Enhancement of Persistent Currents by Hubbard Interactions In Disordered 1D Rings: Avoided Level Crossings Interpretation

We study effects of local electron interactions on the persistent current of one dimensional disordered rings. For different realizations of disorder we compute the current as a function of Aharonov-Bohm flux to zeroth and first orders in the Hubbard interaction. We find that the persistent current is {\em enhanced} by onsite interactions. Using an avoided level crossings approach, we derive analytic formulas which explain the numerical results at weak disorder. The same approach also explains the opposite effect (suppression) found for spinless fermion models with intersite interactions.Comment: uuencoded: 17 pages, text in revtex, 7 figs in postscrip

Scattering of gap solitons by PT-symmetric defects

The resonant scattering of gap solitons (GS) of the periodic nonlinear Schr\"odinger equation with a localized defect which is symmetric under the parity and the time-reversal (PT) symmetry, is investigated. It is shown that for suitable amplitudes ratios of the real and imaginary parts of the defect potential the resonant transmission of the GS through the defect becomes possible. The resonances occur for potential parameters which allow the existence of localized defect modes with the same energy and norm of the incoming GS. Scattering properties of GSs of different band-gaps with effective masses of opposite sign are investigated. The possibility of unidirectional transmission and blockage of GSs by PT defect, as well as, amplification and destruction induced by multiple reflections from two PT defects, are also discussed

Discrete gap solitons in a diffraction-managed waveguide array

A model including two nonlinear chains with linear and nonlinear couplings between them, and opposite signs of the discrete diffraction inside the chains, is introduced. For [$\chi ^{(3)}$] nonlinearity, the model finds two different interpretations in terms of optical waveguide arrays, based on the diffraction-management concept. A straightforward discrete [$\chi ^{(2)}$] model, with opposite signs of the diffraction at the fundamental and second harmonics, is introduced also. Starting from the anti-continuum (AC) limit, soliton solutions in the $\chi ^{(3)}$ model are found, both above the phonon band and inside the gap. Solitons above the gap may be stable as long as they exist, but in the transition to the continuum limit they inevitably disappear. On the contrary, solitons inside the gap persist all the way up to the continuum limit. In the zero-mismatch case, they lose their stability long before reaching the continuum limit, but finite mismatch can have a stabilizing effect on them. A special procedure is developed to find discrete counterparts of the Bragg-grating gap solitons. It is concluded that they exist all the values of the coupling constant, but are stable only in the AC and continuum limits. Solitons are also found in the $\chi ^{(2)}$ model. They start as stable solutions, but then lose their stability. Direct numerical simulations in the cases of instability reveal a variety of scenarios, including spontaneous transformation of the solitons into breather-like states, destruction of one of the components (in favor of the other), and symmetry-breaking effects. Quasi-periodic, as well as more complex, time dependences of the soliton amplitudes are also observed as a result of the instability development.Comment: 18 pages, 27 figures, Eur. Phys. J. D in pres

Stable vortex and dipole vector solitons in a saturable nonlinear medium

We study both analytically and numerically the existence, uniqueness, and stability of vortex and dipole vector solitons in a saturable nonlinear medium in (2+1) dimensions. We construct perturbation series expansions for the vortex and dipole vector solitons near the bifurcation point where the vortex and dipole components are small. We show that both solutions uniquely bifurcate from the same bifurcation point. We also prove that both vortex and dipole vector solitons are linearly stable in the neighborhood of the bifurcation point. Far from the bifurcation point, the family of vortex solitons becomes linearly unstable via oscillatory instabilities, while the family of dipole solitons remains stable in the entire domain of existence. In addition, we show that an unstable vortex soliton breaks up either into a rotating dipole soliton or into two rotating fundamental solitons.Comment: To appear in Phys. Rev.

A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

The Petviashvili's iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: \ $-Mu+u^p=0$, where $M$ is a positive definite self-adjoint operator and $p={\rm const}$. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.Comment: to appear in J. Comp. Phys.; 35 page

Nonlinear dynamics of wave packets in PT-symmetric optical lattices near the phase transition point

Nonlinear dynamics of wave packets in PT-symmetric optical lattices near the phase-transition point are analytically studied. A nonlinear Klein-Gordon equation is derived for the envelope of these wave packets. A variety of novel phenomena known to exist in this envelope equation are shown to also exist in the full equation including wave blowup, periodic bound states and solitary wave solutions.Comment: 4 pages, 2 figure

PT\mathcal{PT}-Symmetric Periodic Optical Potentials

In quantum theory, any Hamiltonian describing a physical system is mathematically represented by a self-adjoint linear operator to ensure the reality of the associated observables. In an attempt to extend quantum mechanics into the complex domain, it was realized few years ago that certain non-Hermitian parity-time ($\mathcal{PT}$) symmetric Hamiltonians can exhibit an entirely real spectrum. Much of the reported progress has been remained theoretical, and therefore hasn't led to a viable experimental proposal for which non Hermitian quantum effects could be observed in laboratory experiments. Quite recently however, it was suggested that the concept of $\mathcal{PT}$-symmetry could be physically realized within the framework of classical optics. This proposal has, in turn, stimulated extensive investigations and research studies related to $\mathcal{PT}$-symmetric Optics and paved the way for the first experimental observation of $\mathcal{PT}$-symmetry breaking in any physical system. In this paper, we present recent results regarding $\mathcal{PT}$-symmetric Optic