Growth rate of modulation instability driven by superregular breathers

Fundamental and Second-Order Superregular Breathers in Vector Fields

We developed an exact theory of the superregular breathers (SRBs) of Manakov equations. We have shown that the vector SRBs do exist both in the cases of focusing and defocusing Manakov systems. The theory is based on the eigenvalue analysis and on finding the exact links between the SRBs and modulation instability. We have shown that in the focusing case the localized periodic initial modulation of the plane wave may excite both a single SRB and the second-order SRBs involving four fundamental breathers.

Exact Analytic Spectra of Asymmetric Modulation Instability in Systems with Self-Steepening Effect

Nonlinear waves become asymmetric when asymmetric physical effects are present within the system. One example is the self-steepening effect. When exactly balanced with dispersion, it leads to a fully integrable system governed by the Chen-Lee-Liu equation. The latter provides a natural basis for the analysis of asymmetric wave dynamics just as nonlinear Schrödinger or Korteweg-de Vries equations provide the basis for analyzing solitons with symmetric profile. In this work, we found periodic wave trains of the Chen-Lee-Liu equation evolved from fully developed modulation instability and analyzed a highly nontrivial spectral evolution of such waves in analytic form that shows strong asymmetry of its components. We present the conceptual basis for finding such spectra that can be used in analyzing asymmetric nonlinear waves in other systems.

Modified linear stability analysis for quantitative dynamics of a perturbed plane wave

We develop linear stability analysis (LSA) to quantitatively predict the dynamics of a perturbed plane wave in nonlinear systems. We take a nonintegrable fiber model with purely fourth-order dispersion as an example to demonstrate this method's effectiveness. For a Gaussian-type initial perturbation with cosine-type modulation on a plane wave, its propagation velocities, periodicity, and localization are predicted successfully, and the range of application is discussed. Importantly, the modulation-instability-induced growth of localized perturbation is proved different from the one of purely periodic perturbation and requires the modification of gain value for more accurate prediction. The method offers a needful supplement and improvement for LSA and paves a way to study the dynamics of a perturbed plane wave in more practical nonlinear systems.

Super-regular breathers in nonlinear systems with self-steepening effect

A family of super-regular (SR) breather solutions in systems with self-steepening effect and in the case of either normal or anomalous dispersion is derived analytically. Derivation is based on the Darboux transformation with a quadratic spectral parameter. In contrast to the SR breather solutions in t-symmetric systems such as the nonlinear Schrödinger equation, the new breathers found in the present work evolve asymmetrically even if started from symmetric initial conditions. The initial stage of this process is modulation instability. Numerical simulations confirm the excitation of the SR breathers when started from the approximate initial conditions leading at first to modulation instability. Our results offer the possibility of experimental observations of SR breather dynamics in systems with self-steepening effects, such as optical frequency-doubling crystals or magnetized plasmas.

Dynamics of perturbations at the critical points between modulation instability and stability regimes

We study numerically the evolutions of perturbations at critical points between modulational instability and stability regimes. It is demonstrated that W-shaped solitons and rogue waves can be both excited from weak resonant perturbations at the critical points. The rogue wave excitation at the critical points indicates that rogue wave comes from modulation instability with resonant perturbations, even when the baseband modulational instability is absent. The perturbation differences for generating W-shaped solitons and rogue waves are discussed in detail. These results can be used to generate W-shaped solitons and rogue waves controllably from weak perturbations.

Excitation conditions of several fundamental nonlinear waves on continuous-wave background

We study the excitation conditions of antidark solitons and nonrational W-shaped solitons in a nonlinear fiber with both third-order and fourth-order effects. We show that the relative phase can be used to distinguish antidark solitons and nonrational W-shaped solitons. The excitation conditions of these well-known fundamental nonlinear waves (on a continuous-wave background) can be clarified clearly by the relative phase and three previously reported parameters (background frequency, perturbation frequency, and perturbation energy). Moreover, the numerical simulations from the nonideal initial states also support these theoretical results. These results provide an important complement for the studies on relationship between modulation instability and nonlinear wave excitations, and are helpful for controllable nonlinear excitations in experiments.