Lie symmetry analysis and exact solutions to N-coupled

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Lie symmetry analysis and exact solutions to N-coupled

Transcript Of Lie symmetry analysis and exact solutions to N-coupled

LIE SYMMETRY ANALYSIS AND EXACT SOLUTIONS TO N -COUPLED NONLINEAR SCHRO¨ DINGER’S EQUATIONS WITH KERR AND PARABOLIC LAW NONLINEARITIES
YAKUP YILDIRIM1, EMRULLAH YAS¸ AR1, HOURIA TRIKI2, QIN ZHOU3, SEITHUTI P. MOSHOKOA4, MALIK ZAKA ULLAH5, ANJAN BISWAS4,5, MILIVOJ BELIC6
1Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey 2Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria 3School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, People’s Republic of China 4Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa
5Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah-21589, Saudi Arabia E-mail: [email protected] 6Science Program, Texas A & M University at Qatar, P.O. Box 23874, Doha, Qatar
Received May 26, 2017

Abstract. This paper addresses N -coupled nonlinear Schro¨dinger’s equation with spatio-temporal dispersion for Kerr and parabolic laws of nonlinearity by the aid of Lie symmetry analysis. We systematically construct similarity reductions to the derived ordinary differential equations by Lie group analysis. These equations lead to exact solutions.
Key words: Lie symmetry analysis; nonlinear Schro¨dinger’s equation; optical solitons; spatio-temporal dispersion.

1. INTRODUCTION

The complex dynamics of optical soliton propagation is governed by the nonlinear Schro¨dinger’s equation (NLSE). There are several advances that have been made with this generic nonlinear evolution equation. While this model has been mostly studied in scalar form, the vector-coupled NLSE has also been addressed in the context of birefringent optical fibers. This paper will focus on N -coupled NLSE that appears in the context of parallel propagation of solitons for large data transmission across trans-continental and trans-oceanic distances. There are several integration algorithms that are applied to reveal results for NLSE in polarization-preserving as well as birefringent optical fibers. This paper will apply one of the most powerful mathematical approaches to study N -coupled NLSE. It is the Lie symmetry analysis. This classic mathematical methodology will never be rusty in any area of

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mathematical physics or other fields of applied mathematics. In the past, several other techniques have been implemented to extract solitons and other solutions to such a system. They are Kudryashov’s method, G /G-expansion scheme, method of undetermined coefficients [1] and several others.
Performance enhancement in soliton propagation across the globe can only be achieved through the dense wavelength division multiplexing (DWDM) technology implementation, which is modeled by N -coupled NLSEs. This paper studies DWDM systems in nonlinear optical fibers with Kerr and parabolic laws of optical nonlinearity. The unique soliton dynamics has been extensively studied in optical fibers, photonic crystal fibers (PCFs), metamaterials, and metasurfaces [1-24]. Therefore, it is about time to focus and pay attention to DWDM optical systems. The details of Lie symmetry analysis along with its implementation to such DWDM systems are studied in this paper.

2. THE MODEL

The dynamics of optical soliton propagation through a DWDM system is governed by the N -coupled NLSEs. There are two types of nonlinear media and associated optical nonlinearities that will be studied in this paper. These are Kerr law nonlinear media and the parabolic law nonlinear media.
For NLSE, in addition to usual group-velocity dispersion (GVD), spatiotemporal dispersion (STD) is included. STD makes the governing model well-posed as opposed to the presence of GVD alone. Thus, the model is discussed in the following subsections based on the types of nonlinearity in question.

2.1. KERR LAW

For Kerr law nonlinearity, the generic DWDM model reads [1]







2N

2

iqt(l) + alqx(lx) + blqx(lt) + cl q(l) + αln q(n) q(l) = 0,

(1)



n=l



where 1 ≤ l ≤ N . The first term in (1) on left-hand side is the evolution term, while al represents the coefficient of GVD. Here bl represents the STD. Then, cl is the coefficient of self-phase modulation (SPM) while αln are the coefficients of crossphase modulation (XPM). The independent variables are x and t, which represents the spatial and temporal variables, respectively. The dependent variable is q(l) (x, t) that gives the soliton profile in every single channel. It must be noted that STD terms are deliberately included for the problem to be well-posed.
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Lie symmetry analysis and exact solutions to N-coupled NLSEs

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2.2. PARABOLIC LAW

For parabolic law nonlinearity, DWDM model reads [1]





2N

2

iqt(l) + alqx(lx) + blqx(lt) + cl q(l) + αln q(n)  q(l)

n=l





4N

2

2

2

+ dl q(l) + q(n) βln q(n) + γln q(l)  q(l) = 0

(2)

n=l

for 1 ≤ l ≤ N . In (2), the SPM terms are the coefficients of cl and dl, while the XPM

coefficients are αln, βln and γln. The remaining parameters have the same definition

as in Kerr law nonlinear medium. In mathematical physics, the generic equations (1)

and (2) fall under the category of nonlinear evolution equations (NLEEs).

3. SYMMETRY ANALYSIS AND SYMMETRY REDUCTIONS
The investigation of exact solutions to NLEEs is a quite important task in the nonlinear science. In this regard, many powerful methods have been developed in the last three decades such as inverse scattering method, Darboux method, Hirota bilinear method, ansatz method, multiple-exp function method, simplest equation method etc. [2–5, 7].
Since the end of the 19th century, the symmetry study plays an important role in almost all the scientific fields. Inspired by Galois’s researches on algebraic equations, S. Lie showed that considered differential equations can be invariant with respect to continuous transformation groups. Obtaining the symmetry reductions and thereby group invariant solutions are possible by Lie generators corresponding to the transformation groups. For the detailed studies on applications of Lie group analysis to differential equations, we suggest to readers to see Refs. [8–11].

3.1. KERR LAW

3.1.1. al and bl = 0 are arbitrary constants

Substituting the complex-valued functions q(l)(x, t) into the system (1) and then decomposing into real and imaginary parts yields a pair of relations. The real part gives







2N

2

−Im qt(l) +alRe qx(lx) +blRe qx(lt) + cl q(l) + αln q(n) Re q(l) = 0



n=l



(3)

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while the imaginary part gives







2N

2

Re qt(l) + alIm qx(lx) + blIm qx(lt) + cl q(l) + αln q(n) Im q(l) = 0



n=l



(4)

where 1 ≤ l ≤ N . Let us consider the Lie group of point transformations

t∗ = t + τ

N

N

x, t, Re q(l) , Im q(l)

l=1

l=1

+O 2

x∗ = x + ξ

N

N

x, t, Re q(l) , Im q(l)

l=1

l=1

+O 2


Re q(l) = Re q(l) + ηl

N

N

x, t, Re q(l) , Im q(l)

l=1

l=1

+O 2



N

N

Im q(l) = Im q(l) + φl x, t, Re q(l) , Im q(l)

+O 2

l=1

l=1

with small parameter 1 and 1 ≤ l ≤ N . The vector field associated with the

above group of transformations can be written as

V = ξ x, t, N Re q(l) , N Im q(l) ∂

∂x

l=1

l=1

+τ x, t, N Re q(l) , N Im q(l) ∂

∂t

l=1

l=1

N
+ ηl
l=1

N

N

x, t, Re q(l) , Im q(l)

l=1

l=1

∂ ∂Re q(l)

N

N

N

(l)

(l)



+ φl x, t, Re q , Im q

∂Im q(l)

(5)

l=1

l=1

l=1

The symmetries of equations (3)-(4) will be generated by the vector field of the form (5). Applying the second prolongation pr(2)V of V to equations (3)-(4) and splitting on the derivatives of Re q(l) , Im q(l) leads to the following overdeter-

mined system of linear partial differential equations:

∂τ ∂τ

∂τ

∂τ

∂x = ∂t = ∂Re q(l) = ∂Im q(l) = 0

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∂ξ ∂ξ

∂ξ

∂ξ

∂x = ∂t = ∂Re q(l) = ∂Im q(l) = 0

∂φl ∂φl

∂φl

N ∂φl

∂x = ∂t = ∂Im q(l) = ∂Re q(n) = 0

n=l

∂φl

φl

Im q(l) φl

∂Re q(l) = Re q(l) , ηl = − Re q(l)

where 1 ≤ l ≤ N . Solving the above equations we obtain the values of ξ, τ , ηl, and

φl

τ = C1; ξ = C2

ηl = −Cl+2Im q(l)

φl = Cl+2Re q(l)

(6)

where C1, C2, and Cl+2 are arbitrary constants and 1 ≤ l ≤ N . As a result we obtain the infinitesimal generators of the corresponding Lie algebra of Eq. (1) are given by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)

∂ ,

∂Im q(l)

∂Re q(l)

where q(l) are complex-valued functions and 1 ≤ l ≤ N .

3.1.2. al is an arbitrary nonzero constant and bl = 0

In this case, Eq. (1) becomes







2N

2

iqt(l) + alqx(lx) + cl q(l) + αln q(n) q(l) = 0

(7)



n=l



The infinitesimal generators of the corresponding Lie algebra of Eq.(7) are given by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)



∂Im q(l)

∂Re q(l)



∂N

VN+3 = 2t ∂t + x ∂x +

l=1

−Re q(l)

∂ − Im q(l) ∂Re q(l)

∂ ∂Im q(l)

∂N

xIm q(l)



xRe q(l)



VN+4 = t ∂x +



+

,

2al ∂Re q(l)

2al ∂Im q(l)

l=1

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where q(l) are complex-valued functions and 1 ≤ l ≤ N .

3.1.3. bl is an arbitrary nonzero constant and al = 0

In this case, Eq. (1) becomes







2N

2

iqt(l) + blqx(lt) + cl q(l) + αln q(n) q(l) = 0

(8)



n=l



The infinitesimal generators of the corresponding Lie algebra of Eq. (8) are given by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)



∂Im q(l)

∂Re q(l)

∂N VN+3 = t ∂t +
l=1

Re q(l)



Im q(l)



− 2 ∂Re q(l) − 2 ∂Im q(l)

∂ VN+4 = x ∂x +

N

blRe q(l) − 2xIm q(l)



blIm q(l) + 2xRe q(l)







2bl

∂Re q(l)

2bl

∂Im q(l)

l=1

where q(l) are complex-valued functions and 1 ≤ l ≤ N .

To obtain the symmetry reductions of equations (3)-(4), we have to solve the

characteristic equation

dx dt dRe q(l) dIm q(l)

==

=

,

(9)

ξτ

ηl

φl

where ξ, τ , ηl, and φl are given by (6) and 1 ≤ l ≤ N . To solve (9), we consider the following cases: (i) V1 + kVl+2 (ii) V2 + µVl+2 Case (i) V1 + kVl+2 Solving the characteristic equation (9), we have the following similarity vari-
ables

ξ=t

q(l)(x, t) = Fl (ξ) exp (i (kx + Gl (ξ)))

(10)

where ξ is a new independent variable and Fl and Gl are new dependent variables and 1 ≤ l ≤ N .

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Substituting equations (10) into system (3)-(4), we immediately obtain the reduced equations, which after separating imaginary and real parts read
(kbl + 1) Fl = 0





N 

clFl3 − kblFlGl +

αlnFn2 Fl − k2alFl − FlGl = 0

(11)

 n=l



where 1 ≤ l ≤ N . We obtain the following solution of ordinary differential equations

(ODEs) (11)

Fl = Cl

N
clCl2 − alk2 + αlnFn2 ξ
n=l
Gl = blk + 1 + κl (12) where Cl and κl are arbitrary constants, and 1 ≤ l ≤ N .
The corresponding solution of the system (1) is given by













q(l)(x, t) = Cl exp

 i kx +





 





N
clCl2 − alk2 + αlnFn2
n=l
blk + 1



t

  





 + κl





 





Case (ii) V2 + µVl+2 Solving the characteristic equation, the similarity variables are
ξ=x

q(l)(x, t) = Fl (ξ) exp {i [µt + Gl (ξ)]} ,

(13)

where ξ is a new independent variable, and Fl and Gl are new dependent variables.

Using (13) in system (3)-(4), we obtain the following system of ODEs

alFlGl + 2alFl Gl + blµFl = 0

(14)





N 

clFl3 − alFlGl2 − blµFlGl +

αlnFn2 Fl − µFl + alFl = 0 (15)

 n=l



Solving the equations (14), we have

G = − blµFl2 − κ2l ,

(16)

l

2alFl2

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where κl are arbitrary constants and 1 ≤ l ≤ N . Using (16) in equations (15), we

have





3 blµFl2 − κ2l 2 blµ blµFl2 − κ2l  N


2

clFl − 4alFl3 +

2alFl

+

αlnFn Fl − µFl + alFl = 0

 n=l



(17)

The corresponding solution of the system (1) is given by

q(l)(x, t) = F (ξ) exp i µt − blµFl2 − κ2l dξ + θ

,

(18)

l

2alFl2

l

where θl are arbitrary constants and 1 ≤ l ≤ N . ξ is given by (13) and Fl are given by (17).

3.2. PARABOLIC LAW

3.2.1. al and bl = 0 are arbitrary constants

The infinitesimal generators of the corresponding Lie algebra of (2) are given

by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)

∂ ,

(19)

∂Im q(l)

∂Re q(l)

where q(l) are complex-valued functions and 1 ≤ l ≤ N .

3.2.2. al is an arbitrary nonzero constant and bl = 0

In this case, Eq. (2) becomes





2N

2

iqt(l) + alqx(lx) + cl q(l) + αln q(n)  q(l)

n=l





4N

2

2

2

+ dl q(l) + q(n) βln q(n) + γln q(l)  q(l) = 0

(20)

n=l

The infinitesimal generators of the corresponding Lie algebra of Eq. (20) are given

by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)



∂Im q(l)

∂Re q(l)

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Lie symmetry analysis and exact solutions to N-coupled NLSEs

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∂N

xIm q(l)



xRe q(l)



VN+3 = t ∂x +



+

,

2al ∂Re q(l)

2al ∂Im q(l)

l=1

where q(l) are complex-valued functions and 1 ≤ l ≤ N .

3.2.3. bl is an arbitrary nonzero constant and al = 0

In this case, Eq. (2) becomes





2N

2

iqt(l) + blqx(lt) + cl q(l) + αln q(n)  q(l)

n=l





4N

2

2

2

+ dl q(l) + q(n) βln q(n) + γln q(l)  q(l) = 0

(21)

n=l

The infinitesimal generators of the corresponding Lie algebra of Eq. (21) are given

by

∂ V1 = ∂x ;

∂ V2 = ∂t

Vl+2 = Re q(l)

∂ − Im q(l)



∂Im q(l)

∂Re q(l)



∂N

xIm q(l)



xRe q(l)



VN+3 = t ∂t − x ∂x +



+

,

bl ∂Re q(l)

bl ∂Im q(l)

l=1

where q(l) are complex-valued functions and 1 ≤ l ≤ N .

The similarity variables for system (2), corresponding to the vector field µ2V1 +

µ1V2 + µ3Vl+2 are given by the vector fields (19)

ξ = µ1x − µ2t

q(l)(x, t) = Fl (ξ) exp {i [µ3x + Gl (ξ)]} ,

(22)

where ξ is the new independent variable and Fl,Gl are the new dependent variables. Using the similarity variables (22) in system (2) and separating the real and

imaginary parts, we obtain

−µ2Fl + 2alµ21Fl Gl − blµ2µ3Fl − blµ1µ2FlGl + 2alµ1µ3Fl

−2blµ1µ2Fl Gl + alµ21FlGl = 0, µ2FlGl −alµ21FlGl2 −2alµ1µ3FlGl −alµ23Fl +alµ21Fl +blµ1µ2FlGl2 +blµ2µ3FlGl
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



N  N 

−blµ1µ2Fl + clFl3 +

αlnFn2 Fl +

γlnFn2 Fl3

 n=l

 n=l







N 

+

βlnFn4 Fl + dlFl5 = 0

(23)

 n=l



We obtain the following solution of the ODE system (23)

Fl = κl

Gl = − (2alµ1µ3 − blµ2µ3 − µ2) ξ + Cl (24) 2µ1 (alµ1 − blµ2)
where Cl, κl, µ1, µ2, µ3,al, and bl are arbitrary constants that satisfy the following
algebraic equations

4alµ21κ4l dl + 4alµ21κ2l cl − 4alµ1µ2µ3 − 4µ1µ2κ2l blcl

−4µ1µ2κ4l bldl + b2l µ22µ23 + 2blµ22µ3 + µ22













N 

N 

N 

+4alµ21

αlnκ2n − 4µ1µ2bl

αlnκ2n + 4alµ21

γlnκ2n κ2l

 n=l



 n=l



 n=l















N 

N 

N 

−4µ1µ2bl

γlnκ2n κ2l + 4alµ21

βlnκ4n − 4µ1µ2bl

βlnκ4n = 0

 n=l



 n=l



 n=l



(25)

The corresponding solution of the system of equations (2) is given by

q(l)(x, t) = κl exp i µ3x − (2alµ1µ3 − blµ2µ3 − µ2) (µ1x − µ2t) + Cl 2µ1 (alµ1 − blµ2)

, (26)

where 1 ≤ l ≤ N.

4. CONCLUSIONS
In this paper, we have considered the dynamics of N -coupled NLSEs with spatio-temporal dispersion by using the Lie symmetry analysis. Two types of nonlinear media that have been considered are those with Kerr law nonlinearity and parabolic law nonlinearity. The systems of equations (1) and (2) under consideration was analyzed by the theory of Lie symmetry method to reduce it to ordinary differential equations. We note that because of the dimensions of the Lie algebras, we did not resort to optimal symmetry technique. Corresponding to each reduction, certain exact solutions of the nonlinear partial differential equations are obtained.
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