Controllable optical rogue waves via nonlinearity management

Zhengping Yang; Wei-Ping Zhong; Milivoj Belić; Yiqi Zhang

doi:10.1364/OE.26.007587

1. Introduction

A rogue wave (or a freak wave) is a localized wave with much higher amplitude than the surrounding waves. It may appear suddenly and disappear without a trace, and it could be highly destructive [1]. Rogue waves have been studied extensively [2–4], as their understanding may be essential for reducing the travel risk on high seas. The concept of rogue waves introduced a unifying idea that can be applied to a variety of extreme wave phenomena in different fields, such as hydrodynamics, plasma physics, and Bose-Einstein condensates [1]. Among different kinds of rogue waves, the optical rogue wave occupies a special place because of its wide potential applications, which have significantly enriched the original concept. In 2007, rogue waves were observed in nonlinear optical fibers, effectively announcing the birth of the field of optical rogue waves [5].

It should be noted that the models describing these physical systems are most often based on the nonlinear Schrödinger (NLS) equation and its variants, with the rogue waves representing the rational solutions [6]. In addition, other mathematical models such as the NLS equation with variable coefficients [7], the derivative NLS equation [8], Hirota’s equation [9,10], Davey-Stewartson equation [11], and other, have also been shown to possess rogue wave solutions. By studying rogue waves in different fields, two common characteristics have been observed: the wave amplitude is at least twice the amplitude of surrounding waves and the localization occurs in both the temporal and spatial domains.

The cause of the emergence of rogue waves is usually considered to be the process of nonlinear modulation instability [12, 13]. Mathematically, modulation instability – also known as the Benjamin-Feir instability – can be described as the instability of nonlinear plane wave modes of NLS equation [14] against weak long-scale perturbations with wavenumbers below some critical value. The localized solutions on the plane wave background can be divided into Akhmediev breathers (or solitons) [15], Kuzenetsov-Ma breathers [16, 17] and Peregrine solitons [18]. The Peregrine soliton is a localized pulse in both space and time, and can be viewed as the limiting case of Kuzenetsov-Ma [17] and Akhmediev breathers [19]. The breathers are localized in one direction and periodic in the other direction. These three types of waves can be considered as the basic first-order rogue waves, based on which the higher-order rogue waves can be constructed. Rogue waves have attracted much interest in recent years; the latest results in the field have been summarized and the next research directions outlined in [20–22]. It should be mentioned that the research on dark rogue waves is also rapidly advancing [23, 24].

The above mentioned exact solutions provide controllable initial excitation conditions for the realization of rogue waves. Of the current physical systems of interest, nonlinear optical fibers represent the mature experimental platform for observing optical rogue waves [25–30]. In 2009, Dudley et al. realized experimentally the Akhmediev breather [25]. In 2010, Kibler et al. confirmed the existence of Peregrine rogue wave by using the frequency-resolved optical gating technique [26]. In more detailed experimental observations, Hammani et al. revealed the spectral evolution of the growth and attenuation of Akhmediev breathers [27]. In addition, the Kuznetsov-Ma breather was also confirmed experimentally in [28]. It should be further mentioned that the experiments on rogue waves in water tanks are also progressing smoothly. In 2011, Chabchoub et al. reported for the first time an experimental rogue wave in a water tank [29]. In an annular wave flume, Toffoli et al. also reported experimentally the existence of rogue waves [30].

In an inhomogeneous Kerr medium, we discovered an exact rogue wave solution of a generalized one-dimensional (1D) NLS equation [31]. By means of an exact solution, one can better understand the formation mechanism of rogue waves. Especially, a controllable optical rogue wave is proposed, which provides an opportunity for an optical beam to control another optical beam. One of the important classes of exact solutions are the rogue waves obtained by the nonlinearity management. Such solutions maintain their overall profiles but are tolerant to changes in the pulse-width and amplitude, according to the management of system parameters. The dynamics of optical rogue waves by the nonlinearity management until now was not reported, to the best of our knowledge. Neither were reported realistic fibers that could provide variable nonlinearity and dispersion coefficients in sufficiently controlled experimental conditions, to enable such a management. There remains only hope that such photonic materials will be invented in the near future.

The present paper is written to explore this management venue theoretically and numerically. In Sec. 2, we extend our previous similarity method, given in [7, 32], to reduce the generalized NLS equation with variable coefficients and an external potential to the standard NLS equation. In Sec. 3, we analyze the formation of several types of optical rogue waves and their interactions, by selecting special nonlinearity coefficients, including the first, second and third-order managed rogue waves. Section 4 provides a brief conclusion.

2. Similarity transformation and rogue wave solution

We concentrate on the generalized dimensionless NLS equation with variable coefficients and an external potential, in the form [33–36]:

i \frac{\partial u}{\partial z} + β (x, z) \frac{\partial^{2} u}{\partial x^{2}} + 2 χ (x, z) {| u |}^{2} u = U (x, z) u,

where

u (z, x)

denotes the optical wave envelope. Here,

β (z, x)

and

χ (z, x)

are the dispersion and the nonlinearity management coefficients of a Kerr medium, and

U (z, x)

is an external potential function, determined by the characteristics of the medium. The variables

x

and

z

may have different physical meanings, depending on the conventions in each field. For example, in fiber optical applications [25], the variable

z

is the distance along the fiber, while

x

is the retarded time in the frame moving with the pulse group velocity. Alternatively, in deep water wave applications [29],

z

is interpreted as the normalized time, and

x

is the distance in the frame moving with the group velocity. In BECs, where Eq. (1) is also known as the Gross-Pitaevskii equation [36],

z

is interpreted as the normalized time, and

x

is the spatial coordinate. The general form of Eq. (1) appears in a number of references [37–40], in which a number of explicit solutions have been constructed by the similarity transformation. Equation (1) also includes many special cases discussed in the literature and its analytical rogue wave solutions were recently more precisely referred to as the nonautonomous rogue waves [33]; they are quite different from the conventional rogue wave concept [41, 42].

We search for a similarity transformation that would reduce Eq. (1) to the standard NLS equation:

i \frac{\partial V}{\partial z} + \frac{\partial^{2} V}{\partial X^{2}} + 2 {| V |}^{2} V = 0,

where

V (z, X)

is the complex field of the optical wave and

X (z, x)

is the similarity variable, to be determined. Equation (2) has exact rogue wave solutions that can be determined by many different methods [18, 19, 25, 26]. Using Darboux transformation, Ling and Zhao [43] have found a simple formula for the general first, second, and third order rogue wave solutions. They gave a classification using parameters s_j (

j

being a non-negative integer,

j = 1, 2, \dots

), where

s_{j}

correspond to the eigenvalues in the Darboux method [44]. Their results show that for the standard first-order rogue waves there are no parameters

s_{j}

, the general second-order rogue wave has one parameter

s_{1}

, while the general third-order rogue wave has two parameters

s_{1}

and

s_{2}

, and so on [43]. A similar procedure is devised in [44] as well.

To connect the rogue wave solution of Eq. (1) with those of Eq. (2), we use the similarity transformation [7, 32]

u (z, x) = V (z, X) e^{A (z, x) + i B (z, x)},

where

A (z, x)

and

B (z, x)

are the real functions of

x

and

z

. Substituting Eq. (3) into Eq. (1) will lead to Eq. (2), provided a system of first-order partial differential equations for

X

,

A

, and

B

is satisfied:

β {(X_{x})}^{2} = 1, χ e^{2 A} = 1

\frac{\partial}{\partial x} (e^{2 A} X_{x}) = 0, - B_{z} + β [A_{x x} + {(A_{x})}^{2} - {(B_{x})}^{2}] = U,

B_{x} = - \frac{1}{2 β} \frac{X_{z}}{X_{x}}, \frac{\partial}{\partial z} e^{2 A} + 2 β \frac{\partial}{\partial x} (e^{2 A} B_{x}) = 0 .

The subscripts mean the partial derivatives with respect to $z$ or $x$ , respectively. Thus, the coefficients of Eq. (1) naturally appear in the similarity transformation and the presumed solution. We restrict our attention to the special case β = 1. After a simple calculation, one obtains the solutions of Eqs. (4):

X = λ (z) F (x) + θ (z),

A = \frac{1}{2} \ln \frac{λ (z)}{F_{x} (x)},

B (z, x) = - \frac{1}{4} λ λ_{z} F^{2} - \frac{1}{2} λ θ_{z} F + ω (z) .

U = \frac{1}{4} λ λ_{z z} F^{2} + \frac{1}{2} λ θ_{z z} F - \frac{1}{4} θ_{z}^{2} + ω_{z} (z) + \frac{3 F_{x x}^{2} - 2 F_{x} F_{x x}}{4 λ^{2} F_{x}^{4}},

F (z, x) = λ (z) \int χ (z, x) d x,

where

λ (z)

,

θ (z)

, and

ω (z)

are real integration constants that may depend on

z

. Here, we introduce an auxiliary variable

F (z, x)

, which depends on the nonlinearity coefficient

χ (z, x)

and an arbitrary integration constant

λ (z)

. As seen from Eqs. (5), as long as we choose a suitable nonlinearity coefficient and an integration constant, that is

χ (z, x) \neq 0

and

λ (z) \neq 0

, the similarity transformation will always be valid, and the two integration “constants”

θ (z)

, and

ω (z)

, the amplitude

\exp [A (z, x)]

and the phase

B (z, x)

of the rogue wave will be calculated by Eqs. (5B) and (5C), and the external potential

U (x, z)

will be determined from Eq. (5D). Hence, the profile of the rogue waves will be effectively controlled.

Collecting the partial solutions together, we obtain the exact rogue wave solution of Eq. (1):

u (z, x) = \sqrt{\frac{λ}{F_{x}}} V (z, X) e^{i [- \frac{λ λ_{z}}{4} F^{2} - \frac{λ θ_{z}}{2} F + ω (z)]} .

Equation (6) describes various rogue waves that can be generated from the exact “seed” rogue wave solutions V(z,X) of Eq. (2). There exist four parameter functions in the model, the auxiliary function

F (z, x)

, and the three integration functions

λ (z)

,

θ (z)

, and

ω (z)

; by selecting these functions appropriately, one can manage the rogue waves, to obtain desirable physical characteristics. We emphasize that Eq. (6) could be used to conveniently study rogue waves of different orders.

3. Rogue wave dynamics in a nonlinearity-managed system

In this section, we study the dynamics of rogue waves with the nonlinearity management. Based on solution (6), we will follow the evolution of the square root of the intensity ( $I = \sqrt{{| u |}^{2}}$ ), and the motion of the whole structure of rogue waves. One can see that the rogue wave solution is characterized by the nonlinearity coefficient $χ (z, x)$ and three arbitrary functions $λ (z)$ , $θ (z)$ , and $ω (z)$ . Therefore, the analytical solution (6) is of practical importance for describing the propagation features of inhomogenous rogue waves, and we can select freely these functions, to control the rogue wave dynamics by designing appropriate system parameters.

First, we analyze the mechanism by which the relevant properties of optical rogue waves can be affected by choosing the nonlinearity coefficient $χ (z, x)$ . We choose the nonlinearity coefficient involving a periodic function $χ (z, x) = {(2 + \cos x)}^{- 1}$ . Considering that different choices of the two arbitrary functions $λ (z)$ and $θ (z)$ bring different solutions, we single out a typical example to illustrate the characteristics of the analytic solution (6):

λ (z) = 1 + A \cos (Ω z), θ (z) = a \cos (ω z),

where

A \in (0, 1)

,

Ω \neq 0

,

a \in (0, 1)

and

ω \neq 0

. Such a choice is realistic, because it avoids the appearance of singularities for all system parameters. The corresponding amplitude and auxiliary variable

F (z, x)

are given with

A = \ln (2 + \cos x) / 2

and

F (x) = \frac{2}{\sqrt{3}} [1 + A \cos (Ω z)] arc \tan [\frac{1}{\sqrt{3}} \tan (\frac{x}{2})]

, while the similarity variable

X (z, x)

can be written as

X (z, x) = \frac{2}{\sqrt{3}} [1 + A \cos (Ω z)] arc \tan [\frac{1}{\sqrt{3}} \tan (\frac{x}{2})] + a \cos (ω z)

. Based on [43], which has given the first, second and third order rogue wave solutions of Eq. (2), we display and discuss solutions given by Eq. (6), using the nonlinearity management supplied by Eq. (7).

In the absence of nonlinearity management, the first-order rogue wave solution has been given for a long time and it appears as the well-known localized profile [43], see Fig. 1(a). We also display in Figs. 1(b)-1(d) the square root of the intensity distributions for Eq. (6) with the nonlinearity management parameters mentioned above. Figure 1(b) depicts the space-periodic rogue wave with parameters $A = a = 0$ ; its profile is similar to the Peregrine rogue wave, but there exists a periodic change along the $x$ -axis. Thus, the wave represents a managed Akhmediev-like breather. Figure 1(c) shows another Akhmediev-like rogue wave with the parameters $A = 0.9$ , $Ω = 4$ , and $a = 0$ ; the wave is further compressed around the peak values. If we choose the parameters $A = 0$ , $a = 1$ , and $ω = 4$ , the profile of the managed rogue wave appears in a periodic half-moon shape [45], see Fig. 1(d). In general, the rogue wave in Eq. (6) evolves periodically along the transverse $x$ -axis, and forms an Akhmediev-like breather, which originates from the periodic modulation of the nonlinearity coefficient.

Fig. 1 First-order rogue waves. (a) Peregrine soliton. (b)-(d) Managed Akhmediev-like breathers. The parameters are given in the text.

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Further, we consider the second-order rogue wave solutions of Eq. (6). In Figs. 2 and 3 we present two typical cases, with different parameters $s_{1}$ . The first case is with the parameter $s_{1} = 0$ , which is the standard second-order rogue wave solution; its profile is given in Fig. 2(a). Without nonlinearity management, the rogue wave is symmetric about both x and z-axes. Obviously, the rogue wave is a localized solution. This is the fundamental 2^nd-order rogue wave, possessing a highest peak, surrounded by four equal small peaks on the sides. If we consider nonlinearity managed analytical solutions of Eq. (6), we will observe that the rogue wave yields periodic patterns along the x-axis. As typical examples, the managed profiles with different modulation parameters are displayed in Figs. 2(b)~2(d). When $A = a = 0$ , the rogue wave becomes a 2^nd-order breather of the fundamental pattern, which is shown in Fig. 2(b), whereas if the modulation parameters are chosen as $A = 0.5$ , $Ω = 0$ , and $a = 0$ , an asymmetric pattern appears in Fig. 2(c). Furthermore, if the modulation parameters are changed to $A = 0$ , $a = 0.5$ , and $ω = 4$ , the profile of the rogue wave becomes more complex, see Fig. 2(d).

Fig. 2 Standard second-order rogue wave solutions with s₁ = 0. (a) Maximum-peak second-order rogue wave. (b)-(d) Managed maximum-peak second-order rogue waves. Other parameters are given in the text.

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Fig. 3 Second-order rogue wave structures with the parameter $s_{1} = - 10 i$ . (a) The standard second-order rogue wave without management. (b) Managed second-order dark rogue wave. Other parameters are given in the text.

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Another interesting case is the solution with the real or imaginary part of the complex parameter being non-zero. Without nonlinearity management, there exist three-peak structures in the distance-spatial distributions [46], in which the position of the three equal peaks depends on the selection of the $s_{1}$ value. As an example, we choose $s_{1}$ to be purely imaginary, such as $s_{1} = - 10 i$ , and the results are shown in Fig. 3, to depict the effect of nonlinearity management.

Figure 3(a) shows a typical example in the absence of nonlinearity management, with three equal peaks located at the three vertices of an equilateral triangle. For the nonlinearity management with the modulation parameters $A = a = 0$ , the pattern of the rogue wave displays a dark breather, with two dark peaks in each periodic unit, see Fig. 3(b).

The general third-order rogue waves need two complex parameters, $s_{1}$ and $s_{2}$ , to describe the structure characterization. Therefore, the third-order rogue waves display more complicated structures than the first and the second order ones. We discuss the cases corresponding to the two complex parameters that are divided into three types. The first type, $s_{1} = s_{2} = 0$ , represents the standard third-order rogue wave; the second type is with the two parameters being real and non-zero simultaneously; the last type is by setting both $s_{1}$ and $s_{2}$ to be purely imaginary. Naturally, more complicated scenarios of third-order rogue waves can be obtained by properly choosing the combinations of complex $s_{1}$ and $s_{2}$ .

First, we construct the standard third-order rogue wave with the parameters $s_{1} = s_{2} = 0$ . Without nonlinearity management, there exists the highest-peak solution, see Fig. 4(a). Generally, if the nonlinearity management is considered, significantly affected patterns of higher-order rogue waves will be obtained, as shown in Fig. 4(b)-4(d), which illustrate some special properties of the third-order rogue waves. We pick the modulation parameters $A = a = 0$ , and the resulting breather is displayed in Fig. 4(b). Compared with Fig. 4(a), the amplitude of the central peak increases in each periodic unit. If we set $A = 0.5$ , $Ω = 0$ , and $a = 0$ , all the peaks are compressed, see Fig. 4(c). If we adjust the modulation parameters to $A = 0$ , $a = 0.5$ , and $ω = 4$ , Fig. 4(d) displays the resulting asymmetric pattern of the third-order rogue wave.

Fig. 4 Third-order rogue wave solutions. (a) The standard maximum-height third-order rogue wave solution. (b)-(d) Managed maximum-height third-order rogue wave solutions. The parameters are given in the text.

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Next, we consider the second case with the two parameters $s_{1}$ and $s_{2}$ being real constants and the modulation parameters $A = a = 0$ . Selecting appropriate $s_{1}$ and $s_{2}$ , one finds the second-order rogue wave with the highest peak and the three first-order rogue waves, which has been found in [43]. Here, we only discuss the impact of the nonlinearity management for the two special cases with parameters $s_{1} = s_{2} = 50$ and $s_{1} = 1$ , $s_{2} = 5000$ . In the absence of nonlinearity management, the left panels in Fig. 5 exhibit six first-order rogue waves. The arrangement of the high peaks depends on the choice of $s_{1}$ and $s_{2}$ ; these rogue waves have been reported in [47]. Because of the nonlinearity management and the selection of parameters, the structure of resulting rogue waves may differ greatly, which is evident in the right panels in Fig. 5. It is interesting to note that the resulting rogue wave for $s_{1} = s_{2} = 50$ [the right panel of Fig. 5(a)], appears completely different from the case of $s_{1} = 1$ and $s_{2} = 5000$ [the right panel of Fig. 5(b)], even though both waves belong to the same type, with real parameters.

Fig. 5 Third-order rogue waves with two real parameters $s_{1} = s_{2} = 50$ (top row) and $s_{1} = 1$ , $s_{2} = 5000$ (bottom row); Left column: without the nonlinearity management, right column: with the nonlinearity management.

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Finally, we study the case with the parameters $s_{1}$ and $s_{2}$ being purely imaginary. To display the peculiar characteristics of the third-order solution of Eq. (6), the intensity of the rogue wave is given in Fig. 6 with the modulation parameters $A = a = 0$ . The parameters are selected as $s_{1} = i$ , $s_{2} = 5000 i$ in the top row, and $s_{1} = i$ , $s_{2} = - 5000 i$ in the bottom row. Each panel in left column shows two first-order rogue waves (a high-peak and a low-peak) and four small humps; the distribution of these peaks and humps are determined by the values of $s_{1}$ and $s_{2}$ . While in the right column, each panel reveals two peaks and two deep valleys in each periodic unit. It is noted that the two peaks are different in distance-spatial distributions in Fig. 6(a) and Fig. 6(b). We also find that the structure of rogue waves with two purely imaginary parameters is completely different from the cases with two real parameters, as in Fig. 5.

Fig. 6 Third-order rogue waves with imaginary parameters. Parameters are given in the text, and the setup is the same as in Fig. 5.

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Thus far, we have constructed the rogue wave solutions of Eq. (1) under conditions $χ (z, x) \neq 0$ and $λ (z) \neq 0$ , by using the similarity transformation. Note that only stable optical rogue waves can be observed experimentally. It is therefore essential to analyze the stability of our analytical solutions against small perturbations, and to further verify the validity of obtained solutions. Here, we perform a direct numerical simulation of Eq. (1) using the beam propagation method [32], to test the stability of solution (6) with initial white noise, as compared to Figs. 3(b) and 5(a), the right panel, in which they are presented as modulated dark breather patterns. The profile plots of rogue wave solution (6) under the perturbation of 10% white noise are displayed in the left and right panels of Fig. 7, respectively. The results of the direct numerical simulation verify that the rogue waves still propagate in a stable way, thereby further confirming the validity of our solutions.

Fig. 7 Dark breather profiles as computed from Eq. (6) under the perturbation with 10% initial white noise. The parameters are the same as in Figs. 3(b) and 5(a), the right panel.

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4. Conclusions

In summary, we have presented a class of rogue wave solutions of the generalized NLS equation with variable coefficients in nonlinear inhomogeneous Kerr media, which may describe the evolution of rogue waves in nonlinear optical fibers. By utilizing the similarity transformation, we have reduced the equation with variable coefficients into the standard NLS equation, the rogue wave solutions of which are then transformed back into the solutions of the original equation. We have also constructed the first-order, second-order, and third-order rogue wave solutions via nonlinearity management. This research brings a better understanding of the phenomena of rogue wave generation, and it might be used for generation of even higher-order rogue waves. We hope that the results with nonlinearity management reported here can be observed in the near future.

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 61775150, and by the Natural Science Foundation of Guangdong Province, China (No. 2017A030313346 and 1814050002436). Work at the Texas A&M University at Qatar was supported by the NPRP 8-028-1-001 project with the Qatar National Research Fund (a member of the Qatar Foundation).

Acknowledgments

MRB acknowledges support from the Al Sraiya Holding Group.

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Controllable optical rogue waves via nonlinearity management

Abstract

1. Introduction

2. Similarity transformation and rogue wave solution

3. Rogue wave dynamics in a nonlinearity-managed system

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (7)

Equations (13)

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