Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media

Abstract

We introduce a model based on the one-dimensional nonlinear Schroedinger equation (NLSE) with the critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact form. In the absence of the gain-loss term, the solitons' stability is investigated analytically too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. If the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

To be published in Phys. Rev. E

References in corpus (19)

• Making Sense of Non-Hermitian Hamiltonians
• Visualization of Branch Points in PT-Symmetric Waveguides
• Unidirectional Nonlinear PT-symmetric Optical Structures
• Physical realization of $\mathcal{P}\mathcal{T}$-symmetric potential scattering in a planar slab waveguide
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• Model of a PT symmetric Bose-Einstein condensate in a delta-functions double well
• Integrable $\mathcal{P}T$-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model
• Bragg solitons in nonlinear PT-symmetric periodic potentials
• A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity
• Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials
• Families of stationary modes in complex potentials
• Nonlinear Schrödinger equation for a PT symmetric delta-functions double well
• Unbreakable PT-symmetry of solitons supported by inhomogeneous defocusing nonlinearity
• Discrete solitons and vortices on two-dimensional lattices of $\mathcal{P}\mathcal{T}$-symmetric couplers
• Solitons and their stability in the nonlocal nonlinear Schroedinger equation with PT-symmetric potentials
• Solitons in PT-symmetric ladders of optical waveguides
• Interaction of sine-Gordon kinks and breathers with a $\mathcal{P}\mathcal{T}$-symmetric defect
• Staggered $\mathcal{P}\mathcal{T}$-symmetric ladders with cubic nonlinearity
• Robust PT symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity

Cited by in corpus (2)

• Singular mean-field states: A brief review of recent results
• Stabilization of single- and multi-peak solitons in the fractional nonlinear Schroedinger equation with a trapping potential