Abstract
The real spectrum of bound states produced by {bf{P}}{bf{T}}-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of {bf{P}}{bf{T}}-symmetric systems for various applications. On the other hand, it is known that the {bf{P}}{bf{T}} symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) {bf{P}}{bf{T}} symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose {bf{P}}{bf{T}} symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a {bf{P}}{bf{T}}-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the {bf{P}}{bf{T}} symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable {bf{P}}{bf{T}} symmetry, while generic soliton families are found in a numerical form. The {bf{P}}{bf{T}}-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.
Highlights
While wave functions of quantum systems may be complex, spectra of their energy eigenvalues must be real, which is usually secured by restricting the underlying Hamiltonian to be Hermitian1
A possibility to implement the concept of the PT symmetry in terms of classical physics was predicted for optical media with symmetrically placed gain and loss elements19–34, which is based on the similarity between the Schrödinger equation in quantum mechanics and the paraxial-propagation equation for optical waveguides
The 1D nonlinear Schrödinger equations (NLSEs) for the amplitude of the electromagnetic field, u(x, z), with the local strength of the self-defocusing nonlinearity, Σ(x), growing from x = 0 towards x = ±∞ faster than |x|, which is capable to maintain bright solitons with unbreakable PT symmetry, is60 i ∂u ∂z
Summary
While wave functions of quantum systems may be complex, spectra of their energy eigenvalues must be real, which is usually secured by restricting the underlying Hamiltonian to be Hermitian. A possibility to implement the concept of the PT symmetry in terms of classical physics was predicted for optical media with symmetrically placed gain and loss elements, which is based on the similarity between the Schrödinger equation in quantum mechanics and the paraxial-propagation equation for optical waveguides. This possibility was implemented in several waveguiding settings, as well as in other photonic media, including exciton-polariton condensates, and in optomechanical systems. Emulation of the PT symmetry was demonstrated in acoustics and electronic circuits, and predicted in atomic Bose-Einstein condensates, magnetism, and chains of coupled pendula
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