Original research articleth-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers
Introduction
Rogue wave, also known as freak wave, monster wave or extreme wave, was initially observed on open oceans to explain a class of special wave with large, unexpected and suddenly appearing surface waves [1], [2]. Thereafter, rogue wave is regarded as a new type of nonlinear phenomena, and is almost ubiquitous in nonlinear evolution systems [3], [4], [5], [6], [7], [8]. Subsequently, a lot of research has been done on optical rogue wave because desktop-based optical experiments have greatly facilitated its exploration [9], [10], [11], [12], [13]. Latest features of optical rogue wave have been revealed in succession. A significant result is that optical rogue wave does not necessarily appear without a warning (namely, the emergence of rogue wave may be predictable), but is often preceded by a short phase of relative order [14]. In addition, optical rogue wave can be used to amplify pulse signal, which is critical for long-distance optical information transmission [15].
It is well-known that nonlinear Schrödinger equations (NLSEs) are the most fundamental models to describe the pulse propagation in nonlinear optical fiber systems [16], [17], [18], [19], [20]. One of the classical NLSEs reads as [21], [22] where represents space–time complex wave envelope in optical fibers, is the longitudinal coordinate, and stands for the time in moving coordinate system, . In general, two independent variables and can be interchanged according to different application scenarios.
In Eq. (1), the coefficients of the dispersion terms and , and the nonlinear Kerr term , are all constants. However, in real optical fiber systems and complex application circumstances, there certainly exist non-uniformities owing to various nonlinear factors and variable environment settings. These non-uniformities often produce nonlinear gain or loss, phase modulation and variable dispersion. As a result, variable coefficient NLSEs (vcNLSEs) were naturally proposed to indicate the inhomogeneous effects in nonlinear optical pulse propagations. The vcNLSE permits ones to reveal more abundant optical wave characteristics under complex conditions [23], [24], [25], [26], [27], [28], [29]. Here, two typical vcNLSEs are taken to illustrate their applications.
Starting with a zero-temperature Bose–Einstein condensates (BECs) of atoms, a generalized one-dimensional nonlinear Schrödinger equation was derived as [30] where, and account for the variable coefficients corresponding to the Kerr nonlinearity, oscillator potential and gain or loss factor, respectively. Due to the involved time-dependent coefficients, Eq. (2) is also classified to nonautonomous models (corresponding to inhomogeneous models contained space-dependent coefficients) [31]. When is set as a constant in Eq. (2), this model can also be used to depict optical pulse in tapered graded-index nonlinear fibers [32], [33].
In this work, we will consider the following generalized inhomogeneous vcNLSE in optical fiber systems with space-dependent variable coefficients [34] where and stand for two types of group velocity dispersion coefficients corresponding to the first- and second-order dispersion terms, respectively; is a function related to the nonlinearity Kerr term.
It is obvious that Eq. (3) can degenerate to a standard vcNLSE as . Further, when , and , Eq. (3) will degenerate to the classical NLSE (1).
There have been many results on Eq. (3). A type of oscillating solitons was studied by the bilinear transformation method [35]. In Ref. [36], the explicit rogue wave solutions were constructed via similarity transformation to the classical NLSE (1), and the first- and second-order rogue waves were excited. The phase shift, oscillation and attenuation of solitons were investigated in Ref. [37]. Several breather structures were observed in Ref. [38].
Darboux transformation (DT) technique is a powerful tool to construct analytical solutions of NLEEs [39], [40], [41], [42], [43]. Its main idea is that, from the Lax pair and initial seed solution of equation, the first-order rouge wave can be obtained firstly, then higher-order solutions can be computed by iterating the lower-order cases again and again. In this paper, the DT iteration algorithm will be performed to construct the th-order rogue wave solution for Eq. (3). Then rogue wave patterns are excited when the variable coefficient functions are set as specific functions.
The rest of the paper is arranged as follows: In Section 2, the solutions of the Lax pair equations are obtained for Eq. (3). In Section 3, the DT is derived for Eq. (3) with a constrain condition, then the th-order rogue wave solutions are constructed via extending the DT. Two types of excited rogue wave patterns are demonstrated graphically in Section 4. Some discussions and conclusions are given in the final section.
Section snippets
Lax pair and its solutions to Eq. (3)
In order to apply the DT iteration algorithm to Eq. (3), it is necessary to get the Lax pair and its corresponding solutions. We constrain and to satisfy the integrability condition as
The Lax pair of Eq. (3) with the condition (4) can be expressed as where with
Choose an initial solution of Eq. (3) as Through computation, we are able to obtain a set of the
One-fold DT and the first-order rogue wave solution
It is easy to verify that and are the solutions of Eqs. (5) corresponding to the initial solution (8) under the conditions and , respectively.
Now, we define the following transformation to Eqs. (5) such that where satisfies with
Excited patterns of rogue waves
Owing to the arbitrariness of the variable dispersion functions and involved in all the rogue wave solutions of Eq. (3), abundant rogue wave patterns can be conveniently excited by properly choosing and . In experiments and applications, these excited rogue waves can be generated by the initial perturbations controlled by the variable dispersion , and Kerr nonlinearity under the integrable condition (4). However, the variable coefficients will be selected
Discussions and conclusions
In this work, a variable coefficient Schrödinger equation was under investigation, which can be used to model the pulse propagation in inhomogeneous optical fiber systems. Through extending DT iteration algorithm, the th-order rogue wave solutions are obtained for the equation for the first time. The solutions are significant to study further characteristics for the equation. Based on the rogue wave solutions, three types of the rogue wave patterns are excited via choosing the variable
Compliance with ethical standards
The authors ensure the compliance with ethical standards for this work.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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