Instruction


FACTA UNIVERSITATIS  
Series: Electronics and Energetics Vol. 30, N

o
 1, March 2017, pp. 39 - 48 

DOI: 10.2298/FUEE1701039V 

SUPER-SECH SOLITON DYNAMICS IN OPTICAL 

METAMATERIALS USING COLLECTIVE VARIABLES

 

Marija Veljković
1
, Daniela Milović

1
, Milivoj Belić

2
, Qin Zhou

3
, 

Seithuti P. Moshokoa
4
, Anjan Biswas

4,5
  

1
Faculty of Electronic Engineering, Department of Telecommunications, University of Niš, Serbia 

2
Science Program, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar 

3
School of Electronics and Information Engineering, Wuhan Donghu University,  

Wuhan-430212, People’s Republic of China 
4
Department of Mathematics and Statistics, Tshwane University of Technology,  

Pretoria-0008, South Africa 
5 Department of Mathematics, Faculty of Science, King Abdulaziz, University Jeddah-21589, 

Saudi Arabia 

Abstract. This paper presents collective variable approach for super-sech soliton dynamics 

in optical metamaterials. The soliton dynamics is governed by the generalized nonlinear 

Schrödinger's equation. The numerical simulations of pulse width, amplitude, chirp and 

frequency are given.  

Key words: solitons, metamaterials, super-sech 

1. INTRODUCTION 

Optical metamaterials as novel type of microstructured material have been  extensively 

studied [1–15]. Metamaterials (MMs) are artificial composite structures with both negative 

permittivity and negative permeability and fascinating physical properties at terahertz and 

optical frequencies. Different waveguide structures using metamaterials are already 

demonstrated in optical region [3]. Optical waveguide can be implemented by slab structure 

with core made of positive-indexed material and claddings of double negative materials. 

These waveguides are engineered using advanced processing technology. However, the 

design of microstructured materials is limited by losses. Nevertheless, the development of 

low-loss metamaterials could be the foundation of switches, modulators and other novel 

optical devices in all-optical integrated information processing systems.  

 The transmission of ultrashort pulses through such promising material exhibit unique 

feature. It is well known that soliton is one of the remarkable nonlinear excitations produced 

by the balance between nonlinearity and group velocity dispersion [9–11, 13–19]. Recent 

                                                           

 Received September 26, 2016 
Corresponding author: Daniela Milović 

Faculty of Electronic Engineering, Department of Telecommunications, University of Niš, 

Aleksandra Medvedeva 14, 18000, Niš, Serbia  
(E-mail: dachavuk@gmail.com) 



40 M. VELJKOVIĆ, D. MILOVIĆ, M. BELIĆ, Q. ZHOU, S. P. MOSHOKOA, A. BISWAS 

researches point out that ultrashort pulses propagating in MMs can be described by a 

modified generalized nonlinear Schrödinger equation (GNLSE) in which the linear and 

nonlinear coefficients can be tailored to attain any combination of signs unachievable in 

ordinary materials [1–13]. Simply engineering the MMs can tailor linear and nonlinear 

effective properties. The nonlinear MMs exhibit a rich spatiotemporal dynamics and 

promising applications which was unthinkable in the past [10–14]. Metamaterials enhance 

nonlinearity by confining electrical field in a small region, so it is a great challenge to 

compensate losses and nonlinearity, using metamaterials as waveguides. In metamaterials, 

linear and nonlinear coefficients of the propagation equation can be set to achieve any 

combination of signs that is not possible in regular materials. This metamaterials properties 

allow propagation of a wider variety of solitary waves, efficient phase-matching and 

modulational instability. Earlier results disclose that similar regular (positive indexed) 

dielectric material dispersion plays a crucial role in supporting short duration soliton pulses. 

The dynamics of soliton propagation through these optical metamaterials is governed by the 

nonlinear Schrödinger'squation (NLSE) with a few perturbation terms. The integrability 

aspect of this model was studied with various forms of nonlinearity [9-15]. Different 

algorithms are used to yield solitons, shock waves and other solution to the model that 

appeared with several integrability conditions.  

1.1. Governing equation  

The dynamics of solitons in optical metamaterials is governed by the model [4-7]  

              

2 2 2

2 2 2

1 2 3

| | (| | ) (| | )

| | | |( )

z tt t t t

tt tt tt

iq aq c q q i q i q q i q q

q q q q q q

  

  


    

  
               (1) 

Equation (1) is the nonlinear Schrödinger's equation (NLSE) that is studied in the 

context of metamaterials. Here in (1), a and b are the group velocity dispersion and the 

self-phase modulation terms respectively. This pair produces the delicate balance 

between dispersion and nonlinearity that accounts for the formation of the stable solitons. 

On the right hand side λ represents the self-steepening term in order to avoid the 

formation of shocks and ν is the nonlinear dispersion, while α represents the intermodal 

dispersion. Then finally, θj for j = 1,2, 3 are the perturbation terms that appears in the 

context of metamaterials [1] 

2. COLLECTIVE VARIABLE APPROACH ALGORITHM  

Algorithm of collective variables principle implies that solution of NLSE is split into 

two components [9, 11, 14]. The first one constitute soliton solution while the second one 

represents the residual radiation. Decomposition of the original soliton field q(z,t) is made 

at position z in the fiber and at time t, as follows:             

 ( , )  ( , )  ( , )q z t f z t g z t   (2) 

The soliton field f is defined as a function that depends on parameters, symbolically 

represented by ,  1,...,jX j n   



 Super-sech Soliton Dynamics in Optical Metamaterials by Using Collective Variables 41 

 
1 2

( , ) ( , , , , ) ( , )
N

q z t f X X X t g z t   ,                                   (3) 

where collection of variables represent soliton amplitude, temporal position, pulse width, 

chirp, frequency and phase of the pulse. 

Introduction of CV in function f increases the degrees of freedom resulting in the 

expansion of available phase space of the system. That is undesirable effect, so there are 

some constraints and residual free energy given by: 

 
2 2

1 2
| | | ( , ,..., ) | .

N
E g dt q f X X X dt

 

 

     (4) 

should be minimized. 

From this definition, let Cj denote the rate of change of residual free energy with 

respect to the j
th 

CV Xj.  

 2| |
j

j j j j

E g g
C g dt g g dt

X X X X


 



 

    
    

     
    (5) 

Second parameter that should be defined is
j

C , the rate of change of Cj with the 

normalized distance. Using 
1 2

( , ) ( , ) { ( , ), ( , ), , ( , ), }
N

g z t q z t f X z t X z t X z t t   in the above 

equation, Cj  can be rewritten as: 

 
j

j j

f f
C g g

X X


 

 
 

            (6) 

Now, parameter
j

C can be presented as: 

 
2

1

2 2
N

j k

j

kj j j k

dC xd f f g f
C gdt dt gdt

dz dz X X z X X z

  
  

  


        
                      

     (7) 

The overhead dot represents the derivative with respect to z and the subscripts Xj 

denote partial derivative.  represents the real part and  means  




 . Thus, 

 

2

2 , , k
j

j j k

Xf g f
C g

X z X X z

    
   
     
 

    (8)  

Dirac's principle implies that if a function is approximately zero, it cannot be set equal 

to zero until its variations with respect to all its parameters are made. Therefore, Cj are 

minimum and the equations of the constraints are obtained as: 

 0
j

C                    (9) 

 0
j

C               (10) 

Substituting (2) into (1), we obtain equations of motion of the residual field g(z, t) 

which upon substitution into (7) gives 



42 M. VELJKOVIĆ, D. MILOVIĆ, M. BELIĆ, Q. ZHOU, S. P. MOSHOKOA, A. BISWAS 

 
2

1

2
N

k

j j

k j k j k

dXf f f
C dt dt R

X X X X dz

 
 

 


   
     

     
                  (11) 

where 

* * * 2

* 2 *

* * 2

* 2

* 2 *

2 2 2 | |

2 | | 2

2 2 ( | | )

2 ( | | )

2 ( | | ) 2 (

j j j

j j

j j

j

j j

X tt X tt X

X X t

X t X t

X t

X t X

j
iaf f dt iaf g dt ibf f g fdt

ibf f g gdt f f dt

f g dt f f g f dt

f f g g dt

R

f f g fdt f



 



 

  

  

 

 

 

 





 

 

       

    

    

  

    

  

 

 



 
2

* 2 * 2

1 1

* 2 * 2

2 2

* 2 * * 2 *

3 3

| | )

2 ( | | ) 2 ( | | )

2 | | 2 | |

2 ( ) 2 ( )

j j

j j

j j

t

X tt X tt

X tt X tt

X tt X tt

f g gdt

i f f g f dt i f f g g dt

i f f g f dt i f f g g dt

i f f g f dt i f f g g dt

 

 

 

 

 

 

 

 

 



     

     

     

 

 

 

 

 

Equation (11) is equivalent to the matrix equation 

 


 


C
C X R

X
            (12) 

 

1 1 1

1 2

2 2 2

1 2

1 2

N

N

NN N

N

C C C

X X X

C C CC

X X X X

CC C

XX X

  

  

  
 

   

 

 

 
 
 
 
 

   
    

 
 
 
 
 
 

, 

1

2

N

X

X

X

 
 
 
 
 
  

X , 

1

2

N

R

R

R

 
 
 
 
 
 

R     (13)        

 

with 

 
2

2
j

k j k j k

C f f f
dt gdt

X X X X X

 
 

 

    
   

      
    (14) 



 Super-sech Soliton Dynamics in Optical Metamaterials by Using Collective Variables 43 

3. SUPER-SECH PARAMETER DYNAMICS  

In this section soliton parameter dynamics in optical metamaterials will be obtained 

by CV approach. We assume the desired form of the function f is: 

   
22

2 5 2
4

6

3

1
sech exp[ ( ) ( ) ]

2

m t
f i

X X
X X X X iXt

X
t i

 
     

 
  (15) 

where X1 stands for soliton amplitude, X2 the center position of the soliton, X3 the pulse 

width, X4 the soliton chirp parameter, X5 the soliton frequency and X6  the soliton phase 

that evolves along with propagation. Also m is the super-sech parameter, where m > 0.  In 

this case N = 6 and matrices have dimension 6x6. 

Equations for all the CV are obtained under lowest order CV theory, bare approximation. 

Applying the bare approximation implies that residual field is set to zero, g(z,t) =0. For 

m = 2 elements of matrix R are as follows:  

              3
1 1 3 4 1 2 3

(
3

35
)

2
R X X X                                                  (16) 

             

4
41

1 3 5

3

2 2 4 2 2 2

1 3 4 3 5

3

2 4 2 2 2

1 3 4 3 5

3

2 2 4 2 2 3

1 5 3 4 5 3 5

3

2 4 2

1 1

2

4

5 3 4

4

512 64

315 35

2 336 35( 6 ) 420

315

2 384 4( 49 6 ) 288

315

2 1008 105( 6 ) 420

315

2 1152 1

( )

( )

( )

2( 49 6 ) 288(

X
bX X X

X

X X X X X

X

X X X X X

X

aX X X X X X X

X

X X X

R

X X



 

 



 

 

   


   


    


   




2

4

2

3 5

3

2 4 2 2

2 1 5 3 4 3 5

3

2

2

4 2 2

3 1 5 3 4 3 5

3

4 2

)

( )

315

2 640 12( 49 6 ) 288

315

2 128 12( 49 6 ) 288

31

( )

5

X

X

X X X X X X

X

X X X X X X

X

 

 

   


   


 (17) 

 

2 2 2 2
2 2 1 1 4 1 1

3 1 4

2 4

1 4
2 3

4 324 48( 31 6 )4
(15 4 )

45 2835

16( 205 24 )
(

2835

( )

)

X X X X
R a X X

X X

 



 

   
  

 
 

        (18) 



44 M. VELJKOVIĆ, D. MILOVIĆ, M. BELIĆ, Q. ZHOU, S. P. MOSHOKOA, A. BISWAS 

2 4 3 2 2 3 2 4 3

1 3 1 3 5 1 3 5

2 2 2 2 4 2 2 2 2

1 3 3 4 3 5

2 2 4 4 2 2

1 1 3 3 4 3

4

4

4 1 4
( 49 6 ) ( 6 ) ( 49 6 )

315 9 315

15120( 15 ) 945 60 7 18900( 6 )

170100

64(58715 7194 ) 1512(60 35 3 ) 2160( 4

( (

9 6

) )

( )

b X X X X X X X X

aX X X X X

X X X X

R

X

X

    

   

    

        

        


      




2 2

5

2 2 4 4 2 2 2 2

2 1 3 3 4 3 5

2 2 4 4 2 2 2 2

3 1 3 3 4 3

4

4

5

170100

5760( 29 3 ) 1512(60 35 3 ) 2160( 49 6 )

170100

5760( 29 3 ) 1512(60

)

( )

35 3 ) 2160( 49 6 )

1

(

7 10

)

0 0

X

X X X X X X

X X X X X X

    

    



       


       


 (19) 

 

2 2 3 2 4 3

1 3 4 1 3 4

2 2 3 2 4 3

1 3 4 5 1 3 4 5 1

2 4 3 2 4 3

1 3 4 5 2 1 3 5 3

5

4

2 8
( 6 ) ( 49 6 )

9 315

4 16
( 6 ) ( 49 6 )

9 315

16 16
( 49 6 ) ( 49 6 )

315 315

X X X X X X

a X X X X X X X X

X X X X X X X X

R    

  

   

     

     

     



 (20) 

 

4 2 4

1 3 1 3 5 1 3 5

2 2 4 2 2 2

1 3 4 3 5

3

2 4 2 2 2

1 1 3 4 3 5

3

2 4 2 2 2

2 1 3 4 3 5

3

2 4 2

3 1 3

4

4

6

4

4

( )

64 8 64

35 3 35

2 336 35( 6 ) 420

315

2 384 4( 49 6 ) 288

315

2 384 4( 49

(

6 ) 288

315

2 384

)

( )

( 4( 49 6 ) 28

bX X X X X X X X

aX X X X X

X

X X X X X

X

X X X X X

X

X X

R

X

 



 

 

 

  

    


   


   


   




2 2

3 5

3

)8

315

X X

X

            (21)  

Finally, the nonlinear dynamical system (DS) reduces to: 

 
2 3 3 2

1 1 4 1 4
2 321 1 4 2

4 ( 755 84 ) 4 ( 1025 228 )
( )

105( 15 )4 315( 15 4 )
X a

X X
X

X X
X

  
 

 

   
 

 


 
      (22) 

 2
52 1 5 1 2 3

(
1

( 21( 2 ) 8 3 2 2 3 )
21

( ))aXX X X                            (23) 

 
2 2 2 2

3 4 1 1 3

23
2

( (2 63 ( 15 4 ) 8 3( 89 12 ) 2( 31 6 )( )

63( 1

)

5

)

4 )

X X a X
X

     



    




   


 (24) 



 Super-sech Soliton Dynamics in Optical Metamaterials by Using Collective Variables 45 

 

4 22

3 41

52 4

3 3

2 4 2 2

1 1 3 4 3

4

3

2 4 2 2

2 1 3 4 3

4

3

2 4 2 2

3 1 3 4 3

4

3

4

2

5

2

5

2

5

2 ( 108 )312
( )

7

2 ( )

315

2 (15840 )

315

2

9 7

(15840 )

315

020

9 7020

9 7020

a pX XbX
b X

pX pX

X r X X X X

pX

X X X X X

pX

X X X X X

pX

X

q

q

q









 
  







 

 

 


  (25) 

  
 

        

2 2

1 42

2 2 2 2

5 1 2 3

5

1
4 (3 11 4

105 6

20 6 2 3 11 4 87 8 207 28 )

X X

X

X  


       

   
 

   



    

              (26) 

2 4 2 4 2 2

3 52 4 2

3

2 2 4 4 2

1 3 4 1 2 3

2 4 2 4

1 2 3

2

6

2 4

3 5

1
(756 (450 75 16 5( 45 30 4 ) )

3780( 45 30 4 )

(81(1680 280 13 )

16((352290 111599 8328 ) 18( 870 305 48 )( ))

36 (3 ( 1470 655 96 ) 3

)

( (

(

3

a X X
X

X X X

X

X b X

   
 

    

      

 

      
  

    

       

   



2 4

2 4 2 4

5 1

2 4 2 4

2 3

30 545 64 )

80( 45 30 4 ) ((6390 5235 672 )

( 810 435 32 ) ( 8010 4365 608 ) )))))

X

  

     

     

  

      

      

 (27) 

where 
4 2

4 30 45p      , 
4 2

96 1045 1050q     ,
2

16(7464 60335)r   ; 

4. RESULTS AND CONCLUSION 

Collective variable approach was applied to solve the evolution equation that governs 

the dynamics of soliton and its propagation through optical metamaterials.  

Numerical investigations on the evolution of pulse parameters have been carried out 

in order to illustrate results of collective variable approach. Results have been obtained 

using standard fourth order Runge-Kutta method for integration of the system of ordinary 

differential equations that resulted from the CV analysis. In figure 1 dynamic of the 

system is presented for the following parameter values:  = 0.25, a = 0.1, b = 20, 

          . As the pulse propagates, the amplitude (X1), 
pulse width (X3), frequency (X5) and chirp (X4) vary periodically.  

The control parameter of the soliton solution as it evolves is the total energy Q. The 

total energy can be expressed as function of the super-sech function parameters  

  

2

1 3
4

3

X X
Q   (28) 



46 M. VELJKOVIĆ, D. MILOVIĆ, M. BELIĆ, Q. ZHOU, S. P. MOSHOKOA, A. BISWAS 

This expression shows that the total energy strongly depends on amplitude (X1) and 

the pulse width (X3). The collective variables method enables a clear analysis of the equations 

and reveals the influence of various parameters.  

 

Fig. 1 Variation of pulse parameters (X1  soliton amplitude, X2  center position of the soliton, 

X3  pulse width, X4  soliton chirp, X5  soliton frequency, X6  soliton phase) with 

propagation distance. 

In conclusion, we have investigated the dynamics of an ultra short pulse in optical 

fibers, using CV approach.This paper could be used for further investigations of solitons 

dynamics and the influence of nonlinear parameters on solitons amplitude, temporal position, 

frequency, phase and chirp.  

 



 Super-sech Soliton Dynamics in Optical Metamaterials by Using Collective Variables 47 

Acknowledgement: This research is funded by Qatar National Research Fund (QNRF) under the 

grant number NPRP 6-021-1-005. The second, third and sixth authors (DM,  MB & AB) thankfully 

acknowledge this support from QNRF. The second author (DM) thankfully acknowledges the support 

from Ministry of Education, Science, and Technological Development of Republic of Serbia [ III 

44006, TR-32051]. The fourth author (QZ) was funded by the National Science Foundation of Hubei 

Province in China under the grant number 2015CFC891.The fifth author (SPM) would like to thank 

the research support provided by the Department of Mathematics and Statistics at Tshwane University 

of Technology and the support from the South African National Foundation under Grant Number 

92052 IRF1202210126. The sixth author (AB) would like to thank Tshwane University of Technology 

during his academic visit on 2016. The authors also declare that there is no conflict of interest. 

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