Amarendra K. Sarma, Mohammad-Ali Miri, Ziad H. Musslimani, Demetrios N. Christodoulides

We investigate the dynamical behavior of continuous and discrete Schr\”odinger systems exhibiting parity-time (PT) invariant nonlinearities. We show that such equations behave in a fundamentally different fashion than their nonlinear Schr\”odinger counterparts. In particular, the PT-symmetric nonlinear Schr\”odinger equation can simultaneously support both bright and dark soliton solutions. In addition, we study a two-element discretized version of this PT nonlinear Schr\”odinger equation. By obtaining the underlying invariants, we show that this system is fully integrable and we identify the PT-symmetry breaking conditions. This arrangement is unique in the sense that the exceptional points are fully dictated by the nonlinearity itself.

http://arxiv.org/abs/1310.7399

Pattern Formation and Solitons (nlin.PS)

Ilario Giordanelli, Gian Michele Graf

We give a partially alternate proof of the reality of the spectrum of the imaginary cubic oscillator in quantum mechanics.

http://arxiv.org/abs/1310.7767

Mathematical Physics (math-ph)

Mohammed F. Saleh, Andrea Marini, Fabio Biancalana

We have investigated the interaction between a strong soliton and a weak probe with certain configurations that allow optical trapping in gas-filled hollow-core photonic crystal fibers in the presence of the shock effect. We have shown theoretically and numerically that the shock term can lead to an unbroken parity-time (PT) symmetry potential in these kinds of fibers. Reciprocity breaking, a remarkable feature of the PT symmetry, is also demonstrated numerically. Our results will open different configurations and avenues for observing PT-symmetry breaking in optical fibers, without the need to resort to cumbersome dissipative structures.

http://arxiv.org/abs/1310.7497

Optics (physics.optics)

Ananya Ghatak, Raka Dona Ray Mandal, Bhabani Prasad Mandal (BHU)

We complexify a 1-d potential which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters becomes imaginary. For one PT-symmetric case we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from left (or right) at certain discrete energies. The penetrating states in the other PT-symmetric configuration are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. Presence of MSS and reflectionlessness are also discussed for the free states in the later case.

http://arxiv.org/abs/1310.7752

Quantum Physics (quant-ph)

H.Xu, P.G.Kevrekidis, Q.Zhou, D.J.Frantzeskakis, V.Achilleos, R.Carretero-Gonzalez

We study the nonlinear Schrodinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

http://arxiv.org/abs/1310.7635

Pattern Formation and Solitons (nlin.PS)