Spontaneous symmetry breaking and switching in planar nonlinear optical antiwaveguides

Stability limits for modes held in alternating trapping-expulsive potentials

We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g<g_{c}^{(num)}≈5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g<0), is accurately predicted by the VA. At g>g_{c}^{(num)}, the collapse threshold is found with the help of full numerical simulations. The relative increase of g_{c} above g_{c}^{(num)} is ≈1.5%. It is a meaningful result, even if its size is small, because the collapse threshold is a universal constant which is difficult to change.

Trapping wave fields in an expulsive potential by means of linear coupling

We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.

Targeted transfer of solitons in continua and lattices

We propose a robust mechanism of targeted energy transfer along a line, as well as on a surface, in the form of transport of coherent solitary-wave structures, driven by a moving, spatially localized external ac field ("arm") in a lossy medium. The efficiency and robustness of the mechanism are demonstrated analytically and numerically in terms of the nonlinear Schrödinger (NLS) equation, and broad regions of stable operation are identified in the model's parameter space. Direct simulations show that the driving arm can manipulate solitons equally well in a lattice NLS model. A salient feature, which is revealed by simulations and explained analytically, is a resonant character of the operation of the driving arm in the lattice medium, both integer and fractional resonances being identified. Numerical experiments also demonstrate that the same solitary-wave-transport mechanism works well in two-dimensional lattice media.