Feng-Hua Qi

Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation.

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.

Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers

In this paper, the nonautonomous Lenells-Fokas (LF) model is investigated. The modulational instability analysis of the solutions with variable coefficients in the presence of a small perturbation is studied. Higher-order soliton, breather, earthwormon, and rogue wave solutions of the nonautonomous LF model are derived via the n-fold variable-coefficient Darboux transformation. The solitons and earthwormons display the elastic collisions. It is found that the nonautonomous LF model admits the higher-order periodic rogue waves, composite rogue waves (rogue wave pair), and oscillating rogue waves, whose dynamics can be controlled by the inhomogeneous nonlinear parameters. Based on the second-order rogue wave, a diamond structure consisting of four first-order rogue waves is observed. In addition, the semirational solutions (the mixed rational-exponential solutions) of the nonautonomous LF model are obtained, which can be used to describe the interactions between the rogue waves and breathers. Our results could be helpful for the design of experiments in the optical fiber communications.

Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects.

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 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 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.

Asymmetric Rogue Waves, Breather-to-Soliton Conversion, and Nonlinear Wave Interactions in the Hirota–Maxwell–Bloch System

We study the nonlinear localized waves on constant backgrounds of the Hirota–Maxwell–Bloch (HMB) system arising from the erbium doped fibers. We derive the asymmetric breather, rogue wave (RW) and semirational solutions of the HMB system. We show that the breather and RW solutions can be converted into various soliton solutions. Under different conditions of parameters, we calculate the locus of the eigenvalues on the complex plane which converts the breathers or RWs into solitons. Based on the second-order solutions, we investigate the interactions among different types of nonlinear waves including the breathers, RWs and solitons.We study the nonlinear localized waves on constant backgrounds of the Hirota–Maxwell–Bloch (HMB) system arising from the erbium doped fibers. We derive the asymmetric breather, rogue wave (RW) and semirational solutions of the HMB system. We show that the breather and RW solutions can be converted into various soliton solutions. Under different conditions of parameters, we calculate the locus of the eigenvalues on the complex plane which converts the breathers or RWs into solitons. Based on the second-order solutions, we investigate the interactions among different types of nonlinear waves including the breathers, RWs and solitons.

Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schrödinger equation

Abstract A variable-coefficient resonant nonlinear Schrodinger (vc-RNLS) equation is considered in this paper. Binary Bell polynomials are employed to obtain the bilinear form and multi-soliton solutions under the integrable conditions. Four types of nonautonomous solitons are derived including the parabolic soliton, compressed soliton, phase-shifted soliton and periodic soliton. Propagation dynamics for each type is analyzed in detail. Nonautonomous resonant and intermediate-state soliton interactions are found to be existent under certain conditions. Specially, periodic soliton interactions are discussed, which shows that the periodic dispersion has no effect on the generation of resonance and intermediate-state solitons. Those analysis might have the applications in optical communication systems with the black hole physics flavor.

Stationary nonlinear waves, superposition modes and modulational instability characteristics in the AB system

We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multi-peak soliton, M-shaped soliton, W-shaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the nonpropagating characteristics of the solitons: (1) Two antidark solitons can produce a W-shaped soliton instead of a higher-order antidark one; (2) the interaction between an antidark soliton and a W-shaped soliton can not only generate a higher-order antidark soliton, but also form a W-shaped soliton pair; and (3) the interactions between an oscillation W-shaped soliton and an oscillation M-shaped soliton show the multi-peak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.

Modulation instability and two types of non-autonomous rogue waves for the variable-coefficient AB system in fluid mechanics and nonlinear optics

Under investigation in this paper is a variable-coefficient AB (vcAB) system, which describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. The modulation instability analysis of solutions with variable coefficients in the presence of a small perturbation is studied. The modified Darboux transformation (mDT) of the vcAB system is constructed via a gauge transformation. The first-order non-autonomous rogue wave solutions of the vcAB system are presented based on the mDT. It is found that the wave amplitude of B exhibits two types of structures, i.e. the double-peak structure appears if the plane-wave solution parameter ω is equal to zero, while selecting ω≠0 yields a single-peak one. Effects of the variable coefficients on the rogue waves are graphically discussed in detail. The periodic rogue wave and composite rogue wave are obtained with different inhomogeneous parameters. Additionally, the nonlinear tunneling of the rogue waves through a c...