Exponential asymptotics for solitons in PT-symmetric periodic potentials

Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT-symmetric multi-soliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.Comment: 17 pages, 4 figure

Light propagation in periodically modulated complex waveguides

Light propagation in optical waveguides with periodically modulated index of refraction and alternating gain and loss are investigated for linear and nonlinear systems. Based on a multiscale perturbation analysis, it is shown that for many non-parity-time ($\mathcal{PT}$) symmetric waveguides, their linear spectrum is partially complex, thus light exponentially grows or decays upon propagation, and this growth or delay is not altered by nonlinearity. However, several classes of non-$\mathcal{PT}$-symmetric waveguides are also identified to possess all-real linear spectrum. In the nonlinear regime longitudinally periodic and transversely quasi-localized modes are found for $\mathcal{PT}$-symmetric waveguides both above and below phase transition. These nonlinear modes are stable under evolution and can develop from initially weak initial conditions.Comment: 6 pages, 4 figure

Nonlinear wave dynamics near phase transition in PT\mathcal{PT}-symmetric localized potentials

Nonlinear wave propagation in parity-time ($\mathcal{PT}$) symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for phase transition to occur are derived based on a generalization of the Krein signature. Using multi-scale perturbation analysis, a reduced nonlinear ODE model is derived for the amplitude of localized solutions near phase transition. Above phase transition, this ODE model predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodically-oscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.Comment: 11 pages, 7 figure

Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides

Many classes of non-parity-time (PT) symmetric waveguides with arbitrary gain and loss distributions still possess all-real linear spectrum or exhibit phase transition. In this article, nonlinear light behaviors in these complex waveguides are probed analytically near a phase transition. Using multi-scale perturbation methods, a nonlinear ordinary differential equation (ODE) is derived for the light's amplitude evolution. This ODE predicts that the first class of these non-PT-symmetric waveguides support continuous families of solitons and robust amplitude-oscillating solutions both above and below phase transition, in close analogy with PT-symmetric systems. For the other classes of waveguides, the light's intensity always amplifies under the effect of nonlinearity even if the waveguide is below phase transition. These analytical predictions are confirmed by direct computations of the full system.Comment: 5 pages, 4 figure

Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials

Continuous families of solitons in generalized nonlinear Sch\"odinger equations with non-PT-symmetric complex potentials are studied analytically. Under a weak assumption, it is shown that stationary equations for solitons admit a constant of motion if and only if the complex potential is of a special form $g^2(x)+ig'(x)$, where $g(x)$ is an arbitrary real function. Using this constant of motion, the second-order complex soliton equation is reduced to a new second-order real equation for the amplitude of the soliton. From this real soliton equation, a novel perturbation technique is employed to show that continuous families of solitons always bifurcate out from linear discrete modes in these non-PT-symmetric complex potentials. All analytical results are corroborated by numerical examples.Comment: 23 pages, 4 figure

Stability of soliton families in nonlinear Schroedinger equations with non-parity-time-symmetric complex potentials

Stability of soliton families in one-dimensional nonlinear Schroedinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian-Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian-Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets $(\lambda, -\lambda, \lambda^*, -\lambda^*)$, similar to conservative systems and PT-symmetric systems. This quartet eigenvalue symmetry is very surprising for non-PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.Comment: 9 pages, 6 figure

All-real spectra in optical systems with arbitrary gain and loss distributions

A method for constructing optical potentials with an arbitrary distribution of gain and loss and completely real spectrum is presented. For each arbitrary distribution of gain and loss, several classes of refractive-index profiles with freely tunable parameters are obtained such that the resulting complex potentials, although being non-parity-time-symmetric in general, still feature all-real spectra for a wide range of tuning parameters. When these refractive indices are tuned below certain thresholds, phase transition can occur, where complex-conjugate pairs of eigenvalues appear in the spectrum. These non-parity-time-symmetric complex potentials generalize the concept of parity-time-symmetric potentials to allow for more flexible gain and loss distributions while still maintaining all-real spectra and the phenomenon of phase transition.Comment: 5 pages, 3 figure

Stability analysis for solitons in PT-symmetric optical lattices

Stability of solitons in parity-time (PT)-symmetric periodic potentials (optical lattices) is analyzed in both one- and two-dimensional systems. First we show analytically that when the strength of the gain-loss component in the PT lattice rises above a certain threshold (phase-transition point), an infinite number of linear Bloch bands turn complex simultaneously. Second, we show that while stable families of solitons can exist in PT lattices, increasing the gain-loss component has an overall destabilizing effect on soliton propagation. Specifically, when the gain-loss component increases, the parameter range of stable solitons shrinks as new regions of instability appear. Thirdly, we investigate the nonlinear evolution of unstable PT solitons under perturbations, and show that the energy of perturbed solitons can grow unbounded even though the PT lattice is below the phase transition point.Comment: 11 pages, 10 figures, 1 tabl

JECL: Joint Embedding and Cluster Learning for Image-Text Pairs

We propose JECL, a method for clustering image-caption pairs by training parallel encoders with regularized clustering and alignment objectives, simultaneously learning both representations and cluster assignments. These image-caption pairs arise frequently in high-value applications where structured training data is expensive to produce, but free-text descriptions are common. JECL trains by minimizing the Kullback-Leibler divergence between the distribution of the images and text to that of a combined joint target distribution and optimizing the Jensen-Shannon divergence between the soft cluster assignments of the images and text. Regularizers are also applied to JECL to prevent trivial solutions. Experiments show that JECL outperforms both single-view and multi-view methods on large benchmark image-caption datasets, and is remarkably robust to missing captions and varying data sizes

A deep learning approach to real-time parking occupancy prediction in spatio-temporal networks incorporating multiple spatio-temporal data sources

A deep learning model is applied for predicting block-level parking occupancy in real time. The model leverages Graph-Convolutional Neural Networks (GCNN) to extract the spatial relations of traffic flow in large-scale networks, and utilizes Recurrent Neural Networks (RNN) with Long-Short Term Memory (LSTM) to capture the temporal features. In addition, the model is capable of taking multiple heterogeneously structured traffic data sources as input, such as parking meter transactions, traffic speed, and weather conditions. The model performance is evaluated through a case study in Pittsburgh downtown area. The proposed model outperforms other baseline methods including multi-layer LSTM and Lasso with an average testing MAPE of 10.6\% when predicting block-level parking occupancies 30 minutes in advance. The case study also shows that, in generally, the prediction model works better for business areas than for recreational locations. We found that incorporating traffic speed and weather information can significantly improve the prediction performance. Weather data is particularly useful for improving predicting accuracy in recreational areas