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Specialty section: This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the

We present exact explicit Peregrine soliton solutions based on a periodic-wave background caused by the interference in the vector cubic-quintic nonlinear Schrödinger equation involving the self-steepening effect. It is shown that such periodic Peregrine soliton solutions can be expressed as a linear superposition of two fundamental Peregrine solitons of different continuous-wave backgrounds. Because of the self-steepening effect, some interesting Peregrine soliton dynamics such as ultrastrong amplitude enhancement and rogue wave coexistence are still present when they are built on a periodic background. We numerically confirm the stability of these analytical solutions against non-integrable perturbations, i.e., when the coefficient relation that enables the integrability of the vector model is slightly lifted. We also demonstrate the interaction of two Peregrine solitons on the same periodic background under some specific parameter conditions. We expect that these results may shed more light on our understanding of the realistic rogue wave behaviors occurring in either the fiber-optic telecommunication links or the crossing seas.

Originally, rogue waves refer to the surface gravity waves occurring in the open ocean whose wave heights are at least twice as high as the significant wave height of the surrounding waves [

Despite the extensive studies, there is still a lot of debate over the physical mechanisms behind rogue waves [

Mathematically, one can associate rogue waves to the rational solutions of the integrable nonlinear wave equation, which are localized on both time and space [

Recently, there has also been an intense research on the so-called periodic Peregrine soliton, by which we mean a Peregrine soliton formed on a periodic background [

In this paper, we present an in-depth study of the formation of Peregrine solitons on a periodic background, within the framework of the vector cubic-quintic NLS (CQ-NLS) equation, which is a two-component version of the scalar NLS-type Gerdjikov–Ivanov equation [

In the context of fiber optics, we write the vector CQ-NLS equation as

Obviously, the above vector system could be reproduced from the compatibility condition,

For our present purposes, we are merely concerned with the fundamental rogue wave solutions, which evolve directly from the MI of continuous wave fields. It is easily shown that the initial plane-wave solutions

Let us now consider the special case where the quartic

Further, we find that when the parameter conditions given by

For given initial parameters, our analytical solutions

First of all, it is obvious that the periodic Peregrine soliton solutions of the vector CQ-NLS equation can be generally expressed as a linear superposition of two fundamental Peregrine solitons of different cw backgrounds, provided that the continuous waves involve a nonvanishing frequency difference. In fact, as one might check, when the frequency difference meets

Peregrine soliton states formed on

Further, we find that the Peregrine solitons formed will possess the following enhancement factors, relative to the average amplitude,

However, there is more to our story intended for the vector CQ-NLS system, which involves the self-steepening effect denoted by the parameter γ. It is found that due to the presence of the self-steepening effect, the enhancement factors of periodic Peregrine solitons, defined by

Peregrine soliton states on a periodic background, with

Of most concern is the case of the combination of negative cubic nonlinearity and positive quintic nonlinearity in our vector model, which admits the existence of periodic Peregrine solitons as well. One may recall that such a competing nonlinearity can often be used to support the formation of stable dissipative solitons in mode-locked fiber lasers [

Peregrine solitons on a periodic background formed in the normal dispersion regime (

Now a natural question arises as to whether these periodic-background Peregrine soliton solutions are robust against numerical noises or even against strong “non-integrable” perturbations by which we mean that the specific relation between the coefficients for quintic nonlinearity and self-steepening terms can be lifted. To answer this question, we perform extensive numerical simulations with respect to our analytical solutions (

Typical simulations of the periodic Peregrine soliton solutions (

Finally, we would like to point out that our solution form defined by

Interaction of two Peregrine solitons on a periodic background, under the same initial parameters as in

In conclusion, we presented exact Peregrine soliton solutions built on a periodic background caused by the interference in the vector CQ-NLS equation involving self-steepening. It is revealed that such periodic Peregrine soliton solutions are indeed a linear superposition of two fundamental Peregrine solitons of different cw backgrounds, provided that the continuous waves possess a nonvanishing frequency difference. With these exact solutions, we demonstrated the coexistence of Peregrine solitons on the same periodic background, under certain parameter conditions. Further, the ultrastrong amplitude enhancement was proved to occur on the periodic background as well, due to the presence of the self-steepening effect. We numerically confirm the stability of these analytical solutions against significant non-integrable perturbations. We also showed the interaction of two Peregrine solitons on the periodic background, which are still a linear superposition of those on the cw background. Basically, such simple superposition rule can be applied to the higher-order rogue wave hierarchy on a periodic background. As one might expect, these findings may shed more light on our understanding of the realistic rogue wave behaviors occurring in either the fiber-optic telecommunication links [

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding authors.

YY, LB, and WW performed the derivations and plotted the figures. SC, FB, and DM proposed the theoretical framework, performed simulations, and wrote and revised the manuscript. All authors contributed to the article and approved the submitted version.

This work was supported by the National Natural Science Foundation of China (Grants No. 11474051 and No. 11974075) and by the Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1872).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.