. A challenging problem is the stability of solitons, as sta- ble pulses in the dual-core system were previously found far from the point of the PT symmetry [18–20]. We pro- duce two families of exact soliton solutions for the PT - symmetric system, which correspond to symmetric and antisymmetric solitons in the ordinary dual-core cou- pler [23]- [26]. For the former family, an exact stability border is found analytically, and verified by simulations. For the PT -antisymmetric solitons, the stability region is identified in a numerical form. In the “supersymmet- ric” limit, when the gain and loss coefficients coincide with the constant of the inter-core coupling, both fami- lies merge into solitons which are subject to a subexpo- nential instability. We demonstrate that this instability can be suppressed by means of a “management” tech- nique, i.e., periodic switching of the gain, loss, and cou- pling. The transmission of light or plasmons in the dual-core waveguide is described by the linearly coupled equations for amplitudes u(z,t) and v(z,t) in the active and passive cores [17]- [20]: iu z + (1/2)u tt + |u| 2 u - iγu + κv = 0, iv z + (1/2)v tt + |v| 2 v + iΓv + κu = 0, (1) where z is the propagation distance and t the reduced time or transverse coordinate in the temporal- or spatial- domain system. Coefficients accounting for the disper- sion or diffraction, Kerr nonlinearity, and inter-core cou- pling, κ, are normalized to be 1, while γ and Γ are coef- ficients of the linear gain and loss in the two cores. The PT symmetry in Eqs. (1) corresponds to Γ = γ . Strictly speaking, the PT -balanced gain and loss correspond to the border between stable and unstable systems: The zero solution, u = v = 0, is unstable at γ> Γ, while the stability region was found at γ< Γ [17, 18]. Obviously, any solution to the nonlinear Schr¨ odinger (NLS) equation (with a frequency shift), iU z +(1/2)U tt + |U | 2 U ±  1 - γ 2 U =0, gives rise to two exact solutions of the PT -symmetric system, provided that γ ≤ 1: v =  iγ ±  1 - γ 2  u = U (z,t) (2) (recall we fix κ ≡ 1). For γ = 0, solutions (2) with + and - amount, respectively, to the symmetric and antisym- metric modes in the dual-core coupler [23]- [26], therefore we call the respective solutions (2) PT -symmetric and PT -antisymmetric ones. In the limit of γ = 1, exact so- lutions (2) reduce to a single one, v = iu = U (z,t). In particular, PT -symmetric and antisymmetric solitons, with arbitrary amplitude η, are generated by the NLS solitons, U (z,t)= η exp  i  η 2 /2 ±  1 - γ 2  z  sech (ηt) . (3) As concerns stability of the solitons, it is again relevant to compare the PT -symmetric system to its counterpart with γ = 0. The exact result for the latter system is that 1 arXiv:1109.5759v1 [physics.optics] 27 Sep 2011