We study the nonlinear waves on constant backgrounds of the higher-order generalized nonlinear Schrödinger (HGNLS) equation describing the propagation of ultrashort optical pulse in optical fibers. We derive the breather, rogue wave, and semirational solutions of the HGNLS equation. Our results show that these three types of solutions can be converted into the nonpulsating soliton solutions. In particular, we present the explicit conditions for the transitions between breathers and solitons with different structures. Further, we investigate the characteristics of the collisions between the soliton and breathers. Especially, based on the semirational solutions of the HGNLS equation, we display the novel interactions between the rogue waves and other nonlinear waves. In addition, we reveal the explicit relation between the transition and the distribution characteristics of the modulation instability growth rate.

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and $\mathsf{W}$-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and $\mathsf{W}$-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.

We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov’s theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.

In this paper, the dynamics of transformed nonlinear waves in the (2+1)-dimensional Ito equation are studied by virtue of the analysis of characteristic line and phase shift. First, the N-soliton solution is obtained via the Hirota bilinear method, from which the breath-wave solution is derived by changing values of wave numbers into complex forms. Then, the transition condition for the breath waves is obtained analytically. We show that the breath waves can be transformed into various nonlinear wave structures including the multi-peak soliton, M-shaped soliton, quasi-anti-dark soliton, three types of quasi-periodic waves, and W-shaped soliton. The correspondence of the phase diagram for such nonlinear waves on the wave number plane is presented. The gradient property of the transformed solution is discussed through the wave number ratio. We study the mechanism of wave formation by analyzing the nonlinear superposition between a solitary wave component and a periodic wave component with different phases. The locality and oscillation of transformed waves can also be explained by the superposition mechanism. Furthermore, the time-varying characteristics of high-dimensional transformed waves are investigated by analyzing the geometric properties (angle and distance) of two characteristic lines of waves, which do not exist in (1+1)-dimensional systems. Based on the high-order breath-wave solutions, the interactions between those transformed nonlinear waves are investigated, such as the completely elastic mode, semi-elastic mode, inelastic mode, and collision-free mode. We reveal that the diversity of transformed waves, time-varying property, and shape-changed collision mainly appear as a result of the difference of phase shifts of the solitary wave and periodic wave components. Such phase shifts come from the time evolution as well as the collisions. Finally, the dynamics of the double shape-changed collisions are presented.