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

Abstract

We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higherorder effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multi-peak solitons, antidark soliton, periodic wave and W-shaped soliton. 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 multi-peak 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 multi-peak soliton. We further derive the breather multiple births by using multiple compression points of Akhmediev breathers in optical fiber systems with periodic dispersion. The number of ABs depends on the amplitude of the modulation but not on its wavelength, which affects their separation distance. In the limiting case, the breather multiple births reduce to the Peregrine combs. We discuss the effects of TOD coefficient on the spatiotemporal characteristics of Peregrine combs. When the amplitude of the modulation is equal to 1, the Peregrine comb is converted into a Peregrine wall that can be seen as intermediate state between rogue wave and W-shaped soliton. 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.

Numerical Investigation of the Dynamics of ‘Hot Spots’ as Models of Dissipative Rogue Waves

“…In most cases, the gain was balanced by a background loss. Although ingenious transformations can provide analytical models of rogue waves, one drawback in performing such transformations is that these variable coefficients may have to fulfill certain conditions [30][31][32][33][34]. As a result, the exact form of the gain cannot be arbitrary.…”

“…Theoretically, Equation 1is a variable coefficient nonlinear Schrödinger equation which has been studied extensively in the literature [30][31][32][33]. Most works focus on the case where the variable coefficients are functions of time (t) only [30], while some concentrate on the case of spatial dependence (x) alone [32].…”

Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects

“…In this section, we shall study the characteristics of superregular breathers induced by higherorder effects, which could describe the nonlinear stage of MI in the presence of higher-order effects. As shown in [41][42][43][44][45][46], the ABs, KMBs or rogue waves can be converted into the stable solitons on constant backgrounds in the higher-order NLS equations, which does not have an analogue in the standard NLS equation. The linear stability analysis indicates that such conversions are strictly associated with the MI analysis that involves an MI region and a stability region, i.e.…”

“…Moreover, Akhmediev and co-workers [41,42] have shown that a breather solution of the third-and fifth-order equations can be converted into a non-pulsating soliton solution that does not have an analogue in the standard NLS equation. Wang et al [43,44] have found that the breather solutions of the fourth-order NLS and variable-coefficient Hirota equations can be converted into four types of nonlinear waves on constant backgrounds, including the multi-peak solitons, antidark soliton, periodic wave and W-shaped soliton. Liu et al [45,46] have discovered that state transitions between the Peregrine rogue wave and W-shaped travelling wave in the Hirota as well as the coupled Hirota equations appear as a result of higher-order effects.…”

Nonautonomous Motion Study on Accelerated and Decelerated Lump Waves for a (3 + 1)‐Dimensional Generalized Shallow Water Wave Equation with Variable Coefficients

“…Interaction solutions for a reduced extended equation (1) equation have been analyzed in Reference [12]. With the inhomogeneities of the media and nonuniformities of the boundaries considered, the variable-coe cient models can often describe more realistic wave propagations in various physical scenes [34][35][36]. Note that the previous studies are mainly focused on the dynamics of lump waves in the constant-coe cient JM-like equations [5][6][7][8][9][10][11][12][13].…”