Localized modes in dissipative lattice media: an overview

Higher-dimensional exceptional points and peakon dynamics triggered by spatially varying Kerr nonlinear media and PT δ(x) potentials

Higher-dimensional PT-symmetric potentials constituted by delta-sign-exponential (DSE) functions are created in order to show that the exceptional points in the non-Hermitian Hamiltonian can be converted to those in the corresponding one-dimensional (1D) geometry, no matter the potentials inside are rotationally symmetric or not. These results are first numerically observed and then are proved by mathematical methods. For spatially varying Kerr nonlinearity, 2D exact peakons are explicitly obtained, giving birth to families of stable square peakons in the rotationally symmetric potentials and rhombic peakons in the nonrotationally symmetric potentials. By adiabatic excitation, different types of 2D peakons can be transformed stably and reciprocally. Under periodic and mixed perturbations, the 2D stable peakons can also travel stably along the spatially moving potential well, which implies that it is feasible to manage the propagation of the light by regulating judiciously the potential well. However, the vast majority of high-order vortex peakons are vulnerable to instability, which is demonstrated by the linear-stability analysis and by direct numerical simulations of propagation of peakon waveforms. In addition, 3D exact and numerical peakon solutions including the rotationally symmetric and the nonrotationally symmetric ones are obtained, and we find that incompletely rotationally symmetric peakons can occur stably in completely rotationally symmetric DSE potentials. The 3D fundamental peakons can propagate stably in a certain range of potential parameters, but their stability may get worse with the loss of rotational symmetry. Exceptional points and exact peakons in n dimensions are also summarized.

Stability and modulation of optical peakons in self-focusing/defocusing Kerr nonlinear media with PT-δ-hyperbolic-function potentials

In this paper, we introduce a class of novel PT- δ-hyperbolic-function potentials composed of the Dirac δ(x) and hyperbolic functions, supporting fully real energy spectra in the non-Hermitian Hamiltonian. The threshold curves of PT symmetry breaking are numerically presented. Moreover, in the self-focusing and defocusing Kerr-nonlinear media, the PT-symmetric potentials can also support the stable peakons, keeping the total power and quasi-power conserved. The unstable PT-symmetric peakons can be transformed into other stable peakons by the excitations of potential parameters. Continuous families of additional stable numerical peakons can be produced in internal modes around the exact peakons (even unstable). Further, we find that the stable peakons can always propagate in a robust form, remaining trapped in the slowly moving potential wells, which opens the way for manipulations of optical peakons. Other significant characteristics related to exact peakons, such as the interaction and power flow, are elucidated in detail. These results will be useful in explaining the related physical phenomena and designing the related physical experiments.

Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity

We present an alternative type of parity-time (PT)-symmetric generalized Scarf-II potentials, which makes possible for non-Hermitian Hamiltonians in the classical linear Schrödinger system to possess fully real spectra with unique features such as the multiple PT-symmetric breaking behaviors and to support one-dimensional (1D) stable PT-symmetric solitons of power-law waveform, namely power-law solitons, in focusing Kerr-type nonlinear media. Moreover, PT-symmetric high-order solitons are also derived numerically in 1D and 2D settings. Around the exactly obtained nonlinear propagation constants, families of 1D and 2D localized nonlinear modes are also found numerically. The majority of fundamental nonlinear modes can still keep steady in general, whereas the 1D multipeak solitons and 2D vortex solitons are usually susceptible to suffering from instability. Likewise, similar results occur in the defocusing Kerr-nonlinear media. The obtained results will be useful for understanding the complex dynamics of nonlinear waves that form in PT-symmetric nonlinear media in other physical contexts.

Stable flat-top solitons and peakons in the PT-symmetric δ-signum potentials and nonlinear media

We discover that the physically interesting PT-symmetric Dirac delta-function potentials can not only make sure that the non-Hermitian Hamiltonians admit fully-real linear spectra but also support stable peakons (nonlinear modes) in the Kerr nonlinear Schrödinger equation. For a specific form of the delta-function PT-symmetric potentials, the nonlinear model investigated in this paper is exactly solvable. However, for a class of PT-symmetric signum-function double-well potentials, a novel type of exact flat-top bright solitons can exist stably within a broad range of potential parameters. Intriguingly, the flat-top solitons can be characterized by the finite-order differentiable waveforms and admit the novel features differing from the usual solitons. The excitation features and the direction of transverse power flow of flat-top bright solitons are also explored in detail. These results are useful for the related experimental designs and applications in nonlinear optics and other related fields.

Impact of near-𝒫𝒯 symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model

We theoretically report the influence of a class of near-parity-time-(𝒫𝒯-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter α₂ or nonlinear gain-loss coefficient β₂, we do find stable exact solitons in the second quadrant of the (α₂, β₂) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.

Breatherlike solitons extracted from the Peregrine rogue wave

Based on the Peregrine solution (PS) of the nonlinear Schrödinger (NLS) equation, the evolution of rational fraction pulses surrounded by zero background is investigated. These pulses display the behavior of a breatherlike solitons. We study the generation and evolution of such solitons extracted, by means of the spectral-filtering method, from the PS in the model of the optical fiber with realistic values of coefficients accounting for the anomalous dispersion, Kerr nonlinearity, and higher-order effects. The results demonstrate that the breathing solitons stably propagate in the fibers. Their robustness against small random perturbations applied to the initial background is demonstrated too.

Localized structures in dissipative media: from optics to plant ecology

Localized structures (LSs) in dissipative media appear in various fields of natural science such as biology, chemistry, plant ecology, optics and laser physics. The proposal for this Theme Issue was to gather specialists from various fields of nonlinear science towards a cross-fertilization among active areas of research. This is a cross-disciplinary area of research dominated by nonlinear optics due to potential applications for all-optical control of light, optical storage and information processing. This Theme Issue contains contributions from 18 active groups involved in the LS field and have all made significant contributions in recent years.