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[INVITED] Soliton propagation through nanoscale waveguides in optical metamaterials
Optics & Laser Technology 77 (2016) 177–186
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Soliton propagation through nanoscale waveguides in optical metamaterials$ Yanan Xu a, Michelle Savescu b, Kaisar R. Khan c, Mohammad F. Mahmood d, Anjan Biswas a,e,n, Milivoj Belic f a
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematics, Kuztown University of Pennsylvania, 15200 Kutztown Road, Kuztown, PA 19530, USA c Department of Electrical Engineering and Computer Science, McNeese State University, Lake Charles, LA 70605, USA d Department of Mathematics, Howard University, Washington, DC 20059, USA e Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 80203, Saudi Arabia f Science Program, Texas A & M University at Qatar, PO Box 23874, Doha, Qatar b
art ic l e i nf o
a b s t r a c t
Article history: Received 4 August 2015 Accepted 21 August 2015
This paper studies the dynamics of soliton propagation through optical metamaterials. The proposed model will be studied with five forms of nonlinearity. They are Kerr law, power law, parabolic law, dualpower law and log-law. The integration scheme that will be adopted is the method of undetermined coefficients. Bright, dark and singular soliton solutions will be obtained. The essential conditions for the existence of these solitons will naturally emerge. & 2015 Published by Elsevier Ltd.
Keywords: Solitons Integrability Metamaterials
1. Introduction The theory of solitons in optical fibers and optical metamaterials is a very fascinating area of research in nonlinear optics [1–25]. Optical metamaterials possess both negative permittivity and negative permeability that cannot be found in nature; but can be engineered by using advanced processing technology [17]. This material has been fabricated using nano-fabrication technology by several research groups [11,17]. They manipulated the periodic structure of photonic crystal as well as resonant ring for negative permeability [11,17]. Recently, by using metamaterials, Shalaev and others demonstrated optical waveguides in visible and infrared regions [17]. One inherent property of optical metamaterials in optical frequency is its loss. Different waveguide structures were proposed using optical metamaterials [17]. As long as optical wave is guided, soliton pulses can evolve owing to delicate balance between dispersion and nonlinearity. However it is always a challenge to compensate for the loss when engineering these types of waveguides using metamaterials. The theoretical results showed that metamaterials enhance nonlinearity by confining electrical field in a small region that allows more light–matter interaction [11,17,19,20]. In metamaterials, linear and nonlinear coefficients of ☆
Invited Article by the Associate editor: Luca Palmieri. Corresponding author at: Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA. E-mail address: [email protected] (A. Biswas). n
http://dx.doi.org/10.1016/j.optlastec.2015.08.021 0030-3992/& 2015 Published by Elsevier Ltd.
the propagation equation can be tuned to achieve any combination of signs that is not possible in regular materials. These properties of metamaterials lead to improved propagation of a wider variety of solitary waves, efficient phase-matching and modulational instability [12,19,20]. Numerical as well as analytical results of soliton propagation in several nanoscale optical waveguides were reported by several authors [12,19,20]. Earlier results reveal that similar regular (positive indexed) dielectric material dispersion plays a pivotal role in supporting short duration soliton pulses. Optical waveguides with selected wavelengths can be implemented in photonic crystal partially filled with gold and nanoparticles. Recently, theoretical results are reportedfor Y-splitter and bend waveguide structures [11]. The dynamics of soliton propagation through these optical metamaterials is governed by the nonlinear Schrödinger's equation (NLSE) with a few perturbation terms. This model was first reported during 2011 [21]. With the advent of such a model, a plethora of results have been reported. The integrability aspect of this model was studied with various forms of nonlinearity. The integration tools that were applied are simplest equation approach, functional variable method, first integral scheme, Kudryashov's method, trial solutions approach, F-expansion scheme and others [5–9,13]. These algorithms yielded solitons, shock waves and other solutions to the model that appeared with several integrability conditions. In addition to these exact soliton solutions, very recently semi-inverse variational principle was applied to extract bright and exotic soliton solutions to the model [23].
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Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
These are analytical soliton solutions although they are not exact. This paper will apply the method of undetermined coefficients that is otherwise conveniently known as ansatz scheme, to retrieve exact soliton solutions. Bright, dark and singular soliton solutions will be recovered that will appear with essential integrability conditions which stems out from the solution structure of the model.
2. Governing equation and mathematical analysis The dynamics of solitons in optical metamaterials is governed by the nonlinear Schrödinger's equation (NLSE) which in the dimensionless form is given by [20]
3λ + 2ν = 2κ ( 3θ1 + θ 2 − θ 3 ).
(8)
This follows from the fact that the amplitude portion P (x, t ) can be written in terms of the wave variable g (x − vt ) with v being the speed of the wave. The two relations (7) and (8) are obtained by setting the coefficients of linearly independent functions from (6) to zero. These two expressions serve as the existence condition for the solitons that is commonly referred to as constraint relation. The speed of the soliton stays the same for all laws of nonlinearity, namely for all forms of the functional F introduced in (1) and for all kinds of solitons. The constraint relation (8) however modifies with power and dual-power laws. It is the real part equation that will be further analyzed in detail for various nonlinear forms of F in the following sections.
iqt + aqxx + F ( q 2 ) q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q 3. Kerr law
+ θ1 (
q 2q
)xx +
θ 2 q 2 qxx
+
⁎ θ 3 q2qxx
(1)
Eq. (1) is the NLSE that is studied in the context of metamaterials. Here in (1), a and b are the group velocity dispersion and the selfphase 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 inter-modal dispersion. This arises from the fact that group velocity of light in multi-mode fibers depends on chromatic dispersion as well as the propagation mode involved. Next, θj for j = 1, 2, 3 are the perturbation terms that appear in the context of metamaterials [1,5–9,13]. Finally, the independent variables are x and t that represent spatial and temporal variables respectively with the dependent variable q (x, t ) being the complex-valued wave profile. The real-valued algebraic functional F must possess smoothness of the complex-valued function F (|q|2) q : C ↦C . Treating the complex plane C as two-dimensional linear space R2, the function F (|q|2) q is k times continuously differentiable provided ∞
F ( |q|2 ) q ∈ ⋃ C k ( ( − n, n) × ( − m, m); R2). m, n = 1
(2)
In order to start with the analysis of (1), the starting hypothesis is
q (x, t ) = P (x,
t ) eiϕ ,
(3)
In (2), P (x, t ) represents amplitude portion of the wave while ϕ (x, t ) is the phase component that is given by
ϕ = − κx + ωt + θ .
(4)
where κ gives the soliton frequency and ω being the soliton wave number while θ represent the phase constant. After substituting (3) into (1) and decomposing into real and imaginary parts lead to 2
( ω + ακ + aκ 2) P − a ∂∂xP2
⎛ ∂P + 6θ1P ⎜ ⎟ ⎝ ∂x ⎠
∂ 2P ( 3θ1 + θ2 + θ3 ) = 0 ∂x2
(5)
∂P ∂P ∂P − ( α + 2 aκ ) = ( 3λ + 2ν − 6θ1κ − 2θ 2 κ + 2θ 3 κ ) P 2 ∂t ∂x ∂x
(9)
For Kerr law nonlinearity the results of bright, dark and singular soliton have been already reported in the past [5,6]. Therefore, this section will just list the results from these earlier published results [5,6]. It is only the singular solitons of second type that will be derived in detail. 3.1. Bright solitons For Kerr law nonlinear medium, bright 1-soliton solution in optical metamaterials is given by [5]
q (x, t ) = A sech [B (x − vt )] ei (−κx + ωt + θ )
(10)
where A is the amplitude and B is the inverse width of the soliton. The relation between amplitude and width is given by
( b − λκ − 5θ1κ 2) A2 − 3θ1A2B2 − 2aB2 = 0.
(11)
The wave number is
ω = aB2 − aκ 2 − ακ
(12)
and the additional constraint condition is
(13)
3.2. Dark solitons
q (x, t ) = A tanh [B (x − vt )] ei (−κx + ωt + θ ) .
(6)
respectively. The imaginary part equation (6) implies the relations
and
⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
For Kerr law, dark soliton solution is given by [5]
and
v = − α − 2aκ
iqt + aqxx + b q 2 q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q
6θ1 + θ 2 + θ 3 = 0.
⎞2
− P 3 {b − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) } + P 2
This law is also known as the cubic nonlinearity and is considered to be the simplest known form of nonlinearity. Most optical fibers that are commercially available nowadays obey this Kerr law of nonlinearity. Therefore, in this first section the attention will be on optical metamaterials with cubic nonlinearity. In this case F (u) = bu for some non-zero constant b [4]. Therefore, the governing equation given by (1) with Kerr law nonlinearity reduces to [5]
(7)
(14)
In this case, the parameters A and B are referred to as free parameters and these are connected as
A2 ( b − λκ − 5θ1κ 2) + 6θ1A2 B2 + 2aB2 = 0 and the wave number is
(15)
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
ω = − ( ακ + aκ 2 + 2aB2 + 6θ1A2 B2)
(16)
together with the constraint condition (13) that also remains valid here.
focussing singularity [4]. In this case, NLSE given by (1) modifies to
iqt + aqxx + b q 2n q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
3.3. Singular solitons (Type-I)
q (x, t ) = A csch [B (x − vt )] ei (−κx + ωt + θ ) .
(17)
where the free parameters A and B are connected as in (11). The wave number is also located in (12) while the same constraint (13) holds here. 3.4. Singular solitons (Type-II)
P (x, t ) = A cothp τ ,
(18)
where
τ = B (x − vt ).
(19)
In this case also, A and B are free parameters and p is the unknown exponent. Substituting this hypothesis into (5) and (6), the real and imaginary parts are
ap (p −
τ−
{2ap2B2
+ ( ω + ακ +
aκ 2
)
} cothp
τ
The starting hypothesis for bright solitons is given by [4,15,16]
P (x, t ) = A sechp τ ,
− A2 B2 {6θ1p2 + p (p − 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p − 2 τ + A2 ⎡⎣ b − λκ + κ 2 ( θ1 + θ 2 + θ 3 )
ap2B2 − ( ω + ακ + aκ 2) − ap (p + 1) B2 sech2 τ + (b − λκ ) A2n sech2np τ A2 { ( θ1 + θ 2 + θ 3 ) κ 2 − p2 B2 ( 9θ1 + θ 2 + θ 3 ) } sech2p τ + A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } sech2p + 2 τ = 0
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} sech2np τ − A2 {2κ ( 3θ1 + θ 2 − θ 3 ) } sech2p τ = 0
}
(
(20)
ω=
and
v + α + 2aκ + A2 {3λ + 2ν − 2κ ( 3θ1 + θ 2 − θ 3 ) } coth2p τ = 0,
1 n
(28)
(21)
1 {aB2 − n2 ( ακ + aκ 2) } , n2
B = nκ
θ1 + θ 2 + θ 3 , 9θ1 + θ 2 + θ 3
and the amplitude–width relation as
p = 1.
n2 (b − λκ ) A2n = a (n + 1) B2.
(22)
Next, setting the coefficients of undetermined coefficients or linearly independent functions, from (20), to zero yields the constraint relation (13), the wave number given by (12) and the relation between the free parameters A and B as in (11). The imaginary part equation leads to (7) and (8). Therefore, singular 1-soliton solution of Type-II in Kerr law medium is
(23)
where the definition of parameters and their respective constraints are all in place.
(29)
width of the soliton
respectively. By virtue of balancing principle for optical solitons, equating the exponents 3p and p + 2 from real part equation (20) gives
ei (−κx + ωt + θ ) .
(27)
Next, setting the coefficients of linearly independent functions, in (26), to zero reveals the wave number
)
− A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p + 2 τ = 0
(26)
and
p=
+ 12θ1p2 B2 + 2p2 B2 3θ1 + θ 2 + θ 3 ⎤⎦ coth3p τ
q (x, t ) = A coth [B (x − vt )]
(25)
respectively. From balancing principle, equating the exponents 2np and 2, from real part (26) gives
+ ap (p + 1) B2 cothp + 2 τ
{
4.1. Bright solitons
which upon substitution, simplifies real and imaginary part equations (5) and (6) to
In this case, the starting hypothesis is given by [15,16]
cothp − 2
(24)
This equation will now be further analyzed in the following sections to obtain four forms of solitons.
In this case, singular soliton solution of first kind is [6]
1) B 2
179
(30)
(31)
From (30) and (31), the amplitude of the soliton is
⎡ a (n + 1) κ 2 (θ1 + θ 2 + θ 3 ) ⎤1/2n A=⎢ ⎥ ⎣ (b − λκ )(9θ1 + θ 2 + θ 3 ) ⎦
(32)
The amplitude and width of the soliton will exist provided
(θ1 + θ 2 + θ 3 )(9θ1 + θ 2 + θ 3 ) > 0,
(33)
and
a (b − λκ )(θ1 + θ 2 + θ 3 )(9θ1 + θ 2 + θ 3 ) > 0,
(34)
which follows from (30) and (32) respectively. The next constraint relation that stems out from the coefficient of sech2p + 2 τ in (26) is 4. Power law
6θ1 + (n + 1) ( 3θ1 + θ 2 + θ 3 ) = 0.
This law of nonlinearity also arises in nonlinear plasmas that solves the problem of small K-condensation in weak turbulence theory. It also arises in the context of nonlinear optics. Physically, various materials, including semiconductors, exhibit power law nonlinearities. For power law, F (u) = bun , where n is the power law nonlinearity parameter. It is important to note that 0 < n < 2 to prevent wave collapse and in particular n ≠ 2 to prevent self-
The imaginary part equation given by (27) leads to the speed v given by (7) as well as the following set of constraints:
(2n + 1) λ + 2nν = 0,
(35)
(36)
and
3θ1 + θ 2 − θ 3 = 0.
(37)
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Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
Thus, bright 1-soliton solution in optical metamaterials with power law nonlinearity is
q (x, t ) = A sech1/ n [B (x − vt )] ei (−κx + ωt + θ )
(38)
with the definition of parameters and necessary constraints in place, as indicated.
For dark soliton solution, the starting hypothesis is given by [4,15,16]
P (x, t ) = A tanhp τ ,
(39)
which upon substituting into the real and imaginary part equations (5) and (6) simplifies to
ap (p −
tanhp − 2
q (x, t ) = A csch1/ n [B (x − vt )] ei (−κx + ωt + θ )
(46)
with the definition of parameters and necessary constraints in place.
4.2. Dark solitons
1) B 2
bright solitons, the same results (29)–(35) fall out. The imaginary part equation (45) leads to (7) (36) and (37), namely the speed and necessary constraints. Thus, singular 1-soliton solution in optical metamaterials with power law nonlinearity is
τ − {ω + ακ +
aκ 2
+ 2ap (p +
1) B2}
4.4. Singular solitons (Type-II) With Kerr law nonlinear medium, singular soliton hypothesis is given by [15,16]
P (x, t ) = A cothp τ ,
(47)
which upon substituting into (5) and (6), the real and imaginary part equations reduce to
tanhp τ + ap (p + 1) B2 tanhp + 2 τ + (b − λκ ) A2n tanh(2n + 1) p τ
ap (p − 1) B2 cothp − 2 τ − {ω + ακ + aκ 2 + 2ap (p + 1) B2} cothp τ
− pA2B2 {6pθ1 + (p − 1) ( 3θ1 + θ 2 + θ 3 ) } tanh3p − 2 τ
+ ap (p + 1) B2 cothp + 2 τ + (b − λκ ) A2n coth(2n + 1) p τ + A2 {12p2 B2θ1 + p (2p + 1) B2 ( 3θ1 + θ 2 + θ 3 )
− pA2B2 {6pθ1 + (p − 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p − 2 τ
+ κ 2 ( θ1 + θ 2 + θ 3 ) } tanh3p τ − pA2B2 {6pθ1 + (p + 1) ( 3θ1 + θ 2 + θ 3 ) } tanh3p + 2 τ = 0
+ A2 {12p2 B2θ1 + p (2p + 1) B2 ( 3θ1 + θ 2 + θ 3 )
(40)
+ κ 2 ( θ1 + θ 2 + θ 3 ) } coth3p τ
and
− pA2B2 {6pθ1 + (p + 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p + 2 τ = 0
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} tanh2np τ − 2κA2 ( 3θ1 + θ 2 − θ 3 ) tanh2p τ = 0
(41)
respectively. Now, setting the coefficient of stand-alone linearly independent function tanhp − 2 τ to zero leads to (22). Again from balancing principle, equating the exponents (2n + 1) p and p þ2 gives (28). From (22) and (28),
n = 1.
(42)
Thus, for dark solitons with power law nonlinearity condenses to Kerr law nonlinearity. The imaginary part equation (41) leads to (7), (36) and (37) with (42). Therefore, dark 1-soliton solution for power law nonlinearity is also given by (14) with the wave number as in (16) and the relation between the free parameters as in (15). The constraint (13) also remains valid, in this case.
(48)
and
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} coth2np τ − 2κA2 ( 3θ1 + θ 2 − θ 3 ) coth2p τ = 0
(49)
respectively. Similarly, as in dark soliton solutions relation (42) falls out. Therefore, singular solitons of second type, with power law nonlinearity, will exist if power law nonlinearity boils down to Kerr law. The imaginary part equation (6) which leads to (7), (36) and (37) along with (40). Therefore, singular 1-soliton solution for power law nonlinearity is also given by (23) with parameter definitions and constraints as in dark solitons with Kerr law.
5. Parabolic law 4.3. Singular solitons (Type-I) The starting hypothesis for singular solitons (Type-I) is given by [15,16]
P (x, t ) = A cschp τ ,
(43)
which upon substitution, simplifies (5) and (6) respectively to
ap2B2 − ( ω + ακ + aκ 2) − ap (p + 1) B2 csch2 τ + (b − λκ ) A2n csch2np τ + A2 { ( θ1 + θ 2 + θ 3 ) κ 2 − p2 B2 ( 9θ1 + θ 2 + θ 3 ) } csch2p τ + A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } csch2p + 2 τ = 0
(
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} csch2np τ (45)
From balancing principle applied to real part equation (44) equating the exponents 2np and 2 gives the same value of p as in (28). Next, from the linearly independent functions, similarly as in
)
iqt + aqxx + b1 q 2 + b2 q 4 q = iαqx + iλ (
(44)
and
− A2 {2κ ( 3θ1 + θ 2 − θ 3 ) } csch2p τ = 0.
This law is alternatively known as the cubic–quintic nonlinearity and is studied in nonlinear interaction between Langmuir waves and electrons. It describes the nonlinear interaction between the high frequency Langmuir waves and the ion-acoustic waves by pondermotive forces. It takes the form F (u) = b1u + b2 u2 for non-zero constants b1 and b2. For parabolic law medium, the NLSE given by (1) changes to
q 2q
)x + iν ( q 2)x q
⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
(50)
The rest of this section will focus on the details of retrieving soliton solutions to this model along with their conditions for existence. 5.1. Bright solitons For bright solitons with parabolic law, the starting hypothesis is
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
[4,15,16]
P (x, t ) =
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
A
( D + cosh τ )p
,
ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + p cosh + D τ ( ) ( D + cosh τ )p + 1
−
−
( D + cosh τ )p + 2
{
}
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
}
+
1
2
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
5p
( D + cosh τ )
P (x, t ) = ( A + B tanh τ )
= 0. (52)
{ 3λ + 2ν − 2κ ( 3θ1 + θ2 − θ3 ) } = 0 (53)
1
p = 2.
(54)
Next, from the undetermined coefficients of linearly independent functions, one recovers the wave number 1 4
( aB2 − 4aκ 2 − 4ακ ),
(63)
ap (p − 1) μ2 ( B2 − A2 ) ( A + B tanh τ )
p−2
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B tanh τ )
p−1
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B tanh τ ) − 2Aaμ2 p (2p + 1) ( A + B tanh τ ) + ap (p + 1) μ2 ( A + B tanh τ ) + ⎡⎣ 2μ2 p2 B2 − 3A2
(
)( 9θ
{
1
p
p+1
p+2
) + θ ) } ⎤⎦ ( A + B tanh τ )
+ θ2 + θ3
+ B2 b1 − λκ + κ 2 ( θ1 + θ 2
3
3p
2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
5θ1 + θ 2 + θ 3 = 0.
⎡ 1 ⎧ ⎨ −D ⎡⎣ θ1 ( b1 − λκ − 4θ1κ 2) + ab2 ⎤⎦ A=⎢ ⎣ 2b2 θ1 ⎩
{
± D2 θ1 ( b − λκ − 4θ1κ 2} + ab2
2
}
− 3ab2 θ1
( b − λκ − 4θ κ ) 1
1
2
(
⎞ ⎤1/2 ⎟⎥ ⎟⎥ D2 − 1 ⎟ ⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦ (57)
)
subject to the conditions
b2 θ1 ⎡⎣ − D θ1 ( b1 − λκ − 4θ1κ 2) + ab2
{
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
2
}
}
(
1
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
( A + B tanh τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) 5 p + b2 B2 ( A + B tanh τ ) = 0
(64)
and 2p
v + α + 2aκ − {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) = 0.
(65)
Again balancing exponents 3p + 1 with p + 2, from (64), leads to the same value of p as given by (54). There are more exponent pairs in (64) that will lead to (54) by virtue of the same principle. The coefficient of stand-alone linearly independent function (A + B tanh τ ) p gives
)
− 3ab2 θ1 D2 − 1 ] > 0,
( b − λκ − 4θ κ ) 1
( A + B tanh τ )3p − 2
(56)
From the remaining linearly independent functions, the amplitude and the width are
2
A = ± B. (58)
and
(62)
τ = μ (x − vt ).
(55)
and the constraint
±
(61)
p
2
( D + cosh τ )2p
respectively. By balancing principle, applied to (52), equating the exponents 3p and p + 1 or the pair 5p and p + 2 leads to
ω=
ei (−κx + ωt + θ )
Here A, B and μ are all free parameters. With (62), Eqs. (5) and (6) respectively reduce to
and
v + α + 2aκ +
A D + cosh [B (x − vt )]
where, in this case
b2 A4
A2
(60)
For dark optical solitons [15,16],
}
( D + cosh τ )3p + 2 −
b1 − λκ − 4θ1κ 2 . aD + θ1A2
5.2. Dark solitons
( D + cosh τ )3p + 1 2
B=A
where the parameter definitions and constraints are all in place.
( D + cosh τ )3p
){ 6p θ B
The width of the soliton is connected to the amplitude by the relation
q (x, t ) =
A2 b1 − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
(
(59)
Now the imaginary part equation (53) in this case gives (7) and (8). Finally, bright 1-soliton solution to optical metamaterials with parabolic law nonlinearity is
ap (p + 1) B2 ( D2 − 1)
A2 B2 D2 − 1
2
}
− 3ab2 θ1 ( D2 − 1)( b1 − λκ − 4θ1κ 2) > 0.
(51)
where A is the amplitude of the soliton and D is a newly introduced parameter and the usual definition of τ is carried over from (19) with the unknown exponent p. Substituting (50) into (5) and (6) gives
−
181
(66)
The undetermined coefficients of the remaining linearly independent functions from (64) yield the same constraint (56) along with the wave number
182
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
ω = aμ2 − ακ − aκ 2,
(67)
and the relation between the free parameters
b2 θ1 ⎡⎣ D θ1 ( b1 − λκ − 4θ1κ 2) + ab2
{
} D { θ ( b − λκ − 4θ κ ) + ab } 2
±
2aμ2 A= , b1 − λκ − 4θ1 ( μ2 + κ 2)
1
2
2
1
2
(
)
− 3ab2 θ1 D2 + 1 ] > 0,
( b − λκ − 4θ κ ) 1
(68)
2
1
which stays valid as long as
(75)
λκ + 4θ1 ( μ2 + κ 2) ≠ b1.
(69)
Next the imaginary part equation (65), relations (7) and (8) are valid. Therefore, dark 1-soliton solution with parabolic law is given by
A {1 ± tanh [μ (x − vt )]} ei (−κx + ωt + θ )
q (x, t ) =
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
For first type of singular solitons, the starting hypothesis is [15,16]
A p, + τ) D sinh (
(76)
The parameters A and B are related as
B=A
5.3. Singular solitons (Type-I)
2
}
+ 3ab2 θ1 ( D2 + 1)( b1 − λκ − 4θ1κ 2) > 0.
(70)
with the respective parameters and constraints as indicated.
P (x, t ) =
and
b1 − λκ − 4θ1κ 2 . aD − θ1A2
(77)
Next, the imaginary part equation (73), relations (7) and (8) are obtained. Finally, singular 1-soliton solution to optical metamaterials with parabolic law nonlinearity is given by
q (x, t ) =
(71)
A D + sinh [B (x − vt )]
ei (−κx + ωt + θ ) (78)
where A, B and consequently D are free parameters. With this hypothesis Eqs. (5) (6) respectively transform to
where the parameter definitions and constraints are all in place.
ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + p ( D + sinh τ ) ( D + sinh τ )p + 1
5.4. Singular solitons (Type-II)
−
For singular solitons of second type [15,16],
ap (p + 1) B2 ( D2 + 1)
( D + sinh τ )
P (x, t ) = ( A + B coth τ )
p+2
−
− p2 B2 ( 3θ1 + θ 2 + θ 3 )
}
2
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
+
){ 6p θ B 2
1
2
b2 A4 5p
( D + sinh τ )
p−1
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B coth τ )
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
− 2Aaμ2 p (2p + 1) ( A + B coth τ )
}
+ ap (p + 1) μ2 ( A + B coth τ )
( D + sinh τ )3p + 2 −
p−2
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B coth τ )
}
( D + sinh τ )3p + 1
(
τ as in (63). Substitution of (79) into
ap (p − 1) μ2 ( B2 − A2 ) ( A + B coth τ )
( D + sinh τ )3p
A2 B2 D2 + 1
(79)
with the same definition of (5) and (6) implies
{
A2 b1 − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) − 6θ1p2 B2 −
p
+ ⎡⎣ 2μ2 p2 B2 − 3A2
(
= 0.
+
(72)
B2
{b
1
− λκ +
)( 9θ
κ2
1
p
p+1
p+2
) + θ ) } ⎤⎦ ( A + B coth τ )
+ θ2 + θ3
( θ1 + θ2
3
3p
2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
and
v + α + 2aκ +
{
A2 3λ + 2ν − 2κ ( 3θ1 + θ 2 − θ 3 )
( D + sinh τ )2p
( A + B coth τ )3p − 2
} = 0. (73)
Proceeding in the same way as in bright solitons, from real part equation (72), relations (54)–(56) are all recovered. The free parameter A in this case is
⎡ 1 ⎧ ⎨ D ⎡⎣ θ1 ( b1 − λκ − 4θ1κ 2) + ab2 ⎤⎦ A=⎢ ⎣ 2b2 θ1 ⎩
{
± D2 θ1 ( b − λκ − 4θ1κ 2} + ab2
( b − λκ − 4θ ) 1
1
κ2
subject to the conditions
2
}
+ 3ab2 θ1
(
⎞ ⎤1/2 ⎟⎥ ⎟⎥ D2 + 1 ⎟ ⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦ (74)
)
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
( A + B coth τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ ) 5p + b2 B2 ( A + B coth τ ) = 0
(80)
and 2p
v + α + 2aκ − {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ ) =0
(81)
respectively. These expressions lead to (7)–(9) as well as (66)–(69). Therefore, singular 1-soliton solution of Type-II with parabolic law in optical metamaterials is given by
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
A ( 1 ± coth [μ (x − vt )] ) ei (−κx + ωt + θ )
q (x, t ) =
(82)
with the respective parameters and constraints are discussed.
183
⎡ (2n + 1) ( D2 − 1)( b − λκ ) ⎤1/2n 1 ⎥ A = ⎢− ⎢⎣ ⎥⎦ 2 (n + 1) b2 D
(89)
provided
b2 D ( D2 − 1) ( b1 − λκ ) < 0,
6. Dual-power law This model describes saturation of nonlinear refractive index and its exact soliton solutions are known. The effective NLSE, with this form of nonlinearity, serves as a basic model to describe spatial solitons in photovoltaic-photorefractive materials such as LiNbO3. Optical nonlinearities in many organic and polymer materials are governed with such form of nonlinearity. The governing NLSE in optical metamaterials for dual-power law nonlinearity is
(90)
and the width is
B=
n ( b1 − λκ ) D (n + 1)
−
(2n + 1) ( D2 − 1) ab2
(91)
that stays valid for
ab2 ( D2 − 1) < 0.
(92)
Finally, bright 1-soliton solution to optical metamaterials with dual-power law nonlinearity is given by
(
) 2 2 = iαqx + iλ ( q q)x + iν ( q )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx ,
iqt + aqxx + b1 q 2n + b2 q 4n q
(83)
q (x, t ) =
A 1/2n
( D + cosh [B (x − vt )] )
ei (−κx + ωt + θ ) (93)
for F (u) = b1u2n + b2 u4n , with non-zero b1 and b2, where n is the power law parameter. This section will now comprehensively derive the soliton solutions to NLSE in the following subsections.
where the parameter definitions and constraints are all in place.
6.1. Bright solitons
With the same starting hypothesis given by (62), the real and imaginary parts (5) and (6) are
With the same starting hypothesis as given by (51), the real part equations (5) and (6) reduce to
−
{
A2 κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
− 2Aaμ2 p (2p + 1) ( A + B tanh τ )
}
+ ap (p + 1) μ2 ( A + B tanh τ )
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
(
){ 6p θ B 2
1
2
(2n + 1) p
( D + cosh τ )
−
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
2
( A + B tanh τ )3p − 2
b2 A4n (4n + 1) p
( D + cosh τ )
{ (2n + 1) λ + 2nν} A2n 2np
( D + cosh τ )
=0
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
= 0. (84)
−
2κ ( 3θ1 + θ 2 − θ 3 ) A2
( D + cosh τ )2p (85)
respectively. The imaginary part equation leads to the constraints given by (7), (36) and (37). By balancing principle applied to real part (84) equating the exponents, (4n + 1) p and p + 2, implies
p=
1 . 2n
1 ( aB2 − 4n2aκ 2 − 4n2ακ ), 4n2
(94)
and 2np
(86)
(87)
and the constraint
3 (2n + 3) θ1 + (2n + 1) ( θ 2 + θ 3 ) = 0.
( A + B tanh τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) (2n + 1) p + ( b1 − λκ ) B2 ( A + B tanh τ ) (4n + 1) p + b2 B2 ( A + B tanh τ ) =0 v + α + 2aκ + {(2n + 1) λ + 2nν} ( A + B tanh τ )
Next, from the undetermined coefficients of linearly independent functions, one recovers the wave number
ω=
p+1
p+2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
and
v + α + 2aκ +
p
( A + B tanh τ )3p
( D + cosh τ )3p + 2
( b1 − λκ ) A2n
p−1
+ {2μ2 p2 ( B2 − 3A2 )( 9θ1 + θ 2 + θ 3 ) + B2κ 2 ( θ1 + θ 2 + θ 3 ) }
}
( D + cosh τ )3p + 1 A2 B2 D2 − 1
p−2
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B tanh τ )
( D + cosh τ )3p
+
−
2
ap (p − 1) μ2 ( B2 − A2 ) ( A + B tanh τ )
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B tanh τ )
ap (p + 1) B2 ( D2 − 1) ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + − p p+1 ( D + cosh τ ) ( D + cosh τ )p + 2 ( D + cosh τ ) −
6.2. Dark solitons
(88)
From the remaining linearly independent functions, the amplitude of the soliton is
2p
− 2κ ( 3θ1 + θ 2 + θ 3 )( A + B tanh τ )
=0
(95)
respectively. Again balancing exponents 3p + 1 with p + 2 leads to the same value of p as given by (54). Next, equating the exponents (2n + 1) p and p + 2 leads to (42). This shows that dark solitons for dualpower law collapse to the case of parabolic law. Therefore, all results from parabolic law dark solitons given by (56) and (66)–(69) remain valid. In addition, the imaginary part equation leads to (7) and (8). Finally, dark 1-soliton solution to dual-power law nonlinearity is given by (70), with all definition of parameters and their respective constraints in place.
184
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
6.3. Singular solitons (Type-I)
2
ap (p − 1) μ2 ( B2 − A2 ) ( A + B coth τ )
p−2
In this case, substituting the starting hypothesis given by (71) into (5) and (6) gives
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B coth τ )
ap (p + 1) B2 ( D2 − 1) ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + − p p+1 ( D + sinh τ ) ( D + sinh τ )p + 2 ( D + sinh τ )
− 2Aaμ2 p (2p + 1) ( A + B coth τ )
−
−
{
A2 κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
{ (
){ 6p θ B 2
1
2
(2n + 1) p
( D + sinh τ )
−
2
( A + B coth τ )3p − 2
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
b2 A4n
( D + sinh τ )(4n+ 1) p
= 0. (96)
and
v + α + 2aκ +
{ (2n + 1) λ + 2nν} A2n 2np
( D + sinh τ )
−
2κ ( 3θ1 + θ 2 − θ 3 ) A2
( D + sinh τ )2p (97)
=0
respectively. The imaginary part equation clearly leads to (7), (36) and (37). While the balancing principle from real part (96) yields (86), the remaining undetermined coefficients lead to (87) and (88) as well. The free parameters A and B are now
⎡ (2n + 1) ( D2 + 1)( b − λκ ) ⎤1/2n 1 ⎥ A = ⎢− 2 (n + 1) b2 D ⎥⎦ ⎢⎣
(98)
provided
b2 D ( b1 − λκ ) < 0,
( A + B coth τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 1 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 2 (2n + 1) p + ( b1 − λκ ) B2 ( A + B coth τ ) (4n + 1) p + b2 B2 ( A + B coth τ ) =0
(103)
and 2np
v + α + 2aκ + {(2n + 1) λ + 2nν} ( A + B coth τ ) 2p
− 2κ ( 3θ1 + θ 2 + θ 3 )( A + B coth τ )
=0
(104)
respectively. Once again, proceeding along the same lines as in the case of dark solitons, (54) and (42) are recovered. Thus, this form of singular solitons of Type-II exists whenever dual-power law reduces to parabolic law nonlinearity. Hence, all results from (7), (36), (37) and (66)–(69) hold. Finally, singular 1-soliton solution for dualpower law is given by (82) along with the parameters and constraints as described.
(99)
and
B=
p+2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
( D + sinh τ )3p + 2
( b1 − λκ ) A2n
p+1
( A + B coth τ )3p
( D + sinh τ )3p + 1 A2 B2 D2 − 1
p
+ {2μ2 p2 ( B2 − 3A2 )( 9θ1 + θ 2 + θ 3 ) + B2κ 2 ( θ1 + θ 2 + θ 3 ) }
( D + sinh τ )3p
+
−
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B coth τ )
+ ap (p + 1) μ2 ( A + B coth τ )
}
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
p−1
7. Log law
n ( b1 − λκ ) D (n + 1)
−
(2n + 1) ( D2 + 1) ab2
(100)
only if
(101)
ab2 < 0.
Finally, singular 1-soliton solution to optical metamaterials with dual-power law nonlinearity is
q (x, t ) =
A
( D + sinh [B (x − vt )] )1/2n
ei (−κx + ωt + θ ) (102)
where the parameter definitions and constraints are all in place.
6.4. Singular solitons (Type-II) With starting hypothesis given by (79), the real and imaginary parts (5) and (6) are
In this case F (u) = b ln u for non-zero constant b. This law permits closed form exact expressions Gaussian beams. The advantage of this model is that the radiation from the periodic soliton is absent as the linearized problem contains discrete spectrum only [4]. For log-law medium, the model given by (1) modifies to
iqt + aqxx + b ( ln q 2 ) q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
(105)
The solutions of NLSE in log-law nonlinear medium lead to Gaussian solitons that are occasionally referred to as Gaussons [15,16]. Therefore, the starting hypothesis for (105) is given by [15,16]
P (x, t ) = Ae−τ
2
(106)
where A is the amplitude and B is the inverse width of the Gausson. Substituting this hypothesis into (5) and (6), the real and imaginary parts respectively simplify to
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
( ω + ακ + aκ 2 + 2aB2 − 2b ln A) − ( 4aB2 − 2b) τ 2 2 + A2 {λκ − κ 2 ( θ1 + θ 2 + θ 3 ) − 2B2 ( 3θ1 + θ 2 + θ 3 ) } e−2τ 2 + A2 {24B2θ1 + 4B2 ( 3θ1 + θ 2 + θ 3 ) } τ 2e−2τ = 0
8. Conclusions
(107)
and
v + α + 2aκ + A {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } = 0.
(108)
From the undetermined coefficients of linearly independent functions in (107), the wave number is
ω = − ακ − aκ 2 − 2aB2 + 2b ln A,
(109)
and the width of Gaussson is given by
B=
λκ − κ 2 ( θ1 + θ 2 + θ 3 ) 2 ( 3θ1 + θ 2 + θ 3 )
(110)
with the condition
{λκ − κ 2 ( θ1 + θ 2 + θ 3 ) } ( 3θ1 + θ 2 + θ 3 ) > 0.
(111)
Substituting the width B from (110), the wave number from (109) reduces to
ω= −
aκλ + 2aκ 2θ1 + ( 3θ1 + θ 2 + θ 3 )( ακ − 2b ln A) 3θ1 + θ 2 + θ 3
,
(112)
which holds provided
Next, setting the coefficient of the fourth linearly independent 2 function in (106), namely τ 2e−2τ to zero, gives
9θ1 + θ 2 + θ 3 = 0.
(114)
Finally, the coefficient of Gausson as
τ2 from (107) leads to the width of
b 2a
(115)
which shows that these Gaussons will exist provided
(116)
ab > 0.
This means that GVD and the nonlinear term in (105) must both carry the same sign for Gaussons to exist. The imaginary part equation (108) yields (7) and (8). Equating the two values of the width B of the soliton from (110) and (115) leads to another constraint between the frequency and coefficients of the model (105) as follows:
aλκ − 2bθ1 = ( b + aκ 2) ( θ1 + θ 2 + θ 3 ).
(117)
From (114) and (111), one can recover
θ1κ ( λ + 8θ1κ ) < 0
(118)
which can be treated as another constraint. Next, substituting (115) into (109) leads to an alternate expression to the wave number:
ω = b ( 2 ln A − 1) − κ ( α + aκ ).
(119)
Thus, the Gausson solution to optical metamaterials with log-law nonlinearity is given by
q (x, t ) = Ae−B
This paper obtained soliton solutions in optical metamaterials with five forms of nonlinear media. For Kerr law nonlinearity, there are three forms of solitons that are already reported earlier; therefore this paper derived only singular soliton (Type-II). For the remaining laws all soliton solutions and their derivations are comprehensively reported in this paper. These solutions come with respective integrability criteria that are listed as constraint conditions. These solutions will be immensely useful in the literature of optical metamaterials. These soliton solutions will be a great asset in all future investigations in this area of nonlinear optics. In the presence of perturbation terms these solitons will dictate the adiabatic parameter dynamics and other such features that will be obtained. The quasi-particle theory of optical soliton interaction will be reported. Later, bifurcation analysis of solitons in optical metamaterials will be carried out. Other integration schemes will be applied to these models and those will reveal additional solutions, such as plane waves and periodic singular solutions. The semi-inverse variational principle will extract exotic solitons such as cosh-Gaussian pulses and bright-dark combo optical solitons. All of these are currently under investigation. The results of those research will be reported gradually and sequentially. Finally, the study will be extended to DWDM systems so that efficient soliton transmission can be conducted in parallel, thus improving performance enhancement. These just form a tip of the iceberg.
(113)
3θ1 + θ 2 + θ 3 ≠ 0.
B=
185
2 (x − vt ) 2 i (−κx + ωt + θ )
e
(120)
where the parameter definitions and constraints are all listed above.
Acknowledgment This research is funded by Qatar National Research Fund (QNRF) under the Grant number NPRP 6-021-1-005. The fifth and sixth authors (A.B. & M.B.) thankfully acknowledge this support from QNRF.
References [1] A.H. Arnous, M. Mirzazadeh, S. Moshokoa, S. Medhekar, A. Biswas, M. Belic, Solitons in optical metamaterials with trial solutions approach and Bäcklund transform of Riccati equation, J. Comput. Theor. Nanosci. (2015), submitted for publication. [2] M.A. Baqir, P.K. Choudhury, Waves in coaxial optical fiber under DB boundaries, Optik 125 (2014) 2950–2953. [3] M.A. Baqir, P.K. Choudhury, Investigation of uniaxial isotropic chiral metamaterial waveguide with perfect electromagnetic conductor loading, Optik 126 (2015) 1228–1232. [4] A. Biswas, S. Konar, Introduction to Non-Kerr Law Optical Solitons, CRC Press, Boca Raton, FL, USA, 2006. [5] A. Biswas, K.R. Khan, M.F. Mahmood, M. Belic, Bright and dark solitons in optical metamaterials, Optik 125 (13) (2014) 3299–3302. [6] A. Biswas, M. Mirzazadeh, M. Savescu, D. Milovic, K.R. Khan, M.F. Mahmood, M. Belic, Singular solitons in optical metamaterials by ansatz method and simplest equation approach, J. Mod. Opt. 61 (19) (2014) 1550–1555. [7] A. Biswas, M. Mirzazadeh, M. Eslami, D. Milovic, M. Belic, Solitons in optical metamaterials by functional variable method and first integral approach, Frequenz 68 (11–12) (2014) 525–530. [8] A. Biswas, M. Mirzazadeh, D. Milovic, K.R. Khan, M.F. Mahmood, M. Belic, Singular and topological solitons in optical metamaterials by Kudryashov's method and G′/G − expansion scheme, J. Comput. Theor. Nanosci. (2015), submitted for publication. [9] G. Ebadi, A. Mohavir, J.V. Guzman, K.R. Khan, M.F. Mahmood, L. Moraru, A. Biswas, M. Belic, Solitons in optical metamaterials by F-expansion scheme, Optoelectron. Adv. Mater.—Rapid Commun. 8 (9–10) (2014) 828–832. [10] H. Erfaninia, A. Rostami, Group velocity reduction in multilayer metamaterial waveguide, Optik 124 (2013) 1230–1233. [11] K.R. Khan, K. Mnaymneh, H. Awad, I. Hasan, T. Hall, Optical wave propagation in photonic crystal metamaterials, Appl. Phys. A 117 (2) (2014) 629–634. [12] K.R. Khan, M. Mahmood, A. Biswas, Coherent super-continuum generation in PCF at visible and near infrared wavelengths, IEEE J. Sel. Top. Quantum Electron. 20 (5) (2014) 7500309. [13] E.V. Krishnan, M. Al Gabshi, Q. Zhou, K.R. Khan, M.F. Mahmood, Y. Xu,
186
[14]
[15]
[16]
[17] [18] [19]
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
A. Biswas, M. Belic, Solitons in optical metamaterials by mapping method, J. Optoelectron. Adv. Mater. 17 (3–4) (2015) 511–516. M. Saha, A.K. Sarma, Modulational instability in nonlinear metamaterials induced by cubic-quintic nonlinearities and higher order dispersive effects, Opt. Commun. 291 (2013) 321–325. M. Savescu, K.R. Khan, P. Naruka, H. Jafari, L. Moraru, A. Biswas, Optical solitons in photonic nano-waveguides with an improved nonlinear Schrödinger's equation, J. Comput. Theor. Nanosci. 10 (5) (2013) 1182–1191. M. Savescu, K.R. Khan, R. Kohl, L. Moraru, A. Yildirim, A. Biswas, Optical soliton perturbation with improved nonlinear Schrödinger's equation in nanofibers, J. Nanoelectron. Optoelectron. 8 (2) (2013) 208–220. V.M. Shalaev, Optical negative-index metamaterials, Nat. Photonics 1 (2007) 41–48. V.G. Vaselago, Electrodynamics of media with simultaneous negative electric permittivity and magnetic permeability, Sov. Phys. Usp. 10 (1967) 509–514. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, D. Fan, Role of anomalous self-steepening effect in modulation instability in negative-index material, Opt. Express 14 (2006) 1568.
[20] S. Wen, W. Wang, W. Su, Y. Xiang, X. Fu, D. Fan, Modulation instability in nonlinear negative-index material, Phys. Rev. E 73 (2006) 036617. [21] Y. Xiang, X. Dai, S. Wen, J. Guo, D. Fan, Controllable Raman soliton self-frequency shift in nonlinear metamaterials, Phys. Rev. A 84 (2011) 033815. [22] Y. Xu, Q. Zhou, A.H. Bhrawy, K.R. Khan, M.F. Mahmood, A. Biswas, M. Belic, Solitons in optical metamaterials by traveling wave hypothesis, Optoelectron. Adv. Mater.—Rapid Commun. 9 (3–4) (2015) 384–387. [23] Y. Xu, J. Vega-Guzman, D. Milovic, M. Mirzazadeh, M. Eslami, M.F. Mahmood, A. Biswas, M. Belic, Bright and exotic solitons in optical metamaterials by semi-inverse variational principle (2015), submitted for publication. [24] X. Zhong, K. Chen, Modulation instability in metamaterials with fourth order linear dispersion, second order nonlinear dispersion and three kinds of saturable nonlinearities, Optik 125 (2014) 6733–6738. [25] Q. Zhou, Q. Zhou, Y. Liu, A. Biswas, A. Bhrawy, K.R. Khan, M.F. Mahmood, M. Belic, Solitons in optical metamaterials with parabolic law nonlinearity and spatio-temporal dispersion, J. Optoelectron. Adv. Mater. 16 (11–12) (2014) 1221–1225.
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Soliton propagation through nanoscale waveguides in optical metamaterials$ Yanan Xu a, Michelle Savescu b, Kaisar R. Khan c, Mohammad F. Mahmood d, Anjan Biswas a,e,n, Milivoj Belic f a
Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA Department of Mathematics, Kuztown University of Pennsylvania, 15200 Kutztown Road, Kuztown, PA 19530, USA c Department of Electrical Engineering and Computer Science, McNeese State University, Lake Charles, LA 70605, USA d Department of Mathematics, Howard University, Washington, DC 20059, USA e Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 80203, Saudi Arabia f Science Program, Texas A & M University at Qatar, PO Box 23874, Doha, Qatar b
art ic l e i nf o
a b s t r a c t
Article history: Received 4 August 2015 Accepted 21 August 2015
This paper studies the dynamics of soliton propagation through optical metamaterials. The proposed model will be studied with five forms of nonlinearity. They are Kerr law, power law, parabolic law, dualpower law and log-law. The integration scheme that will be adopted is the method of undetermined coefficients. Bright, dark and singular soliton solutions will be obtained. The essential conditions for the existence of these solitons will naturally emerge. & 2015 Published by Elsevier Ltd.
Keywords: Solitons Integrability Metamaterials
1. Introduction The theory of solitons in optical fibers and optical metamaterials is a very fascinating area of research in nonlinear optics [1–25]. Optical metamaterials possess both negative permittivity and negative permeability that cannot be found in nature; but can be engineered by using advanced processing technology [17]. This material has been fabricated using nano-fabrication technology by several research groups [11,17]. They manipulated the periodic structure of photonic crystal as well as resonant ring for negative permeability [11,17]. Recently, by using metamaterials, Shalaev and others demonstrated optical waveguides in visible and infrared regions [17]. One inherent property of optical metamaterials in optical frequency is its loss. Different waveguide structures were proposed using optical metamaterials [17]. As long as optical wave is guided, soliton pulses can evolve owing to delicate balance between dispersion and nonlinearity. However it is always a challenge to compensate for the loss when engineering these types of waveguides using metamaterials. The theoretical results showed that metamaterials enhance nonlinearity by confining electrical field in a small region that allows more light–matter interaction [11,17,19,20]. In metamaterials, linear and nonlinear coefficients of ☆
Invited Article by the Associate editor: Luca Palmieri. Corresponding author at: Department of Mathematical Sciences, Delaware State University, Dover, DE 19901-2277, USA. E-mail address: [email protected] (A. Biswas). n
http://dx.doi.org/10.1016/j.optlastec.2015.08.021 0030-3992/& 2015 Published by Elsevier Ltd.
the propagation equation can be tuned to achieve any combination of signs that is not possible in regular materials. These properties of metamaterials lead to improved propagation of a wider variety of solitary waves, efficient phase-matching and modulational instability [12,19,20]. Numerical as well as analytical results of soliton propagation in several nanoscale optical waveguides were reported by several authors [12,19,20]. Earlier results reveal that similar regular (positive indexed) dielectric material dispersion plays a pivotal role in supporting short duration soliton pulses. Optical waveguides with selected wavelengths can be implemented in photonic crystal partially filled with gold and nanoparticles. Recently, theoretical results are reportedfor Y-splitter and bend waveguide structures [11]. The dynamics of soliton propagation through these optical metamaterials is governed by the nonlinear Schrödinger's equation (NLSE) with a few perturbation terms. This model was first reported during 2011 [21]. With the advent of such a model, a plethora of results have been reported. The integrability aspect of this model was studied with various forms of nonlinearity. The integration tools that were applied are simplest equation approach, functional variable method, first integral scheme, Kudryashov's method, trial solutions approach, F-expansion scheme and others [5–9,13]. These algorithms yielded solitons, shock waves and other solutions to the model that appeared with several integrability conditions. In addition to these exact soliton solutions, very recently semi-inverse variational principle was applied to extract bright and exotic soliton solutions to the model [23].
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Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
These are analytical soliton solutions although they are not exact. This paper will apply the method of undetermined coefficients that is otherwise conveniently known as ansatz scheme, to retrieve exact soliton solutions. Bright, dark and singular soliton solutions will be recovered that will appear with essential integrability conditions which stems out from the solution structure of the model.
2. Governing equation and mathematical analysis The dynamics of solitons in optical metamaterials is governed by the nonlinear Schrödinger's equation (NLSE) which in the dimensionless form is given by [20]
3λ + 2ν = 2κ ( 3θ1 + θ 2 − θ 3 ).
(8)
This follows from the fact that the amplitude portion P (x, t ) can be written in terms of the wave variable g (x − vt ) with v being the speed of the wave. The two relations (7) and (8) are obtained by setting the coefficients of linearly independent functions from (6) to zero. These two expressions serve as the existence condition for the solitons that is commonly referred to as constraint relation. The speed of the soliton stays the same for all laws of nonlinearity, namely for all forms of the functional F introduced in (1) and for all kinds of solitons. The constraint relation (8) however modifies with power and dual-power laws. It is the real part equation that will be further analyzed in detail for various nonlinear forms of F in the following sections.
iqt + aqxx + F ( q 2 ) q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q 3. Kerr law
+ θ1 (
q 2q
)xx +
θ 2 q 2 qxx
+
⁎ θ 3 q2qxx
(1)
Eq. (1) is the NLSE that is studied in the context of metamaterials. Here in (1), a and b are the group velocity dispersion and the selfphase 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 inter-modal dispersion. This arises from the fact that group velocity of light in multi-mode fibers depends on chromatic dispersion as well as the propagation mode involved. Next, θj for j = 1, 2, 3 are the perturbation terms that appear in the context of metamaterials [1,5–9,13]. Finally, the independent variables are x and t that represent spatial and temporal variables respectively with the dependent variable q (x, t ) being the complex-valued wave profile. The real-valued algebraic functional F must possess smoothness of the complex-valued function F (|q|2) q : C ↦C . Treating the complex plane C as two-dimensional linear space R2, the function F (|q|2) q is k times continuously differentiable provided ∞
F ( |q|2 ) q ∈ ⋃ C k ( ( − n, n) × ( − m, m); R2). m, n = 1
(2)
In order to start with the analysis of (1), the starting hypothesis is
q (x, t ) = P (x,
t ) eiϕ ,
(3)
In (2), P (x, t ) represents amplitude portion of the wave while ϕ (x, t ) is the phase component that is given by
ϕ = − κx + ωt + θ .
(4)
where κ gives the soliton frequency and ω being the soliton wave number while θ represent the phase constant. After substituting (3) into (1) and decomposing into real and imaginary parts lead to 2
( ω + ακ + aκ 2) P − a ∂∂xP2
⎛ ∂P + 6θ1P ⎜ ⎟ ⎝ ∂x ⎠
∂ 2P ( 3θ1 + θ2 + θ3 ) = 0 ∂x2
(5)
∂P ∂P ∂P − ( α + 2 aκ ) = ( 3λ + 2ν − 6θ1κ − 2θ 2 κ + 2θ 3 κ ) P 2 ∂t ∂x ∂x
(9)
For Kerr law nonlinearity the results of bright, dark and singular soliton have been already reported in the past [5,6]. Therefore, this section will just list the results from these earlier published results [5,6]. It is only the singular solitons of second type that will be derived in detail. 3.1. Bright solitons For Kerr law nonlinear medium, bright 1-soliton solution in optical metamaterials is given by [5]
q (x, t ) = A sech [B (x − vt )] ei (−κx + ωt + θ )
(10)
where A is the amplitude and B is the inverse width of the soliton. The relation between amplitude and width is given by
( b − λκ − 5θ1κ 2) A2 − 3θ1A2B2 − 2aB2 = 0.
(11)
The wave number is
ω = aB2 − aκ 2 − ακ
(12)
and the additional constraint condition is
(13)
3.2. Dark solitons
q (x, t ) = A tanh [B (x − vt )] ei (−κx + ωt + θ ) .
(6)
respectively. The imaginary part equation (6) implies the relations
and
⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
For Kerr law, dark soliton solution is given by [5]
and
v = − α − 2aκ
iqt + aqxx + b q 2 q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q
6θ1 + θ 2 + θ 3 = 0.
⎞2
− P 3 {b − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) } + P 2
This law is also known as the cubic nonlinearity and is considered to be the simplest known form of nonlinearity. Most optical fibers that are commercially available nowadays obey this Kerr law of nonlinearity. Therefore, in this first section the attention will be on optical metamaterials with cubic nonlinearity. In this case F (u) = bu for some non-zero constant b [4]. Therefore, the governing equation given by (1) with Kerr law nonlinearity reduces to [5]
(7)
(14)
In this case, the parameters A and B are referred to as free parameters and these are connected as
A2 ( b − λκ − 5θ1κ 2) + 6θ1A2 B2 + 2aB2 = 0 and the wave number is
(15)
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
ω = − ( ακ + aκ 2 + 2aB2 + 6θ1A2 B2)
(16)
together with the constraint condition (13) that also remains valid here.
focussing singularity [4]. In this case, NLSE given by (1) modifies to
iqt + aqxx + b q 2n q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
3.3. Singular solitons (Type-I)
q (x, t ) = A csch [B (x − vt )] ei (−κx + ωt + θ ) .
(17)
where the free parameters A and B are connected as in (11). The wave number is also located in (12) while the same constraint (13) holds here. 3.4. Singular solitons (Type-II)
P (x, t ) = A cothp τ ,
(18)
where
τ = B (x − vt ).
(19)
In this case also, A and B are free parameters and p is the unknown exponent. Substituting this hypothesis into (5) and (6), the real and imaginary parts are
ap (p −
τ−
{2ap2B2
+ ( ω + ακ +
aκ 2
)
} cothp
τ
The starting hypothesis for bright solitons is given by [4,15,16]
P (x, t ) = A sechp τ ,
− A2 B2 {6θ1p2 + p (p − 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p − 2 τ + A2 ⎡⎣ b − λκ + κ 2 ( θ1 + θ 2 + θ 3 )
ap2B2 − ( ω + ακ + aκ 2) − ap (p + 1) B2 sech2 τ + (b − λκ ) A2n sech2np τ A2 { ( θ1 + θ 2 + θ 3 ) κ 2 − p2 B2 ( 9θ1 + θ 2 + θ 3 ) } sech2p τ + A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } sech2p + 2 τ = 0
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} sech2np τ − A2 {2κ ( 3θ1 + θ 2 − θ 3 ) } sech2p τ = 0
}
(
(20)
ω=
and
v + α + 2aκ + A2 {3λ + 2ν − 2κ ( 3θ1 + θ 2 − θ 3 ) } coth2p τ = 0,
1 n
(28)
(21)
1 {aB2 − n2 ( ακ + aκ 2) } , n2
B = nκ
θ1 + θ 2 + θ 3 , 9θ1 + θ 2 + θ 3
and the amplitude–width relation as
p = 1.
n2 (b − λκ ) A2n = a (n + 1) B2.
(22)
Next, setting the coefficients of undetermined coefficients or linearly independent functions, from (20), to zero yields the constraint relation (13), the wave number given by (12) and the relation between the free parameters A and B as in (11). The imaginary part equation leads to (7) and (8). Therefore, singular 1-soliton solution of Type-II in Kerr law medium is
(23)
where the definition of parameters and their respective constraints are all in place.
(29)
width of the soliton
respectively. By virtue of balancing principle for optical solitons, equating the exponents 3p and p + 2 from real part equation (20) gives
ei (−κx + ωt + θ ) .
(27)
Next, setting the coefficients of linearly independent functions, in (26), to zero reveals the wave number
)
− A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p + 2 τ = 0
(26)
and
p=
+ 12θ1p2 B2 + 2p2 B2 3θ1 + θ 2 + θ 3 ⎤⎦ coth3p τ
q (x, t ) = A coth [B (x − vt )]
(25)
respectively. From balancing principle, equating the exponents 2np and 2, from real part (26) gives
+ ap (p + 1) B2 cothp + 2 τ
{
4.1. Bright solitons
which upon substitution, simplifies real and imaginary part equations (5) and (6) to
In this case, the starting hypothesis is given by [15,16]
cothp − 2
(24)
This equation will now be further analyzed in the following sections to obtain four forms of solitons.
In this case, singular soliton solution of first kind is [6]
1) B 2
179
(30)
(31)
From (30) and (31), the amplitude of the soliton is
⎡ a (n + 1) κ 2 (θ1 + θ 2 + θ 3 ) ⎤1/2n A=⎢ ⎥ ⎣ (b − λκ )(9θ1 + θ 2 + θ 3 ) ⎦
(32)
The amplitude and width of the soliton will exist provided
(θ1 + θ 2 + θ 3 )(9θ1 + θ 2 + θ 3 ) > 0,
(33)
and
a (b − λκ )(θ1 + θ 2 + θ 3 )(9θ1 + θ 2 + θ 3 ) > 0,
(34)
which follows from (30) and (32) respectively. The next constraint relation that stems out from the coefficient of sech2p + 2 τ in (26) is 4. Power law
6θ1 + (n + 1) ( 3θ1 + θ 2 + θ 3 ) = 0.
This law of nonlinearity also arises in nonlinear plasmas that solves the problem of small K-condensation in weak turbulence theory. It also arises in the context of nonlinear optics. Physically, various materials, including semiconductors, exhibit power law nonlinearities. For power law, F (u) = bun , where n is the power law nonlinearity parameter. It is important to note that 0 < n < 2 to prevent wave collapse and in particular n ≠ 2 to prevent self-
The imaginary part equation given by (27) leads to the speed v given by (7) as well as the following set of constraints:
(2n + 1) λ + 2nν = 0,
(35)
(36)
and
3θ1 + θ 2 − θ 3 = 0.
(37)
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Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
Thus, bright 1-soliton solution in optical metamaterials with power law nonlinearity is
q (x, t ) = A sech1/ n [B (x − vt )] ei (−κx + ωt + θ )
(38)
with the definition of parameters and necessary constraints in place, as indicated.
For dark soliton solution, the starting hypothesis is given by [4,15,16]
P (x, t ) = A tanhp τ ,
(39)
which upon substituting into the real and imaginary part equations (5) and (6) simplifies to
ap (p −
tanhp − 2
q (x, t ) = A csch1/ n [B (x − vt )] ei (−κx + ωt + θ )
(46)
with the definition of parameters and necessary constraints in place.
4.2. Dark solitons
1) B 2
bright solitons, the same results (29)–(35) fall out. The imaginary part equation (45) leads to (7) (36) and (37), namely the speed and necessary constraints. Thus, singular 1-soliton solution in optical metamaterials with power law nonlinearity is
τ − {ω + ακ +
aκ 2
+ 2ap (p +
1) B2}
4.4. Singular solitons (Type-II) With Kerr law nonlinear medium, singular soliton hypothesis is given by [15,16]
P (x, t ) = A cothp τ ,
(47)
which upon substituting into (5) and (6), the real and imaginary part equations reduce to
tanhp τ + ap (p + 1) B2 tanhp + 2 τ + (b − λκ ) A2n tanh(2n + 1) p τ
ap (p − 1) B2 cothp − 2 τ − {ω + ακ + aκ 2 + 2ap (p + 1) B2} cothp τ
− pA2B2 {6pθ1 + (p − 1) ( 3θ1 + θ 2 + θ 3 ) } tanh3p − 2 τ
+ ap (p + 1) B2 cothp + 2 τ + (b − λκ ) A2n coth(2n + 1) p τ + A2 {12p2 B2θ1 + p (2p + 1) B2 ( 3θ1 + θ 2 + θ 3 )
− pA2B2 {6pθ1 + (p − 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p − 2 τ
+ κ 2 ( θ1 + θ 2 + θ 3 ) } tanh3p τ − pA2B2 {6pθ1 + (p + 1) ( 3θ1 + θ 2 + θ 3 ) } tanh3p + 2 τ = 0
+ A2 {12p2 B2θ1 + p (2p + 1) B2 ( 3θ1 + θ 2 + θ 3 )
(40)
+ κ 2 ( θ1 + θ 2 + θ 3 ) } coth3p τ
and
− pA2B2 {6pθ1 + (p + 1) ( 3θ1 + θ 2 + θ 3 ) } coth3p + 2 τ = 0
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} tanh2np τ − 2κA2 ( 3θ1 + θ 2 − θ 3 ) tanh2p τ = 0
(41)
respectively. Now, setting the coefficient of stand-alone linearly independent function tanhp − 2 τ to zero leads to (22). Again from balancing principle, equating the exponents (2n + 1) p and p þ2 gives (28). From (22) and (28),
n = 1.
(42)
Thus, for dark solitons with power law nonlinearity condenses to Kerr law nonlinearity. The imaginary part equation (41) leads to (7), (36) and (37) with (42). Therefore, dark 1-soliton solution for power law nonlinearity is also given by (14) with the wave number as in (16) and the relation between the free parameters as in (15). The constraint (13) also remains valid, in this case.
(48)
and
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} coth2np τ − 2κA2 ( 3θ1 + θ 2 − θ 3 ) coth2p τ = 0
(49)
respectively. Similarly, as in dark soliton solutions relation (42) falls out. Therefore, singular solitons of second type, with power law nonlinearity, will exist if power law nonlinearity boils down to Kerr law. The imaginary part equation (6) which leads to (7), (36) and (37) along with (40). Therefore, singular 1-soliton solution for power law nonlinearity is also given by (23) with parameter definitions and constraints as in dark solitons with Kerr law.
5. Parabolic law 4.3. Singular solitons (Type-I) The starting hypothesis for singular solitons (Type-I) is given by [15,16]
P (x, t ) = A cschp τ ,
(43)
which upon substitution, simplifies (5) and (6) respectively to
ap2B2 − ( ω + ακ + aκ 2) − ap (p + 1) B2 csch2 τ + (b − λκ ) A2n csch2np τ + A2 { ( θ1 + θ 2 + θ 3 ) κ 2 − p2 B2 ( 9θ1 + θ 2 + θ 3 ) } csch2p τ + A2 B2 {6θ1p2 + p (p + 1) ( 3θ1 + θ 2 + θ 3 ) } csch2p + 2 τ = 0
(
v + α + 2aκ + A2n {(2n + 1) λ + 2nν} csch2np τ (45)
From balancing principle applied to real part equation (44) equating the exponents 2np and 2 gives the same value of p as in (28). Next, from the linearly independent functions, similarly as in
)
iqt + aqxx + b1 q 2 + b2 q 4 q = iαqx + iλ (
(44)
and
− A2 {2κ ( 3θ1 + θ 2 − θ 3 ) } csch2p τ = 0.
This law is alternatively known as the cubic–quintic nonlinearity and is studied in nonlinear interaction between Langmuir waves and electrons. It describes the nonlinear interaction between the high frequency Langmuir waves and the ion-acoustic waves by pondermotive forces. It takes the form F (u) = b1u + b2 u2 for non-zero constants b1 and b2. For parabolic law medium, the NLSE given by (1) changes to
q 2q
)x + iν ( q 2)x q
⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
(50)
The rest of this section will focus on the details of retrieving soliton solutions to this model along with their conditions for existence. 5.1. Bright solitons For bright solitons with parabolic law, the starting hypothesis is
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
[4,15,16]
P (x, t ) =
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
A
( D + cosh τ )p
,
ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + p cosh + D τ ( ) ( D + cosh τ )p + 1
−
−
( D + cosh τ )p + 2
{
}
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
}
+
1
2
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
5p
( D + cosh τ )
P (x, t ) = ( A + B tanh τ )
= 0. (52)
{ 3λ + 2ν − 2κ ( 3θ1 + θ2 − θ3 ) } = 0 (53)
1
p = 2.
(54)
Next, from the undetermined coefficients of linearly independent functions, one recovers the wave number 1 4
( aB2 − 4aκ 2 − 4ακ ),
(63)
ap (p − 1) μ2 ( B2 − A2 ) ( A + B tanh τ )
p−2
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B tanh τ )
p−1
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B tanh τ ) − 2Aaμ2 p (2p + 1) ( A + B tanh τ ) + ap (p + 1) μ2 ( A + B tanh τ ) + ⎡⎣ 2μ2 p2 B2 − 3A2
(
)( 9θ
{
1
p
p+1
p+2
) + θ ) } ⎤⎦ ( A + B tanh τ )
+ θ2 + θ3
+ B2 b1 − λκ + κ 2 ( θ1 + θ 2
3
3p
2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
5θ1 + θ 2 + θ 3 = 0.
⎡ 1 ⎧ ⎨ −D ⎡⎣ θ1 ( b1 − λκ − 4θ1κ 2) + ab2 ⎤⎦ A=⎢ ⎣ 2b2 θ1 ⎩
{
± D2 θ1 ( b − λκ − 4θ1κ 2} + ab2
2
}
− 3ab2 θ1
( b − λκ − 4θ κ ) 1
1
2
(
⎞ ⎤1/2 ⎟⎥ ⎟⎥ D2 − 1 ⎟ ⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦ (57)
)
subject to the conditions
b2 θ1 ⎡⎣ − D θ1 ( b1 − λκ − 4θ1κ 2) + ab2
{
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
2
}
}
(
1
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
( A + B tanh τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) 5 p + b2 B2 ( A + B tanh τ ) = 0
(64)
and 2p
v + α + 2aκ − {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) = 0.
(65)
Again balancing exponents 3p + 1 with p + 2, from (64), leads to the same value of p as given by (54). There are more exponent pairs in (64) that will lead to (54) by virtue of the same principle. The coefficient of stand-alone linearly independent function (A + B tanh τ ) p gives
)
− 3ab2 θ1 D2 − 1 ] > 0,
( b − λκ − 4θ κ ) 1
( A + B tanh τ )3p − 2
(56)
From the remaining linearly independent functions, the amplitude and the width are
2
A = ± B. (58)
and
(62)
τ = μ (x − vt ).
(55)
and the constraint
±
(61)
p
2
( D + cosh τ )2p
respectively. By balancing principle, applied to (52), equating the exponents 3p and p + 1 or the pair 5p and p + 2 leads to
ω=
ei (−κx + ωt + θ )
Here A, B and μ are all free parameters. With (62), Eqs. (5) and (6) respectively reduce to
and
v + α + 2aκ +
A D + cosh [B (x − vt )]
where, in this case
b2 A4
A2
(60)
For dark optical solitons [15,16],
}
( D + cosh τ )3p + 2 −
b1 − λκ − 4θ1κ 2 . aD + θ1A2
5.2. Dark solitons
( D + cosh τ )3p + 1 2
B=A
where the parameter definitions and constraints are all in place.
( D + cosh τ )3p
){ 6p θ B
The width of the soliton is connected to the amplitude by the relation
q (x, t ) =
A2 b1 − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
(
(59)
Now the imaginary part equation (53) in this case gives (7) and (8). Finally, bright 1-soliton solution to optical metamaterials with parabolic law nonlinearity is
ap (p + 1) B2 ( D2 − 1)
A2 B2 D2 − 1
2
}
− 3ab2 θ1 ( D2 − 1)( b1 − λκ − 4θ1κ 2) > 0.
(51)
where A is the amplitude of the soliton and D is a newly introduced parameter and the usual definition of τ is carried over from (19) with the unknown exponent p. Substituting (50) into (5) and (6) gives
−
181
(66)
The undetermined coefficients of the remaining linearly independent functions from (64) yield the same constraint (56) along with the wave number
182
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
ω = aμ2 − ακ − aκ 2,
(67)
and the relation between the free parameters
b2 θ1 ⎡⎣ D θ1 ( b1 − λκ − 4θ1κ 2) + ab2
{
} D { θ ( b − λκ − 4θ κ ) + ab } 2
±
2aμ2 A= , b1 − λκ − 4θ1 ( μ2 + κ 2)
1
2
2
1
2
(
)
− 3ab2 θ1 D2 + 1 ] > 0,
( b − λκ − 4θ κ ) 1
(68)
2
1
which stays valid as long as
(75)
λκ + 4θ1 ( μ2 + κ 2) ≠ b1.
(69)
Next the imaginary part equation (65), relations (7) and (8) are valid. Therefore, dark 1-soliton solution with parabolic law is given by
A {1 ± tanh [μ (x − vt )]} ei (−κx + ωt + θ )
q (x, t ) =
{
D2 θ1 ( b − λκ − 4θ1κ 2) + ab2
For first type of singular solitons, the starting hypothesis is [15,16]
A p, + τ) D sinh (
(76)
The parameters A and B are related as
B=A
5.3. Singular solitons (Type-I)
2
}
+ 3ab2 θ1 ( D2 + 1)( b1 − λκ − 4θ1κ 2) > 0.
(70)
with the respective parameters and constraints as indicated.
P (x, t ) =
and
b1 − λκ − 4θ1κ 2 . aD − θ1A2
(77)
Next, the imaginary part equation (73), relations (7) and (8) are obtained. Finally, singular 1-soliton solution to optical metamaterials with parabolic law nonlinearity is given by
q (x, t ) =
(71)
A D + sinh [B (x − vt )]
ei (−κx + ωt + θ ) (78)
where A, B and consequently D are free parameters. With this hypothesis Eqs. (5) (6) respectively transform to
where the parameter definitions and constraints are all in place.
ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + p ( D + sinh τ ) ( D + sinh τ )p + 1
5.4. Singular solitons (Type-II)
−
For singular solitons of second type [15,16],
ap (p + 1) B2 ( D2 + 1)
( D + sinh τ )
P (x, t ) = ( A + B coth τ )
p+2
−
− p2 B2 ( 3θ1 + θ 2 + θ 3 )
}
2
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
+
){ 6p θ B 2
1
2
b2 A4 5p
( D + sinh τ )
p−1
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B coth τ )
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
− 2Aaμ2 p (2p + 1) ( A + B coth τ )
}
+ ap (p + 1) μ2 ( A + B coth τ )
( D + sinh τ )3p + 2 −
p−2
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B coth τ )
}
( D + sinh τ )3p + 1
(
τ as in (63). Substitution of (79) into
ap (p − 1) μ2 ( B2 − A2 ) ( A + B coth τ )
( D + sinh τ )3p
A2 B2 D2 + 1
(79)
with the same definition of (5) and (6) implies
{
A2 b1 − λκ + κ 2 ( θ1 + θ 2 + θ 3 ) − 6θ1p2 B2 −
p
+ ⎡⎣ 2μ2 p2 B2 − 3A2
(
= 0.
+
(72)
B2
{b
1
− λκ +
)( 9θ
κ2
1
p
p+1
p+2
) + θ ) } ⎤⎦ ( A + B coth τ )
+ θ2 + θ3
( θ1 + θ2
3
3p
2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
and
v + α + 2aκ +
{
A2 3λ + 2ν − 2κ ( 3θ1 + θ 2 − θ 3 )
( D + sinh τ )2p
( A + B coth τ )3p − 2
} = 0. (73)
Proceeding in the same way as in bright solitons, from real part equation (72), relations (54)–(56) are all recovered. The free parameter A in this case is
⎡ 1 ⎧ ⎨ D ⎡⎣ θ1 ( b1 − λκ − 4θ1κ 2) + ab2 ⎤⎦ A=⎢ ⎣ 2b2 θ1 ⎩
{
± D2 θ1 ( b − λκ − 4θ1κ 2} + ab2
( b − λκ − 4θ ) 1
1
κ2
subject to the conditions
2
}
+ 3ab2 θ1
(
⎞ ⎤1/2 ⎟⎥ ⎟⎥ D2 + 1 ⎟ ⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦ (74)
)
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
( A + B coth τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ ) 5p + b2 B2 ( A + B coth τ ) = 0
(80)
and 2p
v + α + 2aκ − {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ ) =0
(81)
respectively. These expressions lead to (7)–(9) as well as (66)–(69). Therefore, singular 1-soliton solution of Type-II with parabolic law in optical metamaterials is given by
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
A ( 1 ± coth [μ (x − vt )] ) ei (−κx + ωt + θ )
q (x, t ) =
(82)
with the respective parameters and constraints are discussed.
183
⎡ (2n + 1) ( D2 − 1)( b − λκ ) ⎤1/2n 1 ⎥ A = ⎢− ⎢⎣ ⎥⎦ 2 (n + 1) b2 D
(89)
provided
b2 D ( D2 − 1) ( b1 − λκ ) < 0,
6. Dual-power law This model describes saturation of nonlinear refractive index and its exact soliton solutions are known. The effective NLSE, with this form of nonlinearity, serves as a basic model to describe spatial solitons in photovoltaic-photorefractive materials such as LiNbO3. Optical nonlinearities in many organic and polymer materials are governed with such form of nonlinearity. The governing NLSE in optical metamaterials for dual-power law nonlinearity is
(90)
and the width is
B=
n ( b1 − λκ ) D (n + 1)
−
(2n + 1) ( D2 − 1) ab2
(91)
that stays valid for
ab2 ( D2 − 1) < 0.
(92)
Finally, bright 1-soliton solution to optical metamaterials with dual-power law nonlinearity is given by
(
) 2 2 = iαqx + iλ ( q q)x + iν ( q )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx ,
iqt + aqxx + b1 q 2n + b2 q 4n q
(83)
q (x, t ) =
A 1/2n
( D + cosh [B (x − vt )] )
ei (−κx + ωt + θ ) (93)
for F (u) = b1u2n + b2 u4n , with non-zero b1 and b2, where n is the power law parameter. This section will now comprehensively derive the soliton solutions to NLSE in the following subsections.
where the parameter definitions and constraints are all in place.
6.1. Bright solitons
With the same starting hypothesis given by (62), the real and imaginary parts (5) and (6) are
With the same starting hypothesis as given by (51), the real part equations (5) and (6) reduce to
−
{
A2 κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
− 2Aaμ2 p (2p + 1) ( A + B tanh τ )
}
+ ap (p + 1) μ2 ( A + B tanh τ )
{
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
(
){ 6p θ B 2
1
2
(2n + 1) p
( D + cosh τ )
−
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
2
( A + B tanh τ )3p − 2
b2 A4n (4n + 1) p
( D + cosh τ )
{ (2n + 1) λ + 2nν} A2n 2np
( D + cosh τ )
=0
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
= 0. (84)
−
2κ ( 3θ1 + θ 2 − θ 3 ) A2
( D + cosh τ )2p (85)
respectively. The imaginary part equation leads to the constraints given by (7), (36) and (37). By balancing principle applied to real part (84) equating the exponents, (4n + 1) p and p + 2, implies
p=
1 . 2n
1 ( aB2 − 4n2aκ 2 − 4n2ακ ), 4n2
(94)
and 2np
(86)
(87)
and the constraint
3 (2n + 3) θ1 + (2n + 1) ( θ 2 + θ 3 ) = 0.
( A + B tanh τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ )3p + 1 3p + 2 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B tanh τ ) (2n + 1) p + ( b1 − λκ ) B2 ( A + B tanh τ ) (4n + 1) p + b2 B2 ( A + B tanh τ ) =0 v + α + 2aκ + {(2n + 1) λ + 2nν} ( A + B tanh τ )
Next, from the undetermined coefficients of linearly independent functions, one recovers the wave number
ω=
p+1
p+2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
and
v + α + 2aκ +
p
( A + B tanh τ )3p
( D + cosh τ )3p + 2
( b1 − λκ ) A2n
p−1
+ {2μ2 p2 ( B2 − 3A2 )( 9θ1 + θ 2 + θ 3 ) + B2κ 2 ( θ1 + θ 2 + θ 3 ) }
}
( D + cosh τ )3p + 1 A2 B2 D2 − 1
p−2
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B tanh τ )
( D + cosh τ )3p
+
−
2
ap (p − 1) μ2 ( B2 − A2 ) ( A + B tanh τ )
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B tanh τ )
ap (p + 1) B2 ( D2 − 1) ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + − p p+1 ( D + cosh τ ) ( D + cosh τ )p + 2 ( D + cosh τ ) −
6.2. Dark solitons
(88)
From the remaining linearly independent functions, the amplitude of the soliton is
2p
− 2κ ( 3θ1 + θ 2 + θ 3 )( A + B tanh τ )
=0
(95)
respectively. Again balancing exponents 3p + 1 with p + 2 leads to the same value of p as given by (54). Next, equating the exponents (2n + 1) p and p + 2 leads to (42). This shows that dark solitons for dualpower law collapse to the case of parabolic law. Therefore, all results from parabolic law dark solitons given by (56) and (66)–(69) remain valid. In addition, the imaginary part equation leads to (7) and (8). Finally, dark 1-soliton solution to dual-power law nonlinearity is given by (70), with all definition of parameters and their respective constraints in place.
184
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
6.3. Singular solitons (Type-I)
2
ap (p − 1) μ2 ( B2 − A2 ) ( A + B coth τ )
p−2
In this case, substituting the starting hypothesis given by (71) into (5) and (6) gives
+ 2p (2p − 1) Aaμ2 ( B2 − A2 ) ( A + B coth τ )
ap (p + 1) B2 ( D2 − 1) ω + ακ + aκ 2 − ap2B2 ap (2p + 1) B2D + − p p+1 ( D + sinh τ ) ( D + sinh τ )p + 2 ( D + sinh τ )
− 2Aaμ2 p (2p + 1) ( A + B coth τ )
−
−
{
A2 κ 2 ( θ1 + θ 2 + θ 3 ) − p2 B2 ( 9θ1 + θ 2 + θ 3 )
{ (
){ 6p θ B 2
1
2
(2n + 1) p
( D + sinh τ )
−
2
( A + B coth τ )3p − 2
+ p (p + 1) ( 3θ1 + θ 2 + θ 3 )
− 2μ2 pA ( B2 − A2 ) {2p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
b2 A4n
( D + sinh τ )(4n+ 1) p
= 0. (96)
and
v + α + 2aκ +
{ (2n + 1) λ + 2nν} A2n 2np
( D + sinh τ )
−
2κ ( 3θ1 + θ 2 − θ 3 ) A2
( D + sinh τ )2p (97)
=0
respectively. The imaginary part equation clearly leads to (7), (36) and (37). While the balancing principle from real part (96) yields (86), the remaining undetermined coefficients lead to (87) and (88) as well. The free parameters A and B are now
⎡ (2n + 1) ( D2 + 1)( b − λκ ) ⎤1/2n 1 ⎥ A = ⎢− 2 (n + 1) b2 D ⎥⎦ ⎢⎣
(98)
provided
b2 D ( b1 − λκ ) < 0,
( A + B coth τ )3p − 1 + 2μ2 pA {2p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 1 − μ2 p {p ( 9θ1 + θ 2 + θ 3 ) + ( 3θ1 + θ 2 + θ 3 ) } ( A + B coth τ )3p + 2 (2n + 1) p + ( b1 − λκ ) B2 ( A + B coth τ ) (4n + 1) p + b2 B2 ( A + B coth τ ) =0
(103)
and 2np
v + α + 2aκ + {(2n + 1) λ + 2nν} ( A + B coth τ ) 2p
− 2κ ( 3θ1 + θ 2 + θ 3 )( A + B coth τ )
=0
(104)
respectively. Once again, proceeding along the same lines as in the case of dark solitons, (54) and (42) are recovered. Thus, this form of singular solitons of Type-II exists whenever dual-power law reduces to parabolic law nonlinearity. Hence, all results from (7), (36), (37) and (66)–(69) hold. Finally, singular 1-soliton solution for dualpower law is given by (82) along with the parameters and constraints as described.
(99)
and
B=
p+2
− μ2 p ( B2 − A2 ) {p ( 9θ1 + θ 2 + θ 3 ) − ( 3θ1 + θ 2 + θ 3 ) }
}
( D + sinh τ )3p + 2
( b1 − λκ ) A2n
p+1
( A + B coth τ )3p
( D + sinh τ )3p + 1 A2 B2 D2 − 1
p
+ {2μ2 p2 ( B2 − 3A2 )( 9θ1 + θ 2 + θ 3 ) + B2κ 2 ( θ1 + θ 2 + θ 3 ) }
( D + sinh τ )3p
+
−
− {B2 ( ω + ακ + aκ 2) − 2ap2μ2 ( 3A2 − B2) } ( A + B coth τ )
+ ap (p + 1) μ2 ( A + B coth τ )
}
A2 B2 12θ1p2 D + p (2p + 1) D ( 3θ1 + θ 2 + θ 3 )
p−1
7. Log law
n ( b1 − λκ ) D (n + 1)
−
(2n + 1) ( D2 + 1) ab2
(100)
only if
(101)
ab2 < 0.
Finally, singular 1-soliton solution to optical metamaterials with dual-power law nonlinearity is
q (x, t ) =
A
( D + sinh [B (x − vt )] )1/2n
ei (−κx + ωt + θ ) (102)
where the parameter definitions and constraints are all in place.
6.4. Singular solitons (Type-II) With starting hypothesis given by (79), the real and imaginary parts (5) and (6) are
In this case F (u) = b ln u for non-zero constant b. This law permits closed form exact expressions Gaussian beams. The advantage of this model is that the radiation from the periodic soliton is absent as the linearized problem contains discrete spectrum only [4]. For log-law medium, the model given by (1) modifies to
iqt + aqxx + b ( ln q 2 ) q = iαqx + iλ ( q 2 q)x + iν ( q 2 )x q ⁎ + θ1 ( q 2 q)xx + θ 2 q 2 qxx + θ 3 q2qxx
(105)
The solutions of NLSE in log-law nonlinear medium lead to Gaussian solitons that are occasionally referred to as Gaussons [15,16]. Therefore, the starting hypothesis for (105) is given by [15,16]
P (x, t ) = Ae−τ
2
(106)
where A is the amplitude and B is the inverse width of the Gausson. Substituting this hypothesis into (5) and (6), the real and imaginary parts respectively simplify to
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
( ω + ακ + aκ 2 + 2aB2 − 2b ln A) − ( 4aB2 − 2b) τ 2 2 + A2 {λκ − κ 2 ( θ1 + θ 2 + θ 3 ) − 2B2 ( 3θ1 + θ 2 + θ 3 ) } e−2τ 2 + A2 {24B2θ1 + 4B2 ( 3θ1 + θ 2 + θ 3 ) } τ 2e−2τ = 0
8. Conclusions
(107)
and
v + α + 2aκ + A {3λ + 2ν − 2κ ( 3θ1 + θ 2 + θ 3 ) } = 0.
(108)
From the undetermined coefficients of linearly independent functions in (107), the wave number is
ω = − ακ − aκ 2 − 2aB2 + 2b ln A,
(109)
and the width of Gaussson is given by
B=
λκ − κ 2 ( θ1 + θ 2 + θ 3 ) 2 ( 3θ1 + θ 2 + θ 3 )
(110)
with the condition
{λκ − κ 2 ( θ1 + θ 2 + θ 3 ) } ( 3θ1 + θ 2 + θ 3 ) > 0.
(111)
Substituting the width B from (110), the wave number from (109) reduces to
ω= −
aκλ + 2aκ 2θ1 + ( 3θ1 + θ 2 + θ 3 )( ακ − 2b ln A) 3θ1 + θ 2 + θ 3
,
(112)
which holds provided
Next, setting the coefficient of the fourth linearly independent 2 function in (106), namely τ 2e−2τ to zero, gives
9θ1 + θ 2 + θ 3 = 0.
(114)
Finally, the coefficient of Gausson as
τ2 from (107) leads to the width of
b 2a
(115)
which shows that these Gaussons will exist provided
(116)
ab > 0.
This means that GVD and the nonlinear term in (105) must both carry the same sign for Gaussons to exist. The imaginary part equation (108) yields (7) and (8). Equating the two values of the width B of the soliton from (110) and (115) leads to another constraint between the frequency and coefficients of the model (105) as follows:
aλκ − 2bθ1 = ( b + aκ 2) ( θ1 + θ 2 + θ 3 ).
(117)
From (114) and (111), one can recover
θ1κ ( λ + 8θ1κ ) < 0
(118)
which can be treated as another constraint. Next, substituting (115) into (109) leads to an alternate expression to the wave number:
ω = b ( 2 ln A − 1) − κ ( α + aκ ).
(119)
Thus, the Gausson solution to optical metamaterials with log-law nonlinearity is given by
q (x, t ) = Ae−B
This paper obtained soliton solutions in optical metamaterials with five forms of nonlinear media. For Kerr law nonlinearity, there are three forms of solitons that are already reported earlier; therefore this paper derived only singular soliton (Type-II). For the remaining laws all soliton solutions and their derivations are comprehensively reported in this paper. These solutions come with respective integrability criteria that are listed as constraint conditions. These solutions will be immensely useful in the literature of optical metamaterials. These soliton solutions will be a great asset in all future investigations in this area of nonlinear optics. In the presence of perturbation terms these solitons will dictate the adiabatic parameter dynamics and other such features that will be obtained. The quasi-particle theory of optical soliton interaction will be reported. Later, bifurcation analysis of solitons in optical metamaterials will be carried out. Other integration schemes will be applied to these models and those will reveal additional solutions, such as plane waves and periodic singular solutions. The semi-inverse variational principle will extract exotic solitons such as cosh-Gaussian pulses and bright-dark combo optical solitons. All of these are currently under investigation. The results of those research will be reported gradually and sequentially. Finally, the study will be extended to DWDM systems so that efficient soliton transmission can be conducted in parallel, thus improving performance enhancement. These just form a tip of the iceberg.
(113)
3θ1 + θ 2 + θ 3 ≠ 0.
B=
185
2 (x − vt ) 2 i (−κx + ωt + θ )
e
(120)
where the parameter definitions and constraints are all listed above.
Acknowledgment This research is funded by Qatar National Research Fund (QNRF) under the Grant number NPRP 6-021-1-005. The fifth and sixth authors (A.B. & M.B.) thankfully acknowledge this support from QNRF.
References [1] A.H. Arnous, M. Mirzazadeh, S. Moshokoa, S. Medhekar, A. Biswas, M. Belic, Solitons in optical metamaterials with trial solutions approach and Bäcklund transform of Riccati equation, J. Comput. Theor. Nanosci. (2015), submitted for publication. [2] M.A. Baqir, P.K. Choudhury, Waves in coaxial optical fiber under DB boundaries, Optik 125 (2014) 2950–2953. [3] M.A. Baqir, P.K. Choudhury, Investigation of uniaxial isotropic chiral metamaterial waveguide with perfect electromagnetic conductor loading, Optik 126 (2015) 1228–1232. [4] A. Biswas, S. Konar, Introduction to Non-Kerr Law Optical Solitons, CRC Press, Boca Raton, FL, USA, 2006. [5] A. Biswas, K.R. Khan, M.F. Mahmood, M. Belic, Bright and dark solitons in optical metamaterials, Optik 125 (13) (2014) 3299–3302. [6] A. Biswas, M. Mirzazadeh, M. Savescu, D. Milovic, K.R. Khan, M.F. Mahmood, M. Belic, Singular solitons in optical metamaterials by ansatz method and simplest equation approach, J. Mod. Opt. 61 (19) (2014) 1550–1555. [7] A. Biswas, M. Mirzazadeh, M. Eslami, D. Milovic, M. Belic, Solitons in optical metamaterials by functional variable method and first integral approach, Frequenz 68 (11–12) (2014) 525–530. [8] A. Biswas, M. Mirzazadeh, D. Milovic, K.R. Khan, M.F. Mahmood, M. Belic, Singular and topological solitons in optical metamaterials by Kudryashov's method and G′/G − expansion scheme, J. Comput. Theor. Nanosci. (2015), submitted for publication. [9] G. Ebadi, A. Mohavir, J.V. Guzman, K.R. Khan, M.F. Mahmood, L. Moraru, A. Biswas, M. Belic, Solitons in optical metamaterials by F-expansion scheme, Optoelectron. Adv. Mater.—Rapid Commun. 8 (9–10) (2014) 828–832. [10] H. Erfaninia, A. Rostami, Group velocity reduction in multilayer metamaterial waveguide, Optik 124 (2013) 1230–1233. [11] K.R. Khan, K. Mnaymneh, H. Awad, I. Hasan, T. Hall, Optical wave propagation in photonic crystal metamaterials, Appl. Phys. A 117 (2) (2014) 629–634. [12] K.R. Khan, M. Mahmood, A. Biswas, Coherent super-continuum generation in PCF at visible and near infrared wavelengths, IEEE J. Sel. Top. Quantum Electron. 20 (5) (2014) 7500309. [13] E.V. Krishnan, M. Al Gabshi, Q. Zhou, K.R. Khan, M.F. Mahmood, Y. Xu,
186
[14]
[15]
[16]
[17] [18] [19]
Y. Xu et al. / Optics & Laser Technology 77 (2016) 177–186
A. Biswas, M. Belic, Solitons in optical metamaterials by mapping method, J. Optoelectron. Adv. Mater. 17 (3–4) (2015) 511–516. M. Saha, A.K. Sarma, Modulational instability in nonlinear metamaterials induced by cubic-quintic nonlinearities and higher order dispersive effects, Opt. Commun. 291 (2013) 321–325. M. Savescu, K.R. Khan, P. Naruka, H. Jafari, L. Moraru, A. Biswas, Optical solitons in photonic nano-waveguides with an improved nonlinear Schrödinger's equation, J. Comput. Theor. Nanosci. 10 (5) (2013) 1182–1191. M. Savescu, K.R. Khan, R. Kohl, L. Moraru, A. Yildirim, A. Biswas, Optical soliton perturbation with improved nonlinear Schrödinger's equation in nanofibers, J. Nanoelectron. Optoelectron. 8 (2) (2013) 208–220. V.M. Shalaev, Optical negative-index metamaterials, Nat. Photonics 1 (2007) 41–48. V.G. Vaselago, Electrodynamics of media with simultaneous negative electric permittivity and magnetic permeability, Sov. Phys. Usp. 10 (1967) 509–514. S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, D. Fan, Role of anomalous self-steepening effect in modulation instability in negative-index material, Opt. Express 14 (2006) 1568.
[20] S. Wen, W. Wang, W. Su, Y. Xiang, X. Fu, D. Fan, Modulation instability in nonlinear negative-index material, Phys. Rev. E 73 (2006) 036617. [21] Y. Xiang, X. Dai, S. Wen, J. Guo, D. Fan, Controllable Raman soliton self-frequency shift in nonlinear metamaterials, Phys. Rev. A 84 (2011) 033815. [22] Y. Xu, Q. Zhou, A.H. Bhrawy, K.R. Khan, M.F. Mahmood, A. Biswas, M. Belic, Solitons in optical metamaterials by traveling wave hypothesis, Optoelectron. Adv. Mater.—Rapid Commun. 9 (3–4) (2015) 384–387. [23] Y. Xu, J. Vega-Guzman, D. Milovic, M. Mirzazadeh, M. Eslami, M.F. Mahmood, A. Biswas, M. Belic, Bright and exotic solitons in optical metamaterials by semi-inverse variational principle (2015), submitted for publication. [24] X. Zhong, K. Chen, Modulation instability in metamaterials with fourth order linear dispersion, second order nonlinear dispersion and three kinds of saturable nonlinearities, Optik 125 (2014) 6733–6738. [25] Q. Zhou, Q. Zhou, Y. Liu, A. Biswas, A. Bhrawy, K.R. Khan, M.F. Mahmood, M. Belic, Solitons in optical metamaterials with parabolic law nonlinearity and spatio-temporal dispersion, J. Optoelectron. Adv. Mater. 16 (11–12) (2014) 1221–1225.