$N$th-order rogue wave solutions for a variable coefficient Schrödinger equation in inhomogeneous optical fibers

Highlights

• In investigation is a nonlinear Schrodinger equation with variable coefficients.

• $N$th-order rogue wave solutions are obtained by Darboux Transformation method.

• Novel rogue waves can be excited by choosing specific functions in the solutions.

Abstract

In investigation is a nonlinear Schrödinger equation with variable coefficients, which can describe pulse propagation in inhomogeneous optical fiber systems. The variable coefficients can used to control the dispersion and nonlinearity factors. Through extending Darboux transformation iteration algorithm, analytical $N$th-order rogue wave solutions are obtained for the equation. Based on the two free variable coefficient function involved in the solutions, novel rogue waves can be excited by choosing them as specific functions. Polynomial and periodic functions are taken as examples to how to excite rich rogue wave solutions. The results show that as the orders of the rogue wave solutions increase, the spatio-temporal patterns become more complex, while the amplitudes also increase rapidly. These properties will play critical roles in high-capacity information transmission and signal amplification in optical systems.

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]

$$i\frac{\partial u}{\partial x}+\frac{1}{2}\frac{{\partial }^{2}u}{\partial t^2}+{\left|u\right|}^{2}u=0,$$

where $u(x,t)$ represents space–time complex wave envelope in optical fibers, $x$ is the longitudinal coordinate, and $t$ stands for the time in moving coordinate system, $i=\sqrt{-1}$. In general, two independent variables $x$ and $t$ can be interchanged according to different application scenarios.

In Eq. (1), the coefficients of the dispersion terms $\frac{\partial u}{\partial x}$ and $\frac{{\partial }^{2}u}{\partial t^2}$, and the nonlinear Kerr term ${\left|u\right|}^{2}u$, 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]

$$i\frac{\partial u}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}u}{\partial x^2}-\gamma \left(t\right){\left|u\right|}^{2}u-\frac{1}{2}M\left(t\right)x^{2}u-\frac{1}{2}ig\left(t\right)u=0,$$

where, $\gamma \left(t\right),M\left(t\right)$ and $g\left(t\right)$ 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 $\gamma \left(t\right)$ 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]

$$i\frac{\partial u}{\partial x}+i{\beta }_1\left(x\right)\frac{\partial u}{\partial t}+{\beta }_2\left(x\right)\frac{{\partial }^{2}u}{\partial t^2}+\gamma \left(x\right){\left|u\right|}^{2}u=0,$$

where ${\beta }_1\left(x\right)$ and ${\beta }_2\left(x\right)$ stand for two types of group velocity dispersion coefficients corresponding to the first- and second-order dispersion terms, respectively; $\gamma \left(x\right)$ is a function related to the nonlinearity Kerr term.

It is obvious that Eq. (3) can degenerate to a standard vcNLSE as ${\beta }_1\left(x\right)=0$. Further, when ${\beta }_1\left(x\right)=0$, ${\beta }_2\left(x\right)=\frac{1}{2}$ and $\gamma =1$, 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 $N$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 $N$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.