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Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients
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Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients Angelin Vithya , M.S. Mani Rajan PII: DOI: Reference:
S2468-0133(19)30072-5 https://doi.org/10.1016/j.joes.2019.11.002 JOES 155
To appear in:
Journal of Ocean Engineering and Science
Received date: Revised date: Accepted date:
14 March 2019 6 October 2019 26 November 2019
Please cite this article as: Angelin Vithya , M.S. Mani Rajan , Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients, Journal of Ocean Engineering and Science (2019), doi: https://doi.org/10.1016/j.joes.2019.11.002
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V. on behalf of Shanghai Jiaotong University. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Highlights
We consider the higher order NLS equation with fifth order dispersion.
Attosecond soliton solution is obtained via Darboux transformation based on Lax Pair.
Using GVD and fifth order dispersion, various soliton control is examined.
Fifth order dispersion has strong influence on width, amplitude and phase.
Obtained results are useful to understand the dynamics of oceanic waves.
Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients 1
Angelin Vithya1, M.S. Mani Rajan1* Department of Physics, Anna University, University College of Engineering, Ramanathapuram 623513, India
Abstract In this paper, the variable coefficient nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is investigated to study the impact of fifth order dispersion on attosecond soliton propagation. Based on the Darboux transformation method, soliton solution is constructed with modulated coefficients though Lax pair. We show that the coefficients of GVD and fifth order term are the main keys to control the amplitude, width and phase shift in the model. With the suitable choices of coefficients for the obtained soliton solution, some new types of soliton management through fifth order are presented for the first time. Especially, the influence of fifth order dispersion of the higher order NLS equations with and without modulated coefficients are discussed in detail. We will also highlight the impact of the studies on application in ocean engineering. Obtained results may be potentially useful in the soliton shaping and management in inhomogeneous fibers.
Keywords: Fifth order dispersion, attosecond pulse, soliton control, Soliton management, inhomogeneous fiber, Oceanic waves. *
[email protected]
1. Introduction Research on optical solitons in various media is bolstered due to their ability to propagate very large distances without losses. Therefore, a great deal of research is taking place to investigate the many models of nonlinear equations which have experienced an explosive growth due to the invention of several exciting and attractive concepts such as solitons, dromions, breathers, rogue waves, similaritons and super continuum generation, etc [1-4]. During the last few decades numerous efforts in the
community of physicists and oceanographers have been dedicated to the understanding of oceanic waves. Nonlinear evolution equations (NLEEs) has attracted major attention in various fields such as astrophysics, chemical physics, solid state physics, plasma physics, fiber optics, biology and ocean engineering. In the oceanographic background, even though it is an approximation of the fully nonlinear equations, it contains the two basic ingredients of the surface wave dynamics: nonlinearity and dispersion. Generally nonlinear equations are not easy to solve analytically and numerically. Hence, the physical dynamics of oceanic waves is hidden by such complications. More than often, a simplification of the equations is needed in order to get some feeling on the interesting phenomena that may take place in the water wave dynamics. On the other hand, in an optical fiber, the formation of soliton occurs due to exact balance between group velocity dispersion (GVD) and self-phase modulation (SPM). In order to describe the real phenomena of picosecond soliton propagation in an inhomogeneous optical fiber, nonlinear Schrödinger equation (NLS) with variable coefficients can be used [5]. The NLS equation exists in many branches of physics, such as nonlinear optics, nonlinear quantum field theory, condensed matter physics, plasma physics etc. The bestidentified solutions of the NLS equation are those for solitary waves or solitons. In 1973, the optical soliton was theoretically projected by Hasegawa and Tappert [6] and experimentally proved by Mollenauer et al. [7].
However, upon increasing amplitude of the pulse, we have to paid attention on higherorder effects which are not present in the standard NLS model. Therefore, for attosecond pulse propagation in a fiber, the higher-order terms, such as the self-steepening, self-frequency shift, third, fourth and fifth-order dispersions should be considered [8-10]. These higher order effects may provide certain different properties for the wave propagation behaviors [11,12]. Recently, some authors considered much higher-order vcNLS equation including fourth and fifth-order terms respectively in [13] and [14]. In recent years, the shaping and controlling management of solitons in inhomogeneous media have received wide attention [15-17]. Soliton shaping can be
realized by manipulating the control parameters in the inhomogeneous profiles which offer more realistic description than constant coefficient NLS [18-20]. Soliton interaction has been investigated through the management of higher order coefficients in the Hirota equation [21]. The fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort pulses, up to the attosecond duration, in an optical fiber [22]. The arrangement of the paper is as follows. In Section 2, we will present a theoretical model of fifth order variable coefficient NLS equation. Lax pair of Equation (2) will be constructed through Ablowitz–Kaup–Newell– Segur (AKNS) scheme in Section 3. Section 4 will be reserved to obtain the general soliton solutions by means of the Darboux transformation. In section 5, explicit one soliton solution is presented. The dynamical behaviors of soliton through obtained solution under various soliton management are analyzed in section 6. Finally, our conclusions will be addressed in section 7. 2. Theoretical Model Results of the fifth-order dispersion have been reported in the laser experiments when the pulses are nearer to 20 fs in duration. Also, when the pulse width close to the attosecond range in a high-intensity optical field, the fifth-order dispersion term is essential to be considered [2325]. To our knowledge, that soliton management under the influence of fifth order dispersion has been reported very little. Homogeneous NLS equation with the fifth-order terms can be written as | |
[
| |
( | | )
| |
(1)
Equation (1) describes the propagation of attosecond pulses sin homogeneous optical fiber, where q represents a normalized complex amplitude of the optical envelope, the subscripts z and t denote the partial derivatives with respect to the scaled distance and time respectively, δ, a real constant, is the coefficient of the fifth-order terms, and means the complex conjugate.
Considering the inhomogeneities in the fiber, in this paper, we investigate the following variablecoefficient HNLS equation with the fifth-order dispersion [26], ( )
( )| |
[ ( )
( )| | ( )
( )( | | ) ( )| |
(2)
where the real functions α(z) and δ1(z) are the coefficients of GVD and fifth-order dispersion terms, respectively. β (z) is the Kerr nonlinearity and the real functions δ2 (z), δ3 (z), δ4(z) and δ5(z) are of the fifth-order nonlinearity terms.
To our knowledge, Darboux
transformation for Eq. (2) with the influence of fifth order dispersion has not been investigated in the existing literature. In order to ensure the integrability of Eq. (2), all of these coefficients must satisfy the linear relations. Additionally, the investigation of Eq. (2) is of great interest and have wide range of applications, which could be used to manage soliton in nonlinear optics. However, the inhomogeneous nonlinear Schrödinger equation can’t be solved analytically without any integrable condition. Through Lax integrability analysis, we ensure the following integrable constraints: ( )
( )
( )
( )
( )
( )
( )
(3)
( )
( )
( )
3. Lax Pair In this section, we construct the Lax pair for the system (2) by employing Ablowitz– Kaup–Newell–Segur (AKNS) procedure [27]. Generally, Lax pair is used to define the linear system of equations for ψ. Solving this system of equations determines the Lax pair and leads to integrability condition (3) which give the relation among the variable coefficients. Based on the AKNS, U and V matrices are obtained as follows: ,
(4)
(
) ∑
(5)
(
Where
)
( )
( )| | [ ( )
( )(
( )[ [
)
| | ( )| |
( )(
)
( )
( )
( ) ( ) )
( ) ( ) ( )( [ (
| |
( )
) ( )
( )
( )[ (
)
( )
| |
(
)
( ) ( ) ( )( | | ( )
[ ( {
( )
) ( )) ( )[ (
| |
( )
] (
)
( )
| |
)
}
where λ is an eigen value or isospectral parameter. By substituting U and V into the zero[
curvature equation
, we obtain Eq. (2) under constraints (3). In
compatibility condition, the square brackets represent the usual matrix commutator.
4. Soliton Solution via Darboux Transformation (DT) Since the Darboux transformation method is a powerful technique in the sense that it can be used to construct the solutions for nonlinear evolution equations with effective manner, we espouse this technique to construct soliton solution. Based on the lax pair of equation (3), this section devoted to construct the Darboux Transformation to generate one soliton solution [28]. Darboux transformation has been employed in many pioneering works [29-31]. Compared with other analytical methods, DT has been evidenced as an efficient and effective way to find the nsoliton solutions for all types of NLS equations. [
(
)
(6) (7)
(
)
Here H is the non-singular matrix requiring ( [
),
(
)
[
(
) is a vector function. and
are complex functions and both are functions of z and t, the superscript T
signifies the vector transpose. Hence the basic form of Darboux transformation for N-soliton solution is obtained, ∑
(
)
(
)
(
)
where (
)
(
)
(
)
(
)
(
)
(
|
)|
(
corresponding to
)
(
( )
where k = 1, 2…n, m = 1,2…n and (
(
|
)|
)
(
)
(
)
(8)
( )) is the eigen function of Eq. (6)
Substituting q=0 into Eq. (8), one can get a 1-soliton solution for Eq.
(2). If we take one-soliton solution as the seed solution in Eq. (8), we can derive the twosoliton solution. Thus, we can generate up to an n-soliton solution using recursion. 5. One soliton Solution In the above section, we construct the n-soliton solution for Eq. (2) by applying the Darboux transformation. The one-soliton solution for the system (2) is explicitly constructed by means of Darboux transformation through
. (
∫
∫(
( )
) ( )
)
(
)
(9)
) ( )
∫(
∫(
)
( )
Based on the obtained one soliton solution (9), we can investigate the soliton dynamics through various choice of modulated coefficients which will determine some physical related operations. This analytical solution shows some features of system (2) which will be discussed in the next section.
6. Impact of fifth order dispersion on soliton shaping and management The study on soliton solution for the HNLS equation with variable coefficients is one of the essential and most important tasks of the soliton-based communication system. Moreover, in real soliton application systems, the profiles of dispersion, nonlinearity, gain or loss are varied along the propagation distance. By this motivation, this work is devoted to studying of soliton control or soliton management through fifth order dispersion profiles for obtained soliton
solution because of its potential values. Recently many works have been devoted to soliton management due its potential applications [32-36]. (i)
Influence of fifth order dispersion on the phase shift To study the impact of fifth order dispersion on soliton in periodically varying dispersion
managed fiber, GVD, and fifth order dispersion profiles are selected as follows [37]:
( z) 6Sin(2 z)
(10)
1 ( z ) z
Fig. 1 (a). Propagation of soliton in fiber with the presence of fifth order dispersion. The parameters are η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot. Fig.1 (a) illustrate the propagation of soliton under the presence of higher order dispersion with inhomogeneity. Especially, in the periodically varying dispersion fiber, the fifth order dispersion has strong influence on phase shift of the soliton on the phase-shift. Very
recently, soliton interaction is avoided between solitons in dispersion decreasing fibers through controllable phase shift [38]. Here, we demonstrate the controllable phase shift through fifth order dispersion parameters which may be suggest for controlling the soliton interactions while multi-soliton propagation in a fiber optic communication system.
(ii)
Influence of fifth order dispersion on compression To study the soliton characteristics in the presence of fifth order dispersive effects under
a special periodic modulation coefficient for GVD is chosen as given in [39]
( z) 1 2 Cos(0.3 z)
1 ( z ) z
Fig. 2 (a). Propagation of soliton in fiber with the presence of fifth order dispersion. The parameters are η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot.
Figures 2(a) and 2(b) clearly depicts the sharp compression and strong phase shift on the soliton propagation under the action of inhomogeneous nature of the fiber medium. Obviously, we observed that the presence of the δ1(z) leads to significant compression effect on the soliton. Moreover, snake like nature of the soliton propagation is disappeared with the presence of function δ1(z). In Ref. [40], snake soliton pulses observed in tapered erbium doped fiber under periodic amplification system. (iii)
Influence of fifth order dispersion on shaping Soliton shaping is a new technique which can be used for various applications of solitons
to construct the ultrafast switching devices. Moreover, by selecting the cosine profile for variable coefficients, soliton shaping has been achieved in dispersion decreasing fiber [41].
(a)
(c)
(b)
(d)
Fig. 3 (a). Propagation of soliton in fiber with the choice of ( z) 2.9Tanh(2.83 z) and 1 ( z ) z . The control parameters are selected as η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot. In the absence of fifth order dispersion, initially the direction of soliton is invariant and abruptly change in the direction at z=7. Since the trajectory of the soliton propagation is looks like V-shape, this soliton is known as V shaped soliton. On the other hand, direction of soliton propagation is reversed and shape of trajectory is changed from V to W shaped under the influence of fifth order dispersion. In both cases, fifth order dispersion parameter is set to be the hyperbolic tangent function. We infer that this changes in the trajectory of soliton is due to the influence of inhomogeneous parameters while the amplitude of soliton is unchanged. Recently, V-shaped soliton obtained for sixth order NLS equation with properly tailoring the dispersion profile [42].
(e)
(f)
(g)
(h)
z Fig. 3 (e). Propagation of soliton in fiber with the choice of ( z ) Exp(( ) 2 ln(25)) and L 1 ( z ) z . The control parameters are selected as η1= 0.1, γ1=0.3. (f) Corresponding contour
plot. (g) Soliton propagation with 1 ( z ) 0 and (h) Corresponding contour plot. If we choose the dispersion parameter ( z ) in the form of gaussian function [43] with fifth order dispersion 1 ( z ) is set to be vanished, we observed parabolic soliton as depicted in Figure 3 (e) & (f). Such kind of soliton has also been observed for inhomogeneous higher order NLS equation with fourth order dispersion and named as Boomerang soliton [44]. On the contrary, 1 ( z ) 0 leads to the disappear of parabolic structure and soliton get new structure with classical like structure except large phase shift around z = 0. Thus, the soliton shaping during the propagation can be controlled with ( z ) and 1 ( z ) in nonlinear fiber optics. 7. Conclusions Nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is considered to study the influence of fifth order dispersion on attosecond soliton propagation in real fiber. Through constructed Lax pair, by means of Darboux transformation, soliton solution is attained to know the effect of inhomogeneity. With properly tuning the coefficients in the obtained soliton solution, some new types of soliton management through fifth order dispersion are presented for the first time. Especially, the influence of fifth order dispersion of the higher order NLS equation is clearly observed by comparing 1 ( z ) 0 and 1 ( z ) 0 . We demonstrate that the coefficients of GVD and fifth order term are the main keys to control the amplitude, width and phase shift on the soliton dynamics. Soliton’s intensity, phase and width are important parameters which are used to control the amplification, interaction and compression respectively in the field of fiber optic communication links. Moreover, recently many nonlinear evolution equations have been investigated in various fields [45-54]. Certainly, it is known that NLS dynamics in the presence of fifth order dispersion is significant in the ocean engineering, when the higher order effects play crucial role in the controlling of dynamics. This analytical study on fifth order NLS equation is gives massive information to understand the wave propagation phenomena in ocean engineering. And also having some impact on the design of dispersion-managed optical communication system, and
ultrafast switching devices in communication system. We also expect that obtained results can be used to understand the characteristics of higher order nonlinear waves in ocean engineering. Moreover, validity of considered model (1) can be confirmed in down-scaled experimentations in many research laboratories at various places in the world. Obtained theoretical results are crucially significant to understanding the dynamics of oceanic waves. Therefore, nonlinear evolution equations and their solutions are essential for oceanic wave modeling, while symbolic computation is a key factor for their exact solutions.
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Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients Angelin Vithya , M.S. Mani Rajan PII: DOI: Reference:
S2468-0133(19)30072-5 https://doi.org/10.1016/j.joes.2019.11.002 JOES 155
To appear in:
Journal of Ocean Engineering and Science
Received date: Revised date: Accepted date:
14 March 2019 6 October 2019 26 November 2019
Please cite this article as: Angelin Vithya , M.S. Mani Rajan , Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients, Journal of Ocean Engineering and Science (2019), doi: https://doi.org/10.1016/j.joes.2019.11.002
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V. on behalf of Shanghai Jiaotong University. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Highlights
We consider the higher order NLS equation with fifth order dispersion.
Attosecond soliton solution is obtained via Darboux transformation based on Lax Pair.
Using GVD and fifth order dispersion, various soliton control is examined.
Fifth order dispersion has strong influence on width, amplitude and phase.
Obtained results are useful to understand the dynamics of oceanic waves.
Impact of fifth order dispersion on soliton solution for higher order NLS equation with variable coefficients 1
Angelin Vithya1, M.S. Mani Rajan1* Department of Physics, Anna University, University College of Engineering, Ramanathapuram 623513, India
Abstract In this paper, the variable coefficient nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is investigated to study the impact of fifth order dispersion on attosecond soliton propagation. Based on the Darboux transformation method, soliton solution is constructed with modulated coefficients though Lax pair. We show that the coefficients of GVD and fifth order term are the main keys to control the amplitude, width and phase shift in the model. With the suitable choices of coefficients for the obtained soliton solution, some new types of soliton management through fifth order are presented for the first time. Especially, the influence of fifth order dispersion of the higher order NLS equations with and without modulated coefficients are discussed in detail. We will also highlight the impact of the studies on application in ocean engineering. Obtained results may be potentially useful in the soliton shaping and management in inhomogeneous fibers.
Keywords: Fifth order dispersion, attosecond pulse, soliton control, Soliton management, inhomogeneous fiber, Oceanic waves. *
[email protected]
1. Introduction Research on optical solitons in various media is bolstered due to their ability to propagate very large distances without losses. Therefore, a great deal of research is taking place to investigate the many models of nonlinear equations which have experienced an explosive growth due to the invention of several exciting and attractive concepts such as solitons, dromions, breathers, rogue waves, similaritons and super continuum generation, etc [1-4]. During the last few decades numerous efforts in the
community of physicists and oceanographers have been dedicated to the understanding of oceanic waves. Nonlinear evolution equations (NLEEs) has attracted major attention in various fields such as astrophysics, chemical physics, solid state physics, plasma physics, fiber optics, biology and ocean engineering. In the oceanographic background, even though it is an approximation of the fully nonlinear equations, it contains the two basic ingredients of the surface wave dynamics: nonlinearity and dispersion. Generally nonlinear equations are not easy to solve analytically and numerically. Hence, the physical dynamics of oceanic waves is hidden by such complications. More than often, a simplification of the equations is needed in order to get some feeling on the interesting phenomena that may take place in the water wave dynamics. On the other hand, in an optical fiber, the formation of soliton occurs due to exact balance between group velocity dispersion (GVD) and self-phase modulation (SPM). In order to describe the real phenomena of picosecond soliton propagation in an inhomogeneous optical fiber, nonlinear Schrödinger equation (NLS) with variable coefficients can be used [5]. The NLS equation exists in many branches of physics, such as nonlinear optics, nonlinear quantum field theory, condensed matter physics, plasma physics etc. The bestidentified solutions of the NLS equation are those for solitary waves or solitons. In 1973, the optical soliton was theoretically projected by Hasegawa and Tappert [6] and experimentally proved by Mollenauer et al. [7].
However, upon increasing amplitude of the pulse, we have to paid attention on higherorder effects which are not present in the standard NLS model. Therefore, for attosecond pulse propagation in a fiber, the higher-order terms, such as the self-steepening, self-frequency shift, third, fourth and fifth-order dispersions should be considered [8-10]. These higher order effects may provide certain different properties for the wave propagation behaviors [11,12]. Recently, some authors considered much higher-order vcNLS equation including fourth and fifth-order terms respectively in [13] and [14]. In recent years, the shaping and controlling management of solitons in inhomogeneous media have received wide attention [15-17]. Soliton shaping can be
realized by manipulating the control parameters in the inhomogeneous profiles which offer more realistic description than constant coefficient NLS [18-20]. Soliton interaction has been investigated through the management of higher order coefficients in the Hirota equation [21]. The fifth-order dispersive nonlinear Schrödinger equation is investigated, which describes the propagation of ultrashort pulses, up to the attosecond duration, in an optical fiber [22]. The arrangement of the paper is as follows. In Section 2, we will present a theoretical model of fifth order variable coefficient NLS equation. Lax pair of Equation (2) will be constructed through Ablowitz–Kaup–Newell– Segur (AKNS) scheme in Section 3. Section 4 will be reserved to obtain the general soliton solutions by means of the Darboux transformation. In section 5, explicit one soliton solution is presented. The dynamical behaviors of soliton through obtained solution under various soliton management are analyzed in section 6. Finally, our conclusions will be addressed in section 7. 2. Theoretical Model Results of the fifth-order dispersion have been reported in the laser experiments when the pulses are nearer to 20 fs in duration. Also, when the pulse width close to the attosecond range in a high-intensity optical field, the fifth-order dispersion term is essential to be considered [2325]. To our knowledge, that soliton management under the influence of fifth order dispersion has been reported very little. Homogeneous NLS equation with the fifth-order terms can be written as | |
[
| |
( | | )
| |
(1)
Equation (1) describes the propagation of attosecond pulses sin homogeneous optical fiber, where q represents a normalized complex amplitude of the optical envelope, the subscripts z and t denote the partial derivatives with respect to the scaled distance and time respectively, δ, a real constant, is the coefficient of the fifth-order terms, and means the complex conjugate.
Considering the inhomogeneities in the fiber, in this paper, we investigate the following variablecoefficient HNLS equation with the fifth-order dispersion [26], ( )
( )| |
[ ( )
( )| | ( )
( )( | | ) ( )| |
(2)
where the real functions α(z) and δ1(z) are the coefficients of GVD and fifth-order dispersion terms, respectively. β (z) is the Kerr nonlinearity and the real functions δ2 (z), δ3 (z), δ4(z) and δ5(z) are of the fifth-order nonlinearity terms.
To our knowledge, Darboux
transformation for Eq. (2) with the influence of fifth order dispersion has not been investigated in the existing literature. In order to ensure the integrability of Eq. (2), all of these coefficients must satisfy the linear relations. Additionally, the investigation of Eq. (2) is of great interest and have wide range of applications, which could be used to manage soliton in nonlinear optics. However, the inhomogeneous nonlinear Schrödinger equation can’t be solved analytically without any integrable condition. Through Lax integrability analysis, we ensure the following integrable constraints: ( )
( )
( )
( )
( )
( )
( )
(3)
( )
( )
( )
3. Lax Pair In this section, we construct the Lax pair for the system (2) by employing Ablowitz– Kaup–Newell–Segur (AKNS) procedure [27]. Generally, Lax pair is used to define the linear system of equations for ψ. Solving this system of equations determines the Lax pair and leads to integrability condition (3) which give the relation among the variable coefficients. Based on the AKNS, U and V matrices are obtained as follows: ,
(4)
(
) ∑
(5)
(
Where
)
( )
( )| | [ ( )
( )(
( )[ [
)
| | ( )| |
( )(
)
( )
( )
( ) ( ) )
( ) ( ) ( )( [ (
| |
( )
) ( )
( )
( )[ (
)
( )
| |
(
)
( ) ( ) ( )( | | ( )
[ ( {
( )
) ( )) ( )[ (
| |
( )
] (
)
( )
| |
)
}
where λ is an eigen value or isospectral parameter. By substituting U and V into the zero[
curvature equation
, we obtain Eq. (2) under constraints (3). In
compatibility condition, the square brackets represent the usual matrix commutator.
4. Soliton Solution via Darboux Transformation (DT) Since the Darboux transformation method is a powerful technique in the sense that it can be used to construct the solutions for nonlinear evolution equations with effective manner, we espouse this technique to construct soliton solution. Based on the lax pair of equation (3), this section devoted to construct the Darboux Transformation to generate one soliton solution [28]. Darboux transformation has been employed in many pioneering works [29-31]. Compared with other analytical methods, DT has been evidenced as an efficient and effective way to find the nsoliton solutions for all types of NLS equations. [
(
)
(6) (7)
(
)
Here H is the non-singular matrix requiring ( [
),
(
)
[
(
) is a vector function. and
are complex functions and both are functions of z and t, the superscript T
signifies the vector transpose. Hence the basic form of Darboux transformation for N-soliton solution is obtained, ∑
(
)
(
)
(
)
where (
)
(
)
(
)
(
)
(
)
(
|
)|
(
corresponding to
)
(
( )
where k = 1, 2…n, m = 1,2…n and (
(
|
)|
)
(
)
(
)
(8)
( )) is the eigen function of Eq. (6)
Substituting q=0 into Eq. (8), one can get a 1-soliton solution for Eq.
(2). If we take one-soliton solution as the seed solution in Eq. (8), we can derive the twosoliton solution. Thus, we can generate up to an n-soliton solution using recursion. 5. One soliton Solution In the above section, we construct the n-soliton solution for Eq. (2) by applying the Darboux transformation. The one-soliton solution for the system (2) is explicitly constructed by means of Darboux transformation through
. (
∫
∫(
( )
) ( )
)
(
)
(9)
) ( )
∫(
∫(
)
( )
Based on the obtained one soliton solution (9), we can investigate the soliton dynamics through various choice of modulated coefficients which will determine some physical related operations. This analytical solution shows some features of system (2) which will be discussed in the next section.
6. Impact of fifth order dispersion on soliton shaping and management The study on soliton solution for the HNLS equation with variable coefficients is one of the essential and most important tasks of the soliton-based communication system. Moreover, in real soliton application systems, the profiles of dispersion, nonlinearity, gain or loss are varied along the propagation distance. By this motivation, this work is devoted to studying of soliton control or soliton management through fifth order dispersion profiles for obtained soliton
solution because of its potential values. Recently many works have been devoted to soliton management due its potential applications [32-36]. (i)
Influence of fifth order dispersion on the phase shift To study the impact of fifth order dispersion on soliton in periodically varying dispersion
managed fiber, GVD, and fifth order dispersion profiles are selected as follows [37]:
( z) 6Sin(2 z)
(10)
1 ( z ) z
Fig. 1 (a). Propagation of soliton in fiber with the presence of fifth order dispersion. The parameters are η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot. Fig.1 (a) illustrate the propagation of soliton under the presence of higher order dispersion with inhomogeneity. Especially, in the periodically varying dispersion fiber, the fifth order dispersion has strong influence on phase shift of the soliton on the phase-shift. Very
recently, soliton interaction is avoided between solitons in dispersion decreasing fibers through controllable phase shift [38]. Here, we demonstrate the controllable phase shift through fifth order dispersion parameters which may be suggest for controlling the soliton interactions while multi-soliton propagation in a fiber optic communication system.
(ii)
Influence of fifth order dispersion on compression To study the soliton characteristics in the presence of fifth order dispersive effects under
a special periodic modulation coefficient for GVD is chosen as given in [39]
( z) 1 2 Cos(0.3 z)
1 ( z ) z
Fig. 2 (a). Propagation of soliton in fiber with the presence of fifth order dispersion. The parameters are η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot.
Figures 2(a) and 2(b) clearly depicts the sharp compression and strong phase shift on the soliton propagation under the action of inhomogeneous nature of the fiber medium. Obviously, we observed that the presence of the δ1(z) leads to significant compression effect on the soliton. Moreover, snake like nature of the soliton propagation is disappeared with the presence of function δ1(z). In Ref. [40], snake soliton pulses observed in tapered erbium doped fiber under periodic amplification system. (iii)
Influence of fifth order dispersion on shaping Soliton shaping is a new technique which can be used for various applications of solitons
to construct the ultrafast switching devices. Moreover, by selecting the cosine profile for variable coefficients, soliton shaping has been achieved in dispersion decreasing fiber [41].
(a)
(c)
(b)
(d)
Fig. 3 (a). Propagation of soliton in fiber with the choice of ( z) 2.9Tanh(2.83 z) and 1 ( z ) z . The control parameters are selected as η1= 0.1, γ1=0.3. (b) Corresponding contour plot. (c) Soliton propagation with 1 ( z ) 0 and (d) Corresponding contour plot. In the absence of fifth order dispersion, initially the direction of soliton is invariant and abruptly change in the direction at z=7. Since the trajectory of the soliton propagation is looks like V-shape, this soliton is known as V shaped soliton. On the other hand, direction of soliton propagation is reversed and shape of trajectory is changed from V to W shaped under the influence of fifth order dispersion. In both cases, fifth order dispersion parameter is set to be the hyperbolic tangent function. We infer that this changes in the trajectory of soliton is due to the influence of inhomogeneous parameters while the amplitude of soliton is unchanged. Recently, V-shaped soliton obtained for sixth order NLS equation with properly tailoring the dispersion profile [42].
(e)
(f)
(g)
(h)
z Fig. 3 (e). Propagation of soliton in fiber with the choice of ( z ) Exp(( ) 2 ln(25)) and L 1 ( z ) z . The control parameters are selected as η1= 0.1, γ1=0.3. (f) Corresponding contour
plot. (g) Soliton propagation with 1 ( z ) 0 and (h) Corresponding contour plot. If we choose the dispersion parameter ( z ) in the form of gaussian function [43] with fifth order dispersion 1 ( z ) is set to be vanished, we observed parabolic soliton as depicted in Figure 3 (e) & (f). Such kind of soliton has also been observed for inhomogeneous higher order NLS equation with fourth order dispersion and named as Boomerang soliton [44]. On the contrary, 1 ( z ) 0 leads to the disappear of parabolic structure and soliton get new structure with classical like structure except large phase shift around z = 0. Thus, the soliton shaping during the propagation can be controlled with ( z ) and 1 ( z ) in nonlinear fiber optics. 7. Conclusions Nonlinear Schrödinger equation with fifth order dispersion in the inhomogeneous optical fiber is considered to study the influence of fifth order dispersion on attosecond soliton propagation in real fiber. Through constructed Lax pair, by means of Darboux transformation, soliton solution is attained to know the effect of inhomogeneity. With properly tuning the coefficients in the obtained soliton solution, some new types of soliton management through fifth order dispersion are presented for the first time. Especially, the influence of fifth order dispersion of the higher order NLS equation is clearly observed by comparing 1 ( z ) 0 and 1 ( z ) 0 . We demonstrate that the coefficients of GVD and fifth order term are the main keys to control the amplitude, width and phase shift on the soliton dynamics. Soliton’s intensity, phase and width are important parameters which are used to control the amplification, interaction and compression respectively in the field of fiber optic communication links. Moreover, recently many nonlinear evolution equations have been investigated in various fields [45-54]. Certainly, it is known that NLS dynamics in the presence of fifth order dispersion is significant in the ocean engineering, when the higher order effects play crucial role in the controlling of dynamics. This analytical study on fifth order NLS equation is gives massive information to understand the wave propagation phenomena in ocean engineering. And also having some impact on the design of dispersion-managed optical communication system, and
ultrafast switching devices in communication system. We also expect that obtained results can be used to understand the characteristics of higher order nonlinear waves in ocean engineering. Moreover, validity of considered model (1) can be confirmed in down-scaled experimentations in many research laboratories at various places in the world. Obtained theoretical results are crucially significant to understanding the dynamics of oceanic waves. Therefore, nonlinear evolution equations and their solutions are essential for oceanic wave modeling, while symbolic computation is a key factor for their exact solutions.
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