Watch-hand-like optical rogue waves in three-wave interactions

Fermi-Pasta-Ulam-Tsingou recurrence and cascading mechanism for resonant three-wave interactions

Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics ("sidebands"), "regular" and "staggered" FPUT patterns can arise. A "cascading mechanism" provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of "effective energy" is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.

Combined effects of nonparaxiality, optical activity, and walk-off on rogue wave propagation in optical fibers filled with chiral materials

The generalized nonparaxial nonlinear Schrödinger (NLS) equation in optical fibers filled with chiral materials is reduced to the higher-order integrable Hirota equation. Based on the modified Darboux transformation method, the nonparaxial chiral optical rogue waves are constructed from the scalar model with modulated coefficients. We show that the parameters of nonparaxiality, third-order dispersion, and differential gain or loss term are the main keys to control the amplitude, linear, and nonlinear effects in the model. Moreover, the influence of nonparaxiality, optical activity, and walk-off effect are also evidenced under the defocusing and focusing regimes of the vector nonparaxial NLS equations with constant and modulated coefficients. Through an algorithm scheme of wider applicability on nonparaxial beam propagation methods, the most influential effect and the simultaneous controllability of combined effects are underlined, showing their properties and their potential applications in optical fibers and in a variety of complex dynamical systems.

Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber

The resonant interaction of an optical field with two-level doping ions in a cryogenic optical fiber is investigated within the framework of nonlinear Schrödinger and Maxwell-Bloch equations. We present explicit fundamental rational rogue wave solutions in the context of self-induced transparency for the coupled optical and matter waves. It is exhibited that the optical wave component always features a typical Peregrine-like structure, while the matter waves involve more complicated yet spatiotemporally balanced amplitude distribution. The existence and stability of these rogue waves is then confirmed by numerical simulations, and they are shown to be excited amid the onset of modulation instability. These solutions can also be extended, using the same analytical framework, to include higher-order dispersive and nonlinear effects, highlighting their universality.

Generation of spatio-temporal extreme events in noise-like pulses NPE mode-locked fibre laser

We experimentally study spatio-temporal generation of extreme events in the radiation of NPE mode-locked fibre laser generating noise-like pulses. We show that new pulses starts from high-intensity spatio-temporal structure which consist of mainly 3 subsequent pulses which are both separated over fast and slow evolution time. Statistical analysis of the noise-like pulse evolution over round-trips shows that the pulse width and intensity varies with a period of around 85 round-trips which does not change from pulse to pulse. The intensity probability density function has a heavy tail originated only from events of pulse formation.

Semi-rational solutions of the third-type Davey-Stewartson equation

General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers, and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers, and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.

Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers

We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.

Rogue-wave bullets in a composite (2+1)D nonlinear medium

We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.

Complementary optical rogue waves in parametric three-wave mixing

We investigate the resonant interaction of two optical pulses of the same group velocity with a pump pulse of different velocity in a weakly dispersive quadratic medium and report on the complementary rogue wave dynamics which are unique to such a parametric three-wave mixing. Analytic rogue wave solutions up to the second order are explicitly presented and their robustness is confirmed by numerical simulations, in spite of the onset of modulation instability activated by quantum noise.

Integrable pair-transition-coupled nonlinear Schrödinger equations

We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.