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 instability, which may be suppressed by periodic switching ("management"). c 2018 Optical Society of America OCIS codes: 060.5530, 190.6135, 190.5940, 230.4320 Dissipative media featuring the parity-time (PT ) symmetry have recently drawn a great deal of attention. The introduction of this symmetry in optics followed works extending the canonical quantum theory to nonHermitian Hamiltonians that may exhibit a real spectrum [1]. The Hamiltonian is PT -symmetric if it includes a complex potential V (x) which satisfies constraint V (x) = V * (−x). Such potentials were proposed [2]- [7] and realized [8, 9] in optics, by juxtaposing spatially symmetric patterns of the refractive index and appropriately placed gain and loss elements, see Ref. A medium which is akin to PT systems is a dualcore 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 stable pulses in the dual-core system were previously found far from the point of the PT symmetry [18][19][20]. We produce two families of exact soliton solutions for the PTsymmetric system, which correspond to symmetric and antisymmetric solitons in the ordinary dual-core coupler [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 "supersymmetric" limit, when the gain and loss coefficients coincide with the constant of the inter-core coupling, both families merge into solitons which are subject to a subexponential instability. We demonstrate that this instability can be suppressed by means of a "management" technique, i.e., periodic switching of the gain, loss, and coupling.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]:where z is the propagation distance and t the reduced time or transverse coordinate in the temporal-or spatialdomain system. Coefficients accountin...

A one-dimensional mean-field model of a two-component condensate in the parabolic trap is considered, with the components corresponding to different spin states of the same atom. We demonstrate that the linear coupling (interconversion) between them, induced by a resonant electromagnetic wave, can drive the immiscible binary condensate into a miscible state. This transition is predicted in an analytical form by means of a variational approximation (for an infinitely long system), and is confirmed by direct numerical solutions of the symmetric and asymmetric models (the asymmetry accounts for a possible difference in the chemical potential between the components). We define an order parameter of the system as an off-centre shift of the centre of mass of each component. A numerically found dependence of the order parameter on the linear-coupling strength reveals a second-kind phase transition (in an effectively finite system). The phase transition looks very similar in two different regimes, namely, with a fixed number of atoms, or a fixed chemical potential. An additional transition is possible between double- and single-humped density distributions in the weak component of a strongly asymmetric system. We also briefly consider dynamical states, with the two components oscillating relative to each other. In this case, the components periodically separate even if they should be mixed in the static configuration.

We introduce a system based on dual-core nonlinear waveguides with the balanced gain and loss acting separately in the cores. The system features a "supersymmetry" when the gain and loss are equal to the inter-core coupling. This system admits a variety of exact solutions (we focus on solitons), which are subject to a specific subexponential instability. We demonstrate that the application of a "management", in the form of periodic simultaneous switch of the sign of the gain, loss, and inter-coupling, effectively stabilizes solitons, without destroying the supersymmetry. The management turns the solitons into attractors, for which an attraction basin is identified. The initial amplitude asymmetry and phase mismatch between the components transforms the solitons into quasi-stable breathers.

We find that the recently introduced model of self-trapping supported by a spatially growing strength of a repulsive nonlinearity gives rise to robust vortex-soliton tori, i.e., three-dimensional vortex solitons, with topological charges S≥1. The family with S=1 is completely stable, while the one with S=2 has alternating regions of stability and instability. The families are nearly exactly reproduced in an analytical form by the Thomas-Fermi approximation. Unstable states with S=2 and 3 split into persistently rotating pairs or triangles of unitary vortices. Application of a moderate torque to the vortex torus initiates a persistent precession mode, with the torus' axle moving along a conical surface. A strong torque heavily deforms the vortex solitons, but, nonetheless, they restore themselves with the axle oriented according to the vectorial addition of angular momenta.