State transition induced by higher-order effects and background frequency

Abstract

State transition between the Peregrine rogue wave and w-shaped traveling wave induced by higherorder effects and background frequency is studied. We find that this intriguing transition, described by an exact explicit rational solution, is consistent with the modulation instability (MI) analysis that involves MI region and stability region in low perturbation frequency region. In particular, the link between the MI growth rate and transition characteristic analytically demonstrates that, the size characteristic of transition is positively associated with the reciprocal of zero-frequency growth rate. Further, we investigate the case for nonlinear interplay of multi-localized waves. It is interesting that the interaction of second-order waves in stability region features a line structure, rather than an elastic interaction between two w-shaped traveling waves. [7,8] of weakly modulated plane wave. More specifically, rogue waves exist only in the MI subregion where the instability frequency band is close to the vanishing frequency [9]. In this regard, rogue wave has a consistent structure in the standard nonlinear Schrödinger equation (NLSE) where the MI characteristic remains invariant in nature. Nevertheless, the MI often exhibits some interesting features when the additional physical effects are taken into consideration, such as cross-phase modulation [10, 11], higher-order perturbation terms [12], etc. Thus, it is important to study the rogue wave property induced by the features of MI growth rate distribution.Recent studies demonstrate that rogue wave can exhibit structural diversity beyond the reach of the standard NLSE in presence of higher-order effects [13][14][15][16][17][18][19][20][21][22][23][24]. However, to our knowledge, less attention has been paid to analyzing rogue wave property in combination with the distribution characteristic of corresponding MI growth rate. Here we find that, with some higher-order perturbation terms (the third-order dispersion and delayed nonlinear response term), the MI growth rate shows a non-uniform distribution characteristic in low perturbation frequency region, in particular, it opens up a stability region as background frequency changes (see Fig. 1). Thus we anticipate that there will be some interesting physical properties as a rogue wave evolves and approaches the stability region.As a starting point, we address the problem via a completely integrable higher-order NLSE-the Hirota equa- * Electronic address: zyyang@nwu.edu.cn † Electronic address: zhaolichen3@163.com tion (HE) [25]-which involves the higher-order perturbation effects. In dimensionless form, the HE readswhere z is the propagation variable, t is the retarded time in a moving frame with the group velocity, and E(t, z) is the slowly varying envelope of the wave field. The real parameter β is introduced to be responsible for the thirdorder dispersion and delayed nonlinear response term, respectively. If β = 0, it reduces to the standard NLSE. The existence of rogue waves in the HE has been recently dem...

W-shaped soliton complexes and rogue-wave pattern transitions for the AB system

“…8(b), a stable broad spectrum of the M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT W-shaped soliton by letting b → 0 is presented. This is consistent with spectrum properties of the state transitions induced by the higher-order effects in [29].…”

“…Additionally, recent hydrodynamical experiments and theoretical research confirm that the "super rogue waves" could be simulated by higher-order rational solutions [15][16][17][18][19][20][21][22][23][24][25][26][27]. More recently, with resort to the modulation instability (MI) analysis, state transition between brether/rogue wave and W-shaped soliton in high-order system, and state transition among different types of rogue-wave patterns in coupled system have been strictly demonstrated [28][29][30].…”

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

“…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. For the Sasa-Satsuma equation with the third-order dispersion, self-steepening and delayed nonlinear response, the high-order effects could make the rogue waves twisted [47,48].…”

“…The striking process involves both MI and modulation stability (MS) regimes, in contrast to the rogue waves and W-shaped soliton reported before which involve MI and MS [49], respectively. Indeed, with certain higher-order perturbation terms such as the third-order dispersion and delayed nonlinear response term, the MI growth rate shows a non-uniform distribution characteristic in the low perturbation frequency region, in particular, it opens up a stability region as the background frequency changes [45,46].…”

Rogue waves and W-shaped solitons in the multiple self-induced transparency system

“…Manakov system and coupled Hirota equations) with interacting wave components of different amplitudes and frequencies [26][27][28] have been of continued interest. Most recently, Akhmediev and one of the authors Liu show that the higher-order integrable perturbations and coupled-wave interaction can indeed generate state transition effects from breather/rogue wave to W-shaped soliton [29][30][31][32].…”