Characterization of the Distortion-Perception Tradeoff for Finite Channels with Arbitrary Metrics

Whenever inspected by humans, reconstructed signals should not be distinguished from real ones. Typically, such a high perceptual quality comes at the price of high reconstruction error, and vice versa. We study this distortion-perception (DP) tradeoff over finite-alphabet channels, for the Wasserstein-$1$ distance induced by a general metric as the perception index, and an arbitrary distortion matrix. Under this setting, we show that computing the DP function and the optimal reconstructions is equivalent to solving a set of linear programming problems. We provide a structural characterization of the DP tradeoff, where the DP function is piecewise linear in the perception index. We further derive a closed-form expression for the case of binary sources

Single and double linear and nonlinear flatband chains: spectra and modes

We report results of systematic analysis of various modes in the flatband lattice, based on the diamond-chain model with the on-site cubic nonlinearity, and its double version with the linear on-site mixing between the two lattice fields. In the single-chain system, a full analysis is presented, first, for the single nonlinear cell, making it possible to find all stationary states, viz., antisymmetric, symmetric, and asymmetric ones, including an exactly investigated symmetry-breaking bifurcation of the subcritical type. In the nonlinear infinite single-component chain, compact localized states (CLSs) are found in an exact form too, as an extension of known compact eigenstates of the linear diamond chain. Their stability is studied by means of analytical and numerical methods, revealing a nontrivial stability boundary. In addition to the CLSs, various species of extended states and exponentially localized lattice solitons of symmetric and asymmetric types are studied too, by means of numerical calculations and variational approximation. As a result, existence and stability areas are identified for these modes. Finally, the linear version of the double diamond chain is solved in an exact form, producing two split flatbands in the system's spectrum.Comment: Phys. Rev E, in pres

Solitons supported by localized nonlinearities in periodic media

Nonlinear periodic systems, such as photonic crystals and Bose-Einstein condensates (BECs) loaded into optical lattices, are often described by the nonlinear Schr\"odinger/Gross-Pitaevskii equation with a sinusoidal potential. Here, we consider a model based on such a periodic potential, with the nonlinearity (attractive or repulsive) concentrated either at a single point or at a symmetric set of two points, which are represented, respectively, by a single {\delta}-function or a combination of two {\delta}-functions. This model gives rise to ordinary solitons or gap solitons (GSs), which reside, respectively, in the semi-infinite or finite gaps of the system's linear spectrum, being pinned to the {\delta}-functions. Physical realizations of these systems are possible in optics and BEC, using diverse variants of the nonlinearity management. First, we demonstrate that the single {\delta}-function multiplying the nonlinear term supports families of stable regular solitons in the self-attractive case, while a family of solitons supported by the attractive {\delta}-function in the absence of the periodic potential is completely unstable. We also show that the {\delta}-function can support stable GSs in the first finite gap in both the self-attractive and repulsive models. The stability analysis for the GSs in the second finite gap is reported too, for both signs of the nonlinearity. Alongside the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite and first two finite gaps, with the single {\delta}-function positioned at a minimum or maximum of the periodic potential. In the model with the symmetric set of two {\delta}-functions, we study the effect of the spontaneous symmetry breaking of the pinned solitons. Two configurations are considered, with the {\delta}-functions set symmetrically with respect to the minimum or maximum of the potential

Domain walls and vortices in linearly coupled systems

We investigate 1D and 2D radial domain-wall (DW) states in the system of two nonlinear-Schr\"{o}dinger/Gross-Pitaevskii equations, which are coupled by the linear mixing and by the nonlinear XPM (cross-phase-modulation). The system has straightforward applications to two-component Bose-Einstein condensates, and to the bimodal light propagation in nonlinear optics. In the former case, the two components represent different hyperfine atomic states, while in the latter setting they correspond to orthogonal polarizations of light. Conditions guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal states are established, followed by the study of families of the corresponding DW patterns. Approximate analytical solutions for the DWs are found near the point of the symmetry-breaking bifurcation of the CW states. An exact DW solution is produced for ratio 3:1 of the XPM and SPM coefficients. The DWs between flat asymmetric states, which are mirror images to each other, are completely stable, and all other species of the DWs, with zero crossings in one or two components, are fully unstable. Interactions between two DWs are considered too, and an effective potential accounting for the attraction between them is derived analytically. Direct simulations demonstrate merger and annihilation of the interacting DWs. The analysis is extended for the system including single- and double-peak external potentials. Generic solutions for trapped DWs are obtained in a numerical form, and their stability is investigated. An exact stable solution is found for the DW trapped by a single-peak potential. In the 2D geometry, stable two-component vortices are found, with topological charges s=1,2,3. Radial oscillations of annular DW-shaped pulsons, with s=0,1,2, are studied too. A linear relation between the period of the oscillations and the mean radius of the DW ring is derived analytically.Comment: Physical Review E, in pres

Long-ranged attraction between disordered heterogeneous surfaces

Long-ranged attractions across water between two surfaces that are randomly covered with (mobile) positive and negative charge domains have been attributed to induced correlation of the charges (positive lining up with negative) as the surfaces approach. Here we show, by directly measuring normal forces under a rapid shear field, that these attractions may not in fact be due to such correlations. It is rather the inherent interaction-asymmetry between equally- and between oppositely-charged domains that results in the long-ranged attraction even in the complete absence of any charge correlation