A complex short pulse system in optical-fiber communications: Rogue waves and phase transitions

doi:10.1016/j.aml.2022.108399

Applied Mathematics Letters

Volume 135, January 2023, 108399

https://doi.org/10.1016/j.aml.2022.108399 Get rights and content

Abstract

In this article, we investigate a complex short pulse system (CSPS) in optical-fiber communications with fast varying packets. Via the Darboux transformation technique, new higher-order rogue wave solutions are constructed for the system. Through controlling the parameter, novel and phase transitions from single-peak to multi-ring-folded rogue waves are observed. The results are helpful to understand the formation mechanism of rogue wave compressions.

Introduction

It is well-known that a variety of nonlinear wave models play important roles in modern science, technology and engineering [1], [2], [3], [4]. Nonlinear Schrödinger equations (NLSEs) and their various variants have been the most fundamental mathematical models, especially in nonlinear optics and fibers [5], [6], [7]. However, NLSEs are not available in new ultra-fast fiber system with fast varying packets (FVPs) [8], [9]. Therefore, it has been becoming necessary to find a more accurate model to describe the propagation properties of ultra-short pulse fibers with FVPs.

Fortunately, a new mathematical model, namely short pulse system (SPS), was proposed to describe the propagation of ultra-short pulse in optical-fiber communications by a better approximation to the solution of the Maxwell equation than NLSEs [10], [11]. SPS reads as $q_{x t} = q + \frac{1}{6} {(q^{3})}_{x x},$ where $q = q (x, t)$ is a real-valued function, and represents the electric pulse magnitude.

Based on SPS (1), a complex short pulse system (CSPS) in optical-fiber communications has been established to model a short pulse fiber with the FVPs

Based on SPS (1) and consideration of propagation of optical pulses in complex field, a complex short pulse system (CSPS) in optical-fiber communications has been established to model a short pulse fiber with FVPs which reads [12], [13] $q_{x t} + q + \frac{1}{2} {({| q |}^{2} q_{x})}_{x} = 0,$ where $q = q (x, t)$ is a complex-valued function, and represents the electric pulse magnitude in complex field.

Via compared with SPS (1) and CSPS (2), we find out that CSPS (2) is more suitable to manipulate multiple wave interactions because of the pulse waves’ complex representation. Furthermore, CSPS (2) possesses an excellent feature: it is integrable in sense of Lax integrability [14]. Stemming from its integrability, some interesting researches have been achieved, e.g., multi-soliton, multi-breather and higher-order rogue wave solutions [14], multi-soliton interactions [15], periodic solutions and soliton solutions [16].

Taking into account the importance of CSPS in short pulse optical-fiber communications with FVPs, a further study on the system becomes necessary. So far, certain particular properties for CSPS (2), e.g., multiple-value, multiple-ring, connection between ring number and order number, has still remained unknown. In this work, we will explore new rogue wave solutions and their properties for CSPS (2) by employing Darboux transformation (DT) technique.

Section snippets

The new higher-order rogue wave solutions for CSPS (2)

In order to construct the rogue wave solutions of Eq. (2), we first introduce the following transformations to Eq. (2) $ρ^{- 1} = \sqrt{1 + {| q_{x} |}^{2}}, d y = ρ^{- 1} d x - \frac{1}{2} ρ^{- 1} {| q |}^{2} d t, d s = - d t,$ where $y$ and $s$ are two new dependent variables, $ρ$ is a new independent variable.

It is seen that, the following equations $q_{y s} = ρ q, ρ_{s} + \frac{1}{2} {({| q |}^{2})}_{y} = 0,$ and Eq. (2) are equivalent. Next, our attention will be focused on finding the rogue wave solutions of Eqs. (4).

From Ref. [14], we get the Lax pair of Eqs. (4) as $Ψ_{y} = U Ψ, Ψ_{s} = V Ψ,$ where $U = - \frac{1}{λ} σ_{1} (i ρ I + Q_{y}), V = \frac{1}{4} i λ$

The phase transition of the rogue waves

In general, phase can represent physical state at different time, and perform certain critical actions [22], [23], [24]. In this section, we study the phase transition of the rogue waves for the system (2).

For all the solutions (36), (37), (38) obtained newly, we notice that there are two parameters $β$ and $γ$ . But $β$ and $γ$ must satisfy the condition (13). This means that only one of $β$ and $γ$ will work. Here, we take $γ$ and the third-order rogue wave solution to observe the phase transitions.

Discussions and conclusions

In this article, we first obtain the first-, second- and third-order rogue wave solutions to CSPS (2) via the DT technique. Then, we study the phase transitions and multi-ring-folded effect of the rogue wave via adjusting the parameter involved in the solutions.

It is noticeable that there are two differences between the rogue wave solutions obtained in Ref. [14] and our new solutions (36), (37), (38) in this work: (i) The constraint to the parameter $γ$ determined by the conditions (13) is

Compliance with ethical standards

The authors ensure the compliance with ethical standards for this work.

CRediT authorship contribution statement

Bang-Qing Li: Data check, Visualization, Software, Writing – review & editing. Yu-Lan Ma: Conceptualization, Methodology, Writing – original draft.

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A complex short pulse system in optical-fiber communications: Rogue waves and phase transitions

Abstract

Introduction

Section snippets

The new higher-order rogue wave solutions for CSPS (2)

The phase transition of the rogue waves

Discussions and conclusions

Compliance with ethical standards

CRediT authorship contribution statement

Appl. Math. Lett.

Chaos Solitons Fract.

Chin. J. Phys.

Phys. Rep.-Rev. Sect. Phys. Lett.

Phys. Rep.

Physica D

Physica D

Commun. Nonlinear Sci. Numer. Simul.

Optik

Commun. Nonlinear Sci. Numer. Simul.

Comput. Math. Appl.

Appl. Math. Comput.