An instability criterion for nonlinear standing waves on nonzero backgrounds

A nonlinear Schr\"odinger equation with repulsive (defocusing) nonlinearity is considered. As an example, a system with a spatially varying coefficient of the nonlinear term is studied. The nonlinearity is chosen to be repelling except on a finite interval. Localized standing wave solutions on a non-zero background, e.g., dark solitons trapped by the inhomogeneity, are identified and studied. A novel instability criterion for such states is established through a topological argument. This allows instability to be determined quickly in many cases by considering simple geometric properties of the standing waves as viewed in the composite phase plane. Numerical calculations accompany the analytical results.Comment: 20 pages, 11 figure

Vectorial dissipative solitons in vertical-cavity surface-emitting Lasers with delays

We show that the nonlinear polarization dynamics of a vertical-cavity surface-emitting laser placed into an external cavity leads to the formation of temporal vectorial dissipative solitons. These solitons arise as cycles in the polarization orientation, leaving the total intensity constant. When the cavity round-trip is much longer than their duration, several independent solitons as well as bound states (molecules) may be hosted in the cavity. All these solutions coexist together and with the background solution, i.e. the solution with zero soliton. The theoretical proof of localization is given by the analysis of the Floquet exponents. Finally, we reduce the dynamics to a single delayed equation for the polarization orientation allowing interpreting the vectorial solitons as polarization kinks.Comment: quasi final resubmission version, 12 pages, 9 figure

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model

Dispersive, superfluid-like shock waves in nonlinear optics

In most classical fluids, shock waves are strongly dissipative, their energy being quickly lost through viscous damping. But in systems such as cold plasmas, superfluids, and Bose-Einstein condensates, where viscosity is negligible or non-existent, a fundamentally different type of shock wave can emerge whose behaviour is dominated by dispersion rather than dissipation. Dispersive shock waves are difficult to study experimentally, and analytical solutions to the equations that govern them have only been found in one dimension (1D). By exploiting a well-known, but little appreciated, correspondence between the behaviour of superfluids and nonlinear optical materials, we demonstrate an all-optical experimental platform for studying the dynamics of dispersive shock waves. This enables us to observe the propagation and nonlinear response of dispersive shock waves, including the interaction of colliding shock waves, in 1D and 2D. Our system offers a versatile and more accessible means for exploring superfluid-like and related dispersive phenomena.Comment: 21 pages, 6 figures Revised abstrac

Nonlinearity and Topology

The interplay of nonlinearity and topology results in many novel and emergent properties across a number of physical systems such as chiral magnets, nematic liquid crystals, Bose-Einstein condensates, photonics, high energy physics, etc. It also results in a wide variety of topological defects such as solitons, vortices, skyrmions, merons, hopfions, monopoles to name just a few. Interaction among and collision of these nontrivial defects itself is a topic of great interest. Curvature and underlying geometry also affect the shape, interaction and behavior of these defects. Such properties can be studied using techniques such as, e.g. the Bogomolnyi decomposition. Some applications of this interplay, e.g. in nonreciprocal photonics as well as topological materials such as Dirac and Weyl semimetals, are also elucidated

Exciton-polariton topological insulator

The authors thank R. Thomale for fruitful discussions. S.K. acknowledges the European Commission for the H2020 Marie Skłodowska-Curie Actions (MSCA) fellowship (Topopolis). S.K., S.H. and M.S. are grateful for financial support by the JMU-Technion seed money program. S.H. also acknowledges support by the EPSRC ”Hybrid Polaritonics” Grant (EP/M025330/1). The Würzburg group acknowledges support by the ImPACT Program, Japan Science and Technology Agency and the State of Bavaria. T.C.H.L. and R. G. were supported by the Ministry of Education (Singapore) Grant No. 2017-T2-1-001Topological insulators—materials that are insulating in the bulk but allow electrons to flow on their surface—are striking examples of materials in which topological invariants are manifested in robustness against perturbations such as defects and disorder1. Their most prominent feature is the emergence of edge states at the boundary between areas with different topological properties. The observable physical effect is unidirectional robust transport of these edge states. Topological insulators were originally observed in the integer quantum Hall effect2 (in which conductance is quantized in a strong magnetic field) and subsequently suggested3,4,5 and observed6 to exist without a magnetic field, by virtue of other effects such as strong spin–orbit interaction. These were systems of correlated electrons. During the past decade, the concepts of topological physics have been introduced into other fields, including microwaves7,8, photonic systems9,10, cold atoms11,12, acoustics13,14 and even mechanics15. Recently, topological insulators were suggested to be possible in exciton-polariton systems16,17,18 organized as honeycomb (graphene-like) lattices, under the influence of a magnetic field. Exciton-polaritons are part-light, part-matter quasiparticles that emerge from strong coupling of quantum-well excitons and cavity photons19. Accordingly, the predicted topological effects differ from all those demonstrated thus far. Here we demonstrate experimentally an exciton-polariton topological insulator. Our lattice of coupled semiconductor microcavities is excited non-resonantly by a laser, and an applied magnetic field leads to the unidirectional flow of a polariton wavepacket around the edge of the array. This chiral edge mode is populated by a polariton condensation mechanism. We use scanning imaging techniques in real space and Fourier space to measure photoluminescence and thus visualize the mode as it propagates. We demonstrate that the topological edge mode goes around defects, and that its propagation direction can be reversed by inverting the applied magnetic field. Our exciton-polariton topological insulator paves the way for topological phenomena that involve light–matter interaction, amplification and the interaction of exciton-polaritons as a nonlinear many-body system.PostprintPeer reviewe

Analysis of Temporal Dissipative Solitons in a Delayed Model of a Ring Semiconductor Laser

Temporal dissipative solitons are short pulses observed in periodic time traces of the electric field envelope in active and passive optical cavities. They sit on a stable background, so that their trajectory comes close to a stable steady state solution between the pulses. A common approach to predict and study these solitons theoretically is based on the use of Ginzburg–Landau-type partial differential equations, which, however, cannot adequately describe the dynamics of many realistic laser systems. Here, for the first time, we demonstrate the formation of temporal dissipative soliton solutions in a time-delay model of a ring semiconductor cavity with coherent optical injection, operating in anomalous dispersion regime, and perform bifurcation analysis of these solutions