Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation

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Self-similar rogue waves in an inhomogeneous generalized nonlinear Schrödinger equation Shally Loomba a,∗ , Rama Gupta a , Harleen Kaur a , M.S. Mani Rajan b a b

Department of Physics, Panjab University, Chandigarh 160014, India Department of Physics, Anna University, Madurai region, Ramanathapuram 623513, India

a r t i c l e

i n f o

Article history: Received 1 June 2013 Received in revised form 4 May 2014 Accepted 14 May 2014 Available online xxxx Communicated by A.P. Fordy Keywords: Rogue wave Similarity transformation Recurrence Annihilation

a b s t r a c t We present an explicit analytical form of first and second order rogue waves for distributive nonlinear Schrödinger equation (NLSE) by mapping it to standard NLSE through similarity transformation. Upon obtaining the rogue wave solutions, we study the propagation of rogue waves through a periodically distributed system for the two cases when Wronskian of dispersion and nonlinearity is (i) zero, (ii) not equal to zero. For the former case, we discuss a mechanism to control their propagation and for the latter case we depict the interesting features of rogue waves as they propagate through dispersion increasing and decreasing fiber. © 2014 Elsevier B.V. All rights reserved.

1. Introduction In the past decades, significant research has been carried out in the study of nonlinear Schrödinger equation (NLSE) because it is an ubiquitous and eloquent model to describe various physical systems involving nonlinear phenomena, plasma physics, Bose– Einstein condensation (BEC), etc. Initially the classical soliton concept was developed for nonlinear and dispersive autonomous systems, where time has not appeared explicitly in the nonlinear evolution equations. However, in real experiments, solitons cannot be autonomous, which are quite different from the conventional soliton concept. To emphasize the novel features of these soliton like solutions in nonautonomous systems, they were named as nonautonomous solitons. Recently, different aspects of soliton dynamics were described by the nonautonomous NLSE models [1–3]. In the context of nonlinear fiber optics, concept of optical solitons was developed and has been explored theoretically [4] and experimentally [5], because of its enormous potential applications in optical communication systems. Optical solitons are formed due to the balance between the group velocity dispersion (GVD) and nonlinear change of the index of refraction (Kerr effect) [6] and its propagation in a homogeneous optical fiber is described by NLS equation. In reality optical fibers cannot be considered homogeneous due to

*

Corresponding author. Tel.: +91 7299971130. E-mail address: [email protected] (S. Loomba).

http://dx.doi.org/10.1016/j.physleta.2014.05.028 0375-9601/© 2014 Elsevier B.V. All rights reserved.

manufacturing defects [7]. Thus, more attention has been paid to study the propagation of nonautonomous optical solitons in an inhomogeneous optical fiber, which is governed by the variable coefficient NLS (vc-NLS)-type equations. In addition, previous studies in nonlinear optics showed that vc-NLS type equations can be used to describe the soliton control or soliton management in the field of soliton application, and have been extensively investigated [8,9]. Therefore, it is quiet significant to study the fiber system with variable coefficients, such as group velocity dispersion (GVD), distributed nonlinearity, and gain or loss [10–15]. In general, the shape-preserving optical waves are self-similar structures maintaining their identity upon interactions [16,17]. The recent technological developments in the field of nonlinear optics have led to the discovery of a new class of ultrashort pulses – the optical similaritons. Optical similaritons arise when the interaction of nonlinearity, dispersion and gain in a high-power fiber amplifier causes the shape of an arbitrary input pulse to converge asymptotically to a pulse whose shape is self-similar [18]. To date, many researchers have analyzed these waves from different aspects and revealed their interesting features [19–21]. As we know, the so-called Peregrine solution of the NLSE could mathematically serve as the weakly nonlinear prototype of freakwave [22]. Recently, rational solutions of NLSE [23] and higherorder NLSE [24] were discussed as a general way to represent the rogue waves. Fascinatingly, the NLSE not only gives a suitable picture of rogue waves in the ocean, but it is also the governing equation for pulse propagation in optical fibers and matter waves

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in BECs. Thus, nonlinear optical fibers and BECs provide a good platform for the study of rogue waves [25,26]. In nonlinear optics, Solli et al. [27] have observed the rogue waves in a photonic crystal fiber and termed them as “optical rogue waves”. From this concept, we can consider exceptionally high amplitude optical pulses as optical rogue waves. Dudley et al. [28,29] investigated control of optical rogue waves in supercontinuum generation. Dynamics of optical rogue waves through Darboux transformation have been discussed in [30–33]. Nowadays there is a great deal of research being carried out in studying the dynamical behavior of self-similar rogue waves as they propagate through nonlinear optical fibers. In [34], He et al. found new types of rogue wave in NLS–Maxwell–Bloch (MB) system and the management of rogue wave triplets in a nonautonomous system has been investigated in [35]. Several researchers have obtained the self-similar rogue waves for the variants of NLSE by using self-similarity transformation [36–39]. Dai et al. have discussed the controllable behavior of self-similar rogue waves like recurrence, annihilation, postponement, etc., by employing such transformations [40–43]. Being motivated by the ongoing research, we wish to obtain more exciting nonautonomous rogue waves through similarity transformation method for generalized NLSE which have not yet been investigated. The paper is organized as follows. In Section 2, we work out the transformation and the conditions which reduces nonautonomous NLSE to standard NLSE. In Section 3, the explicit first and second order rogue wave solutions are presented for nonautonomous NLSE. In Section 4, we study the propagation of rogue waves through a periodically distributed fiber system for the two cases when Wronskian (W ) of dispersion (D) and nonlinearity i.e. W [ R , D ] = 0 and W [ R , D ] = 0. Conclusion is given in Section 5.

By using the gauge and similarity transformation [40,47]

U (τ , Z ) = A ( Z )Ψ





χ , ρ ( Z ) e iΦ(τ , Z ) ,

(2)

with

τ − τc ( Z )

χ (τ , Z ) =

(3)

,

w

where χ (τ , Z ) is the similarity variable, w and τc ( Z ) are the dimensionless width and position of the rogue wave center. The quadratically chirped phase is given by

Φ(τ , Z ) = C ( Z )

τ2

+ B ( Z )τ + d1 ( Z ), (4) 2 where C ( Z ), B ( Z ) and d1 ( Z ) are parameters related to the phasefront curvature, the frequency shift, and the phase offset, respectively. For the given D ( Z ), R ( Z ) and α ( Z ) the gain Γ ( Z ) and the nonlinear focus length P ( Z ) can be obtained by using the following integrability conditions



Γ ( z) =

 − δP D ,

W [R, D] RD

(5)

with

W [R, D] = R D Z − D R Z , P z = −δ P 2 D .

(6)

The functional form of the parameters R ( Z ) and D ( Z ) must be chosen in such a way so that the gain Γ ( Z ) does not become singular. On inserting the transformation given in Eq. (2) with Eq. (3) in Eq. (1) along with the phase Φ , it reduces to the standard NLSE

∂ψ 1 ∂ 2ψ + + |ψ|2 ψ = 0, (7) ∂ρ 2 ∂χ 2 where ρ ( Z ) represents the effective propagation distance and is i

2. Model equation and gauge and similarity transformation To begin with consider the model which is given by Serkin and Hasegawa to formulate the effect of varying dispersion with external harmonic oscillator potential [44]

i

∂U D( Z ) ∂2U + + R ( Z )|U |2 U ∂Z 2 ∂τ 2  ∂U i + i α ( Z ) + δ D ( Z ) P ( Z )τ − Γ ( Z ) U = 0, ∂τ 2

(1)

where U ( Z , τ ) represents the normalized slowly varying complex envelope of the pulse, τ and Z are the retarded time and the normalized propagation distance in the nondimensional form, respectively. D ( Z ), R ( Z ), P ( Z ) and Γ ( Z ) are real functions and account for the varying dispersion corresponding to which it has a harmonic oscillator potential form, varying nonlinearity, nonlinear focus length and gain/loss respectively. The parameter α ( Z ) denotes the velocity of propagation. In the above equation it is mandatory that the radius of curvature of the wavefront must be an oscillating function of the propagating distance in order to have the oscillating self-focusing light beam in nonlinear Kerr-like media. We would like to mention that the model Eq. (1) is nonautonomous in nature due to the explicit presence of time τ . In this work we will demonstrate with the aid of similarity transformation that it can support nonautonomous rogue waves which can maintain their overall shapes but allow their widths, amplitudes and the pulse center to change according to the management of system parameters such as dispersion, nonlinearity and gain. This model has been used to understand the soliton interaction under soliton dispersion management [45,46]. For α ( Z ) = P ( Z ) = 0, Eq. (1) reduces to the one given in [42] which has been employed to study the controllable characteristics of rogue waves.

given as

ρ( Z ) =

Z 0

D ( S )dS w2

(8)

.

It is worth mentioning here that the form of ρ is governed by the dispersion parameter D ( Z ) as w represents the width of the pulse which is constant. The rogue wave center τc ( Z ) is given as follows



Z

τc ( Z ) = τ0 + C 02



  α ( S ) + D ( S ) B ( S ) dS .

(9)

0

The parameters A ( Z ), B ( Z ), C ( Z ) and d1 ( Z ) are obtained by collecting the similar terms after the substitution of Eq. (2) with Eq. (3) in Eq. (1) along with the phase Φ and demanding the coefficients of real and imaginary parts of each term to be separately equal to zero. In the explicit form they can be defined as

A( Z ) =



1 w

D R

Z B( Z ) =

,

0

 d1 ( Z ) = −

α( Z ) B ( Z ) +

C ( Z ) = −δ P ( Z ),

δ P ( S )dS , D ( Z ) B ( Z )2 2

 dZ.

(10)

The nonlinear focus length P ( Z ) can be obtained by using Eq. (6) as

P (Z) = −

1

(c 0 − δ

Z 0

D ( Z  )d Z  )

,

(11)

where c 0 is an integration constant and has to be chosen in such a way so that P ( Z ) should be nonsingular.

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Fig. 2. Intensity profile (a) for first order rogue waves of Eq. √ (1), (b) for second order rogue waves of Eq. (1) with D ( Z ) = R ( Z ) = a cos(κ Z ), κ = 2 and w = 1.

Fig. 1. The profile of Γ ( Z ) for D ( Z ) = R ( Z ) = a cos(κ Z ), ρ0 = 0, c0 = 1.5.

√

κ = 2 and w = 1, a = 1,

3. First and second order rogue waves As Eq. (1) is reduced to standard NLSE whose abundant solutions are well known such as single soliton or multisoliton, breathers, rogue wave solutions, etc., we can obtain the exact solutions for Eq. (1) by using the transformation given in Eq. (2). In this paper, we are interested in first and second order rogue wave solutions. The intensity of first (I 1 ) and second (I 2 ) order rogue waves for Eq. (1) are given as [48]

I 1 = |U |2 =

1 D

1+8

1 + 4(ρ − ρ0 )2 − 4χ 2



(1 + 4(ρ − ρ0 )2 + 4χ 2 )2

 h21 1 D (l − k)2 , I2 = 2 + 2 2 w R

w2 R l

(12)

,

(13)

l

where

l=

1 3

3

χ 2 + (ρ − ρ0 )2 +

1 4

3

χ 2 − 3(ρ − ρ0 )2 +

3 64

12χ 2

+ 44(ρ − ρ0 )2 + 1, (14)    3 3 3 − , (15) k = χ 2 + (ρ − ρ0 )2 + χ 2 + 5(ρ − ρ0 )2 + 4 4 4    2 15 h1 = ρ −3χ 2 + (ρ − ρ0 )2 + 2 χ 2 + (ρ − ρ0 )2 − . (16) 8

Here ρ0 is an arbitrary constant. The effective propagation distance ρ and the similarity variable χ can be obtained from Eq. (8) and Eq. (3), respectively. To understand the pulse propagation, various forms of distributed parameters can be chosen according to the specific problem. Here, we are exemplifying it by considering some important systems that are currently being used in the literature and will study the propagation behavior of rogue waves for these systems. 4. Periodically distributed systems Various researchers are focusing on periodic systems because of their potential applications in long distance communication. To investigate the dynamics of rogue waves in periodically distributed systems, we consider two different forms of the parameters, one corresponds to the W [ R , D ] = 0 and the other one for W [ R , D ] = 0. Case (i) Wronskian W [ R , D ] = 0 Consider the following forms of nonlinearity and dispersion parameter

R ( Z ) = D ( Z ) = a cos[κ Z ].

(17)

Fig. 3. (a) Recurrence, (b) annihilation for first order rogue waves of Eq. (1). The parameters are w = δ = 1, and ρ0 = 15 with (a) κ = 0.05 and (b) κ = 0.15.

For these periodic choices of the parameters, the corresponding P ( Z ) and Γ ( Z ) can be obtained by using Eq. (11) and Eq. (5), respectively. The profile of Γ ( Z ) is plotted in Fig. 1. Clearly the gain/loss profile is periodic in nature and for the above choice of parameters yields the zero value of Wronskian W [ R , D ]. So the pulse does not suffer any broadening and compression but an overall phase change, which is depicted in Fig. 2(a) and Fig. 2(b) for first and second order rogue waves, respectively. The periodic choice of dispersion and nonlinearity parameters is of practical relevance as it has been used to study nonautonomous solitons in external potentials [49] and indicates the improved stability of solitons [50]. In particular, periodic form of dispersion finds applications in enhancing signal to noise ratio, to reduce Gordon Hauss time jitter and in suppressing the phase matched condition for four wave mixing for single mode optical fibers [51]. Now we discuss the mechanism which is used to control the rogue waves. For this one has to consider the relation between an effective propagation distance ρ and the original propagation distance Z , given by Eq. (8). This is a general mechanism to control the propagation of rogue waves and is valid for both first and second order optical rogue waves. In this Letter we are demonstrating it by considering first order rogue waves. On using Eq. (8) an effective propagation distance comes out to be

ρ=

a w2κ

sin(κ Z ).

(18)

ρ is periodic in nature and its maximum value ρmax is    a   ρmax =  2 . (19) w κ Clearly

If we discuss rogue waves in the context of Eq. (7) maximum amplitude appears at ρ = ρ0 and then disappears, while the situation is different in the case of Eq. (1). If ρ < ρ0 , rogue waves do not have an appropriate propagation distance to get excited and even get annihilated. For ρ > ρ0 , rogue waves recur periodically. Thus the rogue waves can be controlled by regulating the values of ρmax and ρ0 . If | wa2 κ | > ρ0 , the first order rogue waves get excited from their ρ w2κ

initial value at z = κ1 arcsin( 0 a ) and recur periodically as shown in Fig. 3(a). Fig. 3(b) shows that the rogue wave is restrained and excites partially for | wa2 κ | < ρ0 . Sectional view for recurrence and

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Fig. 4. Sectional view of Fig. 3(a) (solid) and Fig. 3(b) (dotted), with

τ = 0.5.

annihilation is shown in Fig. 4. So we have demonstrated that by properly managing ρmax and ρ0 we can get the excitation or the annihilation of rogue waves at a specified location. The similar analysis of rogue waves for the periodic choices of the parameters has been reported in [42] by using lens-type transformation. However, while studying the dynamical properties of the system in [42] the gain term is explicitly made zero, whereas in our study the gain/loss parameter Γ ( Z ) is Z dependent which corresponds to more realistic systems. Consequently, our results hold greater practicality. Case (ii) Wronskian W [ R , D ] = 0 For this case we have taken the nonlinearity and dispersion parameters as follows

R ( Z ) = γ cos(κ Z ),

D( Z ) =

γ d0

cos(κ Z ) exp(σ Z ).

(20)

Fig. 5. The profile of Γ ( Z ) for (a) dispersion increasing fiber case with c 0 = −2, σ = −0.1.

where the parameters κ and d0 are related to Kerr nonlinearity and the initial peak power in the system, respectively. The parameter σ can take positive and negative values. The positive (negative) value of σ stands for dispersion increasing (decreasing) fibers. For these choices of the parameters, the corresponding nonlinear focus length P ( Z ) and the gain/loss function Γ ( Z ) can be obtained by using Eq. (11) and Eq. (5), respectively. The profiles of Γ ( Z ) for the dispersion increasing and dispersion decreasing fiber cases have been plotted in Fig. 5. To analyze the propagation characteristics of rogue waves, we have plotted the intensity profiles for first and second order rogue waves for both dispersion increasing and decreasing fibers in Fig. 6(a), Fig. 6(b), Fig. 7(a) and Fig. 7(b), respectively. It can be clearly seen from the intensity profiles that as the pulse propagates through the fiber its width remains constant, which is a useful feature for various practical purposes. Moreover, we found that the amplitude of the rogue wave increases (decreases) for dispersion increasing (decreasing) fiber on a constant background due to the presence of Z dependent gain/loss term Γ ( Z ), depicted in Fig. 5. This is in contrast to the other known results, where the increase in the amplitude occurs at the cost of the width of the pulse. 5. Conclusion We have mapped an inhomogeneous NLSE to the standard NLSE by using a similarity transformation as the first and second order rogue wave solutions for the pure NLSE are known. We have analytically obtained the first and second order rogue wave solutions for inhomogeneous NLSE by doing reverse transformation. The dynamics of rogue waves can be controlled by just managing the distributed parameters and we have demonstrated the management of rogue waves for periodically distributed system. In general our results are valid for any choice of the nonlinear and dispersion

κ = 1, γ = 0.5 and σ = 0.1, (b) dispersion decreasing fiber case with c0 = 1, κ = 1 and

Fig. 6. Intensity profile (a) for first order rogue waves of Eq. (1), (b) for second order rogue waves of Eq. (1), with c 0 = −2, κ = 1, γ = 0.5 and σ = 0.1.

Fig. 7. Intensity profile (a) for first order rogue waves of Eq. (1), (b) for second order rogue waves of Eq. (1), with c 0 = 1, κ = 1 and σ = −0.1.

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parameters, specific choices of these parameters can be made depending upon the requirement of the system under study. These results can be applied in the context of BECs as the governing equation is same. The analytical results obtained here may prove to be useful to study the propagation of rogue waves experimentally through optical fibers and in BECs as well.

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