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Numerical solutions for some generalized coupled nonlinear evolution equations
Mathematical and Computer Modelling 56 (2012) 268–277
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Numerical solutions for some generalized coupled nonlinear evolution equations A.A.M. Arafa ∗ , S.Z. Rida Department of mathematics, Faculty of Science, South Valley University, Qena, Egypt
article
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Article history: Received 29 March 2011 Received in revised form 24 December 2011 Accepted 27 December 2011 Keywords: The decomposition method Coupled evolution equation Numerical solution Fractional calculus
abstract In this paper, the Adomian decomposition method (ADM) is presented for the numerical solutions of the coupled evolution equations of fractional order. The ADM in applied mathematics can be used as an alternative method for obtaining analytic and approximate solutions for different types of fractional differential equations. The fractional derivatives are described in the Caputo sense. The given solutions are compared with the traveling wave solutions. The results obtained are then graphically represented. Finally, numerical results demonstrate the accuracy, efficiency and high rate of convergence of this method. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The optical soliton has been proposed as an information carrier in fiber communication systems [1]. Because of the fiber loss, the soliton disperses as it propagates along the fiber. The propagation of optical solitons has been the object of extensive theoretical and experimental research in optical fibers due to their enormous potential applications in telecommunication and ultrafast signal routing systems [2–4]. The propagation of optical solitons in the picosecond regime is governed by nonlinear Schrödinger (NLS)-type equations which arise from many physical fields such as fluid dynamics [5], plasma physics [6,7], solid-state physics [8], Bose–Einstein condensation [9] and fiber optics [10]. Hasegawa and Tappert [11] modeled the propagation of a guided mode in a perfect nonlinear mono-mode fiber by the NLS equation. They pointed out that the solitons could be propagated on glass fibers, with negative group velocity dispersion (GVD), under the action of the optical Kerr effect [12], in which there is a change of refractive index induced by the optical field. The GVD in glass fibers with Ge-doped cores is controlled mainly by the material dispersion, but it can be influenced by the index (doping) profile. An experimental verification of soliton propagation was first given by Molleneur et al. [13]. This work confirmed the predictions of the NLS equation and proved that the properties of fiber propagation are described to a remarkable degree by this equation. Respectively, in previous theoretical research, Raman pump waves were assumed to be either uniformly depleted, or undepleted. It is well known that nonlinear dispersive waves in polarization preserving single-mode fibers obey the so-called nonlinear Schrödinger equation [14,15], which is a model of the evolution of a one-dimensional packet of surface waves on sufficiently deep water. It is widely used in basic models of nonlinear waves in many areas of physics. It arises from the study of nonlinear wave propagation in dispersive and inhomogeneous media, such as plasma phenomena and non-uniform dielectric media. It is a member of the class of integrable equations. It is widely used in basic models of nonlinear waves in many areas of physics. It is a generic equation describing the evolution of the slowly varying amplitude of a nonlinear wave train in weakly nonlinear, strongly dispersive, and hyperbolic systems [16].
∗
Corresponding author. E-mail addresses: [email protected] (A.A.M. Arafa), [email protected] (S.Z. Rida).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.12.046
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In this article, we implement the Adomian decomposition method (ADM) [17,18] on the nonlinear generalized coupled evolution equations, which take the form
∂αu ∂ 2u + a + b(|u|2 + c |v|2 )u + Φ1 (u, v) = 0 ∂tα ∂ x2 ∂αv ∂ 2v i α + a 2 + b(|v|2 + c |u|2 )v + Φ2 (u, v) = 0. ∂t ∂x i
(1.1) (1.2)
This fractional system of equations is obtained by replacing the first time derivative term by a fractional derivative of order α > 0. The derivatives are understood in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of α → 1, the fractional system equations reduce to the standard system of partial differential equations. We introduce several examples to illustrate the main ideas of this work. The Adomian decomposition method will be applied for computing solutions to the systems of fractional partial differential equations considered in this paper. The method provides the solutions in the form of a power series with easily computed terms. It has many advantages over the classical techniques, mainly that it provides an efficient numerical solution with high accuracy and minimal calculations. 2. Fractional calculus There are several definitions of fractional derivatives. These definitions include Riemann–Liouville, Wely, Reize, Caputa, and Nishimoto fractional operators. The commonly used definition for a general fractional derivative is the Riemann–Liouville definition of the fractional derivative operator Iaα . Definition 2.1. Let α ∈ R+ . The operator Jaα defined on the usual Lebesque space L1 [a, b] by Jaα f (x) =
1
Γ (α)
x
(x − t )α−1 f (t )dt
0
Ja0 f (x) = f (x) for a ≤ x ≤ b is called the Riemann–Liouville fractional integral operator of order α . The properties of the operator Jaα can be found in Ref. [19]; we mention the following properties: for f ∈ L1 [a, b], α, β ≥ 0, and γ > −1: 1. 2. 3.
J α J β f (x) = J α+β f (x) J α J β f (x) = J β J α f (x) Γ (γ +1) J α xγ = Γ (α+γ +1) xα+γ .
Therefore, we will introduce a modified fractional operator Dα proposed by Caputo which is used in the up-coming paper and plays the most important role in the theory of differential and integral equations of fractional order. The main advantages of Caputo’s approach are that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer order differential equations. The Caputo definitions are founds in detail in Ref. [20]. Definition 2.2. The fractional derivative of f (x) in the Caputo sense is defined as Dα f (x) = J m−α Dm f (x) =
x
1
Γ (m − α)
(x − t )m−α+1 f (m) (t )dt
0
for m − 1 < α ≤ m, m ∈ N , x > 0. For the Caputo derivative we have Dα C = 0, Dα t n =
C is constant
0,
(n ≤ α − 1)
(n > α − 1)
Γ (n + 1) n−α t , Γ (n − α + 1)
.
Definition 2.3. For m to be the smallest integer that exceeds α , the Caputo fractional derivative of order α > 0 is defined as D α u( x , t ) =
1
∂ α u(x, t ) = Γm(m − α) ∂tα ∂ u(x, t ) , ∂tm
t
(t − τ ) 0
m−α+1
∂ µ (x, τ ) dτ , ∂τ m
for m − 1 < α < m for α = m ∈ N
.
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3. Analysis of the method We first consider the system of generalized coupled PDEs written in an operator form Dαt u + Lx v + N1 (u, v) = Φ1
(3.1)
Dαt v + Lx u + N2 (u, v) = Φ2 with initial data u(x, 0) = f1 (x)
(3.2)
v(x, 0) = f2 (x) α
where Lx = ∂∂x2 , and Dαt = ∂∂t α is the fractional operator; without loss of generality the first order partial differential operators N1 and N2 are nonlinear operators, and Φ1 and Φ2 are inhomogeneous terms. Applying the inverse operator J α to the system (3.1) and using the initial data (3.2) yields 2
u(x, t ) = f1 (x) + J α (Φ1 − Lx v − N1 (u, v))
(3.3)
v(x, t ) = f2 (x) + J α (Φ2 − Lx u − N2 (u, v)).
The Adomian decomposition method suggests that the linear terms u(x, t ) and v(x, t ) are decomposed by an infinite series of components u(x, t ) =
∞
un ( x , t )
n =0
v(x, t ) =
∞
(3.4)
v n ( x, t )
n=0
and the nonlinear operators N1 (u, v) and N2 (u, v) by the infinite series of so called Adomian polynomials N1 (u, v) =
∞
An
n=0
N2 (u, v) =
∞
(3.5) Bn
n=0
where un (x, t ) and vn (x, t ); n ≥ 0 are the components of u(x, t )and v(x, t ) that will be elegantly determined, and An and Bn , n ≥ 0, are Adomian polynomials that can be generated for all forms of nonlinearity. Substituting (3.4) and (3.5) into (3.3) gives ∞
un (x, t ) = f1 (x) + J
α
Φ1 − Lx
n =0
∞
vn (x, t ) = f2 (x) + J
α
Φ2 − Lx
n =0
∞
v n ( x, t ) −
∞
n=0
n =0
∞
∞
un ( x , t ) −
n=0
An (3.6)
Bn
.
n =0
Following Adomian analysis, the nonlinear system (3.1) is transformed into a set of recursive relations given by u0 (x, t ) = f1 (x) un+1 (x, t ) = J α (Φ1 − Lx vn − An ),
n≥0
(3.7)
n ≥ 0.
(3.8)
v0 (x, t ) = f2 (x) vn+1 (x, t ) = J α (Φ2 − Lx un − Bn ),
It is an essential feature of the decomposition method that the zeroth components u0 (x, t ) and v0 (x, t ) are defined always by all terms that arise from initial data and from integrating the inhomogeneous terms. Having defined these the zeroth pair (u0 , v0 ) can be determined recurrently by using (3.7) and (3.8). The remaining pairs (un , vn ), n ≥ 1, can be easily determined in a parallel manner. Additional pairs for the decomposition series normally account for higher accuracy. Having determined the components of u(x, t ) and v(x, t ), the solution (u, v) of the system follows immediately in the form of a power series expansion upon using (3.4). The series obtained can be summed up in many cases to give a closed form solution. For concrete problems, the n term approximants can be used for numerical purposes. Comparing the scheme presented above with existing techniques such as the characteristics method and Riemann invariants, it is clear that the decomposition method introduces a fundamentally qualitative difference in approach, because no assumptions are made. The approach is straightforward and
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rapid convergence is guaranteed. To give a clear overview of the content of this work, several illustrative examples have been selected to demonstrate the efficiency of the method. In this paper the nonlinear terms by an infinite series of polynomials take the forms u|u|2 =
∞
An
(3.9)
Bn
(3.10)
n=0
u|v|2 =
∞ n =0
v2 u =
v|v|2 =
Cn
∞
(3.11)
Dn
(3.12)
En
(3.13)
n =0
v|u|2 =
∞ n =0
∞
u2 v =
Fn
(3.14)
Gn
(3.15)
n =0
∞
vvt =
n=0
where An , Bn , Cn , Dn , En , Fn and Gn are the Adomian polynomials defined as follows. The Adomian polynomials for the nonlinear term u|u|2 are given by A0 = |u0 |2 u0 A1 = 2|u0 |2 u1 + u20 u¯ 1 2
u0 u21
A2 = 2|u0 | u2 + ¯
(3.16) 2
+ | u1 | u0 +
¯ .
u20 u2
2
Those for the nonlinear term u|v| are given by B0 = |v0 |2 u0 B1 = |u0 |2 u1 + u0 v1 u¯ 0 + u0 v0 v¯ 1
(3.17)
B2 = u0 v0 v¯ 2 + u0 |v1 | + u1 (v0 v¯ 1 + v1 v¯ 0 ) + u2 |v0 | . 2
2
Those for the nonlinear term v 2 u are given by C0 = v02 u0 C1 = 2v0 v1 u0 + v02 u1 C2 =
(3.18)
v + 2u1 v0 v1 + 2u0 v0 v2 +
u2 02
v .
u0 12
Those for the nonlinear term v|v| are given by 2
D0 = |v0 |2 v0 D1 = 2|v0 |2 v1 + v02 v¯ 1
(3.19)
D2 = 2|v0 | v2 + v¯ v + |v1 |2 v0 + v02 v¯ 2 . 2
2 0 1
Those for the nonlinear term v|u|2 are given by E0 = |u0 |2 v0 E1 = |v0 |2 v1 + v0 u1 v¯ 0 + v0 u0 u¯ 1
(3.20)
E2 = v0 u0 u¯ 2 + v0 |u1 |2 + v1 (u0 u¯ 1 + u1 u¯ 0 ) + v2 |u0 |2 . Those for the nonlinear term u2 v are given by F0 = u20 v0 F1 = 2u0 u1 v0 + u20 v1 F2 = v
2 2 u0
+ 2v1 u0 u1 + 2v0 u0 u2 + v
(3.21) 2 0 u1
.
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And those for the nonlinear term vvt are given by G0 = v0 v0t G1 = v0 v1t + v1 v0t
(3.22)
G2 = v0 v2t + v1 v1t + v2 v0t .
4. Solution of weakly birefringent optical fibers with time-fractional order Let us consider a single-mode fiber with weak birefringence. The fiber is assumed to be straight (no twists) so as to avoid any effects arising from optical activity [21]. If u is the field envelope of the polarization along the x axis of the birefringent fiber and v the envelope along the y axis, then u and v obey the following pair of coupled evolution equations of fractional order:
∂ α u β ∂ 2 u w 0 n2 i α − + ∂t 2 ∂ x2 2c i
∂αv β ∂ 2v w0 n2 − + α ∂t 2 ∂ x2 2c
2
2
|u| + |v| 3
2
u+
1 3
v ue 2
−4iρ t
=0
(4.1)
2 1 |v|2 + |u|2 v + v 2 u e4iρ t = 0 3
(4.2)
3
where w0 = 2π c /λ0 is the carrier angular frequency of the light field, λ0 is its wave length, n2 is the nonlinear Kerr coefficient of glass, β is the fiber dispersive coefficient evaluated at w0 , and ρ is related to the birefringence of the fiber. Consider the generalized coupled evolution equations in an operator form: iDαt u − α
iDt v −
β 2
β 2
uxx +
vxx +
w0 n2 2c
2 1 |u|2 + |v|2 u + v 2 u e−4iρ t = 0
(4.3)
2 1 2 4iρ t 2 2 |v| + |u| v + v u e =0
(4.4)
3
w 0 n2 2c
3
3
3
0 < α ≤ 1, t > 0 with the initial data u(x, 0) = a0 sech
v(x, 0) = b0 tanh
x
(4.5)
x0
x
.
x0
(4.6)
The exact solution, for the special case α → 1, is given by u(x, t ) = a0 sech
v(x, t ) = b0 tanh
x
ei(σ −ρ)t
(4.7)
ei(σ +ρ)t .
(4.8)
x0
x x0
We may follow the same analysis as applied before. Operating with J α on (4.3) and (4.4) we obtain u(x, t ) = u(x, 0) + iJ α
v(x, t ) = v(x, 0) + iJ
α
β 2
β 2
uxx −
vxx −
w0 n2 2c
w 0 n2 2c
|u|2 +
2
2 3 2
1 |v|2 u + v 2 u e−4iρ t 3
2
|v| + |u| 3
1
v + v ue 3
2
4iρ t
.
(4.9)
(4.10)
The decomposition method represents the linear terms u(x, t ) and v(x, t ) by the decomposition series u(x, t ) =
∞
un ( x , t )
(4.11)
vn (x, t ).
(4.12)
n=0
v(x, t ) =
∞ n=0
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Substituting the nonlinear terms by an infinite series of polynomials ((3.9)–(3.14)), and ((4.11)–(4.12)) into ((4.9)–(4.10)) yields ∞
un (x, t ) = u(x, 0) + iJ
α
∞ w0 n2
∞
i
6c
un ( x , t )
−
n=0
xx
w0 n2
∞
2c
An +
n=0
∞ 2
3 n =0
Bn
Cn J α ( e−4iρ t )
(4.13)
n=0
vn (x, t ) = v(x, 0) + iJ
α
∞ w0 n2
i
6c
β
2
n =0
−
∞
2
n =0
−
β
∞
v n ( x, t )
−
n=0
xx
w0 n2
2c
∞
Dn +
n =0
∞ 2
3 n =0
En
Fn J α ( e4iρ t ).
(4.14)
n=0
To accelerate the convergence of the solution, the decomposition method will be employed. The decomposition form, although it introduces a slightly different approach in the definition of the recursive relation, in fact results in a qualitative tool in minimizing the size of the calculations. The decomposition method defines the recursive relations in the form u0 (x, t ) = u0 (x, 0) un+1 (x, t ) = iJ
α
β 2
(unxx ) −
w0 n2
An +
2c
2 3
−
Bn
w0 n2
iCn J α ( e−4iρ t ),
6c
n≥0
(4.15)
and
v0 (x, t ) = v0 (x, 0) w 0 n2 2 w 0 n2 α β vn+1 (x, t ) = iJ (vnxx ) − D n + En − iFn J α ( e4iρ t ), 2
2c
3
6c
n ≥ 0.
(4.16)
By using Eqs. ((3.16)–(3.21)), and Eqs. ((4.15)–(4.16)), we can calculate some of the terms of the decomposition series (4.11) and (4.12) as u0 = f (x)
u1 = i f1 (x)
Γ (α + 1)
u2 = i f3 (x)
tα
t
− f2 (x)
∞ k =0
2α
− f4 (x)
Γ (2α + 1)
(−4iρ)k t α+k Γ (α + k + 1)
(−4iρ)k t α+k Γ (α + k + 1)
∞ k=0
and
v0 = g (x) v 1 = i g 1 ( x)
Γ (α + 1)
v 2 = i g 3 ( x)
tα
t
− g2 (x)
∞ k=0
2α
Γ (2α + 1)
− g4 (x)
(4iρ)k t α+k Γ (α + k + 1)
∞ k=0
(4iρ)k t α+k Γ (α + k + 1)
,
where f (x) = a0 sech f 1 ( x) = f 2 ( x) = f 3 ( x) = f 4 ( x) =
β 2
6c 2
(2)
f1
w0 n2 6c
x
x0
f (2) −
w 0 n2 β
w 0 n2
2c
2
|f |2 f + |g |2 f
3
2
g f
−
w 0 n2 2c
2
2|f | f1 + f f1 + (|g | f1 + f (g g¯1 + g1 g¯ ))
(2gg1 f + g 2 f1 )
2
2
3
2
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and g (x) = b0 tanh g1 (x) = g2 (x) = g3 (x) =
β 2
x0
w 0 n2
2c
2
|g |2 g + |f |2 g
3
2
f g
6c 2
x
g (2) −
w 0 n2 β
(2)
g1
−
w 0 n2
2c
2
2|g | g1 + g g¯1 + (|f | g1 + g (f f¯1 + f1 f¯ )) 2
2
2
3
w0 n2
(2ff1 f + f 2 g1 ) 6c and so on. Substituting u0 , u1 , u2 , u3 , . . . and v0 , v1 , v2 , v3 , . . . into (4.11) and (4.12) gives the solutions u(x, t ) and v(x, t ) in a series form by g4 (x) =
u(x, t ) = u0 + u1 + u2 + u3 + · · ·
(4.17)
v(x, t ) = v0 + v1 + v2 + v3 + · · · . 5. Solution of the coupled soliton and a forward pump wave in a fiber with time-fractional order
With the assumption of uniform depletion, the pump wave is assumed to be a c w wave along the fiber. Chi and Wen [22] shows the effects caused by a small relative group velocity between a forward pump wave and the soliton. We consider the following coupled dimensionless wave equations:
∂ α u 1 ∂ 2u + as 2 + ηs (|u|2 + 2|v|2 )u = −iΓs u + iG |v|2 u (5.1) ∂tα 2 ∂x α ∂ v ∂u 1 ∂ 2v λs i −V − ap 2 + ηp (|v|2 + 2|u|2 )v = −iΓp v − i G |u|2 v (5.2) α ∂t ∂x 2 ∂x λf where u is the soliton and v is a forward pump wave; these are due to the Kerr effect. For Raman scattering, λs and λf are the wavelengths of the soliton and the pump wave, V is the relative velocity, as , ap , ηs , ηp , Γs , Γp , and G are arbitrary constants. i
Consider the generalized coupled dimensionless wave equations in an operator form: Dαt u = i
1 2
α
as D2xx u + ηs |u|2 u + (2ηs + G) |v|2 u − Γs u
Dt v = i −
1 2
ap D2xx
(5.3)
λs 2 v + vvt + ηp |v| v + 2ηp − G |u| v − Γp v λf 2
(5.4)
0 < α ≤ 1, t > 0. The initial conditions are taken from [22] as u(x, 0) =
as
ηs
sech(x)
(5.5)
v(x, 0) = b0
where
as
ηs
(5.6)
and b0 are the amplitudes of the soliton and pump wave.
We may follow the same analysis as applied before. Operating with J α on (5.3) and (5.4), we obtain u(x, t ) = u(x, 0) + iJ α
1 2
as D2xx u + ηs |u|2 u + (2ηs + G) |v|2 u − Γs u
λs 2 v(x, t ) = v(x, 0) + iJ − v + vvt + ηp |v| v + 2ηp − G |u| v − Γp v . 2 λf The decomposition method represents the linear terms u(x, t ) and v(x, t ) by the decomposition series α
u(x, t ) =
∞
∞ n=0
1
ap D2xx
2
(5.8)
un ( x , t )
n=0
v(x, t ) =
(5.7)
(5.9)
vn (x, t ).
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275
Substituting the nonlinear terms by an infinite series of polynomials ((3.9)–(3.12) and (3.14)), and (5.9) into ((5.7)–(5.8)) yields ∞
un (x, t ) = u(x, 0) + iJ α
2
n =0
∞
vn (x, t ) = v(x, 0) + iJ
n =0
α
1
as 1
− ap 2
∞
un (x, t )
+ ηs
n =0
xx
∞
An + (2ηs + G)
n =0
xx
∞
Bn − Γs u
(5.10)
n=0
∞ λs + Gn + ηp Dn + 2ηp − G En − Γp v . (5.11) λf n =0 n =0 n=0 ∞
v n ( x, t )
n=0
∞
∞
To accelerate the convergence of the solution, the decomposition method will be employed. The decomposition form, although it introduces a slightly different approach in the definition of the recursive relation, in fact results in a qualitative tool in minimizing the size of the calculations. The decomposition method defines the recursive relations in the form u0 (x, t ) = u(x, 0) un+1 (x, t ) = iJ α
1 2
as unτ τ + ηs An + (2ηs + G) Bn − Γs u ,
n≥0
(5.12)
and
v0 (x, t ) = v(x, 0) 1 λs α vn+1 (x, t ) = iJ − ap vnτ τ + Gn + ηp Dn + 2ηp − G En − Γp v , 2 λf
n ≥ 0.
(5.13)
By using Eqs. ((3.16)–(3.17), (3.19)–(3.20), and (3.22)), and Eqs. ((5.12)–(5.13)) we can calculate some of the terms of the decomposition series (5.9) as u0 = f (x) u1 = if1 (x) u2 = if2 (x)
tα
Γ (α + 1) t 2α
Γ (2α + 1)
and
v0 = g (x) v1 = ig1 (x) v2 = ig2 (x)
tα
Γ (α + 1) t 2α
Γ (2α + 1)
where f (x) = u(x, 0) = f 1 ( x) = f 2 ( x) =
1 2 1 2
as /ηs sech(x)
as f (2) + ηs |f |2 f + (2ηs + G) |g |2 f − Γs f (2)
as f 1
+ ηs 2 |f |2 f1 + f 2 f1 + (2ηs + G) |g |2 f1 + f (g g¯1 + g1 g¯ )
and g (x) = v(x, 0) = b0
λs G |f |2 g λf 1 λs (2) (1) (1) 2 2 g2 (x) = − ap g1 + gg1 + g1 g + ηp 2 |g | g1 + g g¯1 + 2ηp − G (|f |2 g1 + g (f f¯1 + f1 f¯ )) 2 λf
1 g1 (x) = − ap g (2) + gg (1) + ηp |g |2 g + 2
2ηp −
and so on. Substituting u0 , u1 , u2 , u3 , . . . and v0 , v1 , v2 , v3 , . . . into Eq. (5.9) gives the solutions u(x, t ) and v(x, t ) in a series form by u(x, t ) = u0 + u1 + u2 + u3 + · · ·
v(x, t ) = v0 + v1 + v2 + v3 + · · · .
(5.14)
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6. Conclusions The basic goal of this work has been to employ the Adomian decomposition method form as a reasonable basis for studying the solution of the nonlinear generalized coupled evolution equations. The goal has been achieved by obtaining analytical approximations with a high degree of accuracy. Figs. 1 and 2 show the computational results obtained for approximation solutions for α → 1 of system (4.1)–(4.2) by using the ADM in comparison with the exact solution. Fig. 3 shows the evolution results for u(x, t ) and v(x, t ) at α = 0.99. Figs. 4 and 5 show the computational results obtained for the approximation solution for α → 1 of the system (5.1)–(5.2) and the evolution results for u(x, t ) and v(x, t ) at α = 0.99. We find that the ADM provides an efficient numerical solution with high accuracy.
a
b
4 × 109 5 × 109 | u3 | 3 × 109 2.5 × 109 2 × 109 -10
4 × 109 5 × 109 | u3 | 3 × 109 2.5 × 109 2 × 109 -10
0.1 0.075 0.05t -5 x
5
0.05t -5
0.025
0
0.1 0.075
x
0.025
0 5
10 0
10 0
Fig. 1. The numerical results for u(x, t ) at α = 1 (a), and comparison with the exact solution (b). as = 23.7, ap = −36.2, ηs = 6.1 × 10−10 , ηp = 6.4 × 10−10 , Γs = 3, Γp = 5, G = 0.05, V = 590, λs = 1.3, λf = 1.23, b0 = 3.
a
| v7 |
b
1×1010 5×109 5×109 2.5×109 0 -10
0.1
| v7 |
0.075 0.05t -5 x
1×1010 5×109 5×109 2.5×109 0 -10
5
0.075 0.05t -5
0.025
0
0.1
x
0.025
0 5
10 0
10
Fig. 2. The numerical results for v(x, t ) at α = 1 (a), and comparison with the exact solution (b).
a
|
u3 |
b
4×109 5×109
0.1
3×109
2.5×109
0.75
2×109 -10
0.05t -5
0.025
0 x
5
10 0
1×1010 | v7 | 5×109 5×109 2.5×109 0 -10
0.1 0.75 0.05t -5
0.025
0 x
Fig. 3. The numerical results for u(x, t ) and v(x, t ) at α = 0.99.
5
10 0
A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
a
277
b
2×108 | v | 1.5×108 1×108 5×107 0
3 2
400000 300000 | u | 200000 100000 0
t
-10
2
0 10
t
-10
1 x
3
1 x
0 10
0
0
Fig. 4. The numerical results for u(x, t ) and v(x, t ) at α → 1. λ0 = 0.62, β = 5.4, n2 = 1.2 × 10−22 , ρ = 2.35 × 10−2 , σ = 0.08, x0 = 7.6, a0 = 3.9 × 109 , b0 = 1.3 × 1010 .
a
b
400000 | u | 300000 200000 100000 0
2×108 3 2 1
3
1×108 2 t
0 t
-10
|v|
-10
0 x
1
0 10
x 0
10
0
Fig. 5. The numerical results for u(x, t ) and v(x, t ) at α = 0.99.
The computational size has been reduced and rapid convergence has been guaranteed. The decomposition method introduces a significant improvement in this field over existing techniques. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
A. Hasegawa, Y. Kodama, Proc. IEEE 69 (1981) 1145. G.L. Lamb Jr., Elements of Soliton Theory, A Wiley-Inter-science Publication, Wiley, New York, 1980. H. Toda, Y. Furukawa, T. Kinoshita, Y. Kodama, A. Hasegawa, IEEE Photon. Technol. Lett. 9 (1997) 1415. T. Xu, C.Y. Zhang, G.M. Wei, J. Li, X.H. Meng, B. Tian, Eur. Phys. J. B 55 (2007) 323. D.U. Matine, H.C. Yuen, J. Phys. Fluids 23 (1980) 881. E.A. Kuznetsov, A.M. Rubenchik, V.E. Zakharov, Phys. Rep. 142 (1986) 103. T. Colin, Physica D 64 (1993) 215. I.A. Movaghar, Howard, Synth. Met. 27 (1988) A61. J. Zhang, J. Stat. Phys. 101 (2000) 731. G.P. Agrawal, Academic Press, New York, 1995. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 142. A. Hasegawa, Springer, Berlin, 1989. L.F. Mollenaur, R.H. Stolen, J.P. Gordon, Phy. Rev. Lett. 45 (1980) 1095. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 142. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23 (1973) 171. O. Bang, P.L. Christiansen, F.K. Rasmussen, Appl. Anal. 57 (1995) 3–15. G. Adomain, Kluwer A academic publisher, Boston, 1994. G. Adomain, Math. Anal. Appl. 135 (1988) 501–544. Y. Luchko, R. Grorefio, Preprint series A08-98, Fachbreich mathematic and informatics, Fre ic university, Berlin, 1998. M. Caputo, J. Roy. Astral. Soc. 13 (1967) 529–539. R. Ulrich, A. Simon, Appl. Opt. 18 (1979) 2241. S. Chi, S. Wen, Opt. Lett. 14 (1989) 84–86.
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Numerical solutions for some generalized coupled nonlinear evolution equations A.A.M. Arafa ∗ , S.Z. Rida Department of mathematics, Faculty of Science, South Valley University, Qena, Egypt
article
info
Article history: Received 29 March 2011 Received in revised form 24 December 2011 Accepted 27 December 2011 Keywords: The decomposition method Coupled evolution equation Numerical solution Fractional calculus
abstract In this paper, the Adomian decomposition method (ADM) is presented for the numerical solutions of the coupled evolution equations of fractional order. The ADM in applied mathematics can be used as an alternative method for obtaining analytic and approximate solutions for different types of fractional differential equations. The fractional derivatives are described in the Caputo sense. The given solutions are compared with the traveling wave solutions. The results obtained are then graphically represented. Finally, numerical results demonstrate the accuracy, efficiency and high rate of convergence of this method. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The optical soliton has been proposed as an information carrier in fiber communication systems [1]. Because of the fiber loss, the soliton disperses as it propagates along the fiber. The propagation of optical solitons has been the object of extensive theoretical and experimental research in optical fibers due to their enormous potential applications in telecommunication and ultrafast signal routing systems [2–4]. The propagation of optical solitons in the picosecond regime is governed by nonlinear Schrödinger (NLS)-type equations which arise from many physical fields such as fluid dynamics [5], plasma physics [6,7], solid-state physics [8], Bose–Einstein condensation [9] and fiber optics [10]. Hasegawa and Tappert [11] modeled the propagation of a guided mode in a perfect nonlinear mono-mode fiber by the NLS equation. They pointed out that the solitons could be propagated on glass fibers, with negative group velocity dispersion (GVD), under the action of the optical Kerr effect [12], in which there is a change of refractive index induced by the optical field. The GVD in glass fibers with Ge-doped cores is controlled mainly by the material dispersion, but it can be influenced by the index (doping) profile. An experimental verification of soliton propagation was first given by Molleneur et al. [13]. This work confirmed the predictions of the NLS equation and proved that the properties of fiber propagation are described to a remarkable degree by this equation. Respectively, in previous theoretical research, Raman pump waves were assumed to be either uniformly depleted, or undepleted. It is well known that nonlinear dispersive waves in polarization preserving single-mode fibers obey the so-called nonlinear Schrödinger equation [14,15], which is a model of the evolution of a one-dimensional packet of surface waves on sufficiently deep water. It is widely used in basic models of nonlinear waves in many areas of physics. It arises from the study of nonlinear wave propagation in dispersive and inhomogeneous media, such as plasma phenomena and non-uniform dielectric media. It is a member of the class of integrable equations. It is widely used in basic models of nonlinear waves in many areas of physics. It is a generic equation describing the evolution of the slowly varying amplitude of a nonlinear wave train in weakly nonlinear, strongly dispersive, and hyperbolic systems [16].
∗
Corresponding author. E-mail addresses: [email protected] (A.A.M. Arafa), [email protected] (S.Z. Rida).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.12.046
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269
In this article, we implement the Adomian decomposition method (ADM) [17,18] on the nonlinear generalized coupled evolution equations, which take the form
∂αu ∂ 2u + a + b(|u|2 + c |v|2 )u + Φ1 (u, v) = 0 ∂tα ∂ x2 ∂αv ∂ 2v i α + a 2 + b(|v|2 + c |u|2 )v + Φ2 (u, v) = 0. ∂t ∂x i
(1.1) (1.2)
This fractional system of equations is obtained by replacing the first time derivative term by a fractional derivative of order α > 0. The derivatives are understood in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of α → 1, the fractional system equations reduce to the standard system of partial differential equations. We introduce several examples to illustrate the main ideas of this work. The Adomian decomposition method will be applied for computing solutions to the systems of fractional partial differential equations considered in this paper. The method provides the solutions in the form of a power series with easily computed terms. It has many advantages over the classical techniques, mainly that it provides an efficient numerical solution with high accuracy and minimal calculations. 2. Fractional calculus There are several definitions of fractional derivatives. These definitions include Riemann–Liouville, Wely, Reize, Caputa, and Nishimoto fractional operators. The commonly used definition for a general fractional derivative is the Riemann–Liouville definition of the fractional derivative operator Iaα . Definition 2.1. Let α ∈ R+ . The operator Jaα defined on the usual Lebesque space L1 [a, b] by Jaα f (x) =
1
Γ (α)
x
(x − t )α−1 f (t )dt
0
Ja0 f (x) = f (x) for a ≤ x ≤ b is called the Riemann–Liouville fractional integral operator of order α . The properties of the operator Jaα can be found in Ref. [19]; we mention the following properties: for f ∈ L1 [a, b], α, β ≥ 0, and γ > −1: 1. 2. 3.
J α J β f (x) = J α+β f (x) J α J β f (x) = J β J α f (x) Γ (γ +1) J α xγ = Γ (α+γ +1) xα+γ .
Therefore, we will introduce a modified fractional operator Dα proposed by Caputo which is used in the up-coming paper and plays the most important role in the theory of differential and integral equations of fractional order. The main advantages of Caputo’s approach are that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer order differential equations. The Caputo definitions are founds in detail in Ref. [20]. Definition 2.2. The fractional derivative of f (x) in the Caputo sense is defined as Dα f (x) = J m−α Dm f (x) =
x
1
Γ (m − α)
(x − t )m−α+1 f (m) (t )dt
0
for m − 1 < α ≤ m, m ∈ N , x > 0. For the Caputo derivative we have Dα C = 0, Dα t n =
C is constant
0,
(n ≤ α − 1)
(n > α − 1)
Γ (n + 1) n−α t , Γ (n − α + 1)
.
Definition 2.3. For m to be the smallest integer that exceeds α , the Caputo fractional derivative of order α > 0 is defined as D α u( x , t ) =
1
∂ α u(x, t ) = Γm(m − α) ∂tα ∂ u(x, t ) , ∂tm
t
(t − τ ) 0
m−α+1
∂ µ (x, τ ) dτ , ∂τ m
for m − 1 < α < m for α = m ∈ N
.
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A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
3. Analysis of the method We first consider the system of generalized coupled PDEs written in an operator form Dαt u + Lx v + N1 (u, v) = Φ1
(3.1)
Dαt v + Lx u + N2 (u, v) = Φ2 with initial data u(x, 0) = f1 (x)
(3.2)
v(x, 0) = f2 (x) α
where Lx = ∂∂x2 , and Dαt = ∂∂t α is the fractional operator; without loss of generality the first order partial differential operators N1 and N2 are nonlinear operators, and Φ1 and Φ2 are inhomogeneous terms. Applying the inverse operator J α to the system (3.1) and using the initial data (3.2) yields 2
u(x, t ) = f1 (x) + J α (Φ1 − Lx v − N1 (u, v))
(3.3)
v(x, t ) = f2 (x) + J α (Φ2 − Lx u − N2 (u, v)).
The Adomian decomposition method suggests that the linear terms u(x, t ) and v(x, t ) are decomposed by an infinite series of components u(x, t ) =
∞
un ( x , t )
n =0
v(x, t ) =
∞
(3.4)
v n ( x, t )
n=0
and the nonlinear operators N1 (u, v) and N2 (u, v) by the infinite series of so called Adomian polynomials N1 (u, v) =
∞
An
n=0
N2 (u, v) =
∞
(3.5) Bn
n=0
where un (x, t ) and vn (x, t ); n ≥ 0 are the components of u(x, t )and v(x, t ) that will be elegantly determined, and An and Bn , n ≥ 0, are Adomian polynomials that can be generated for all forms of nonlinearity. Substituting (3.4) and (3.5) into (3.3) gives ∞
un (x, t ) = f1 (x) + J
α
Φ1 − Lx
n =0
∞
vn (x, t ) = f2 (x) + J
α
Φ2 − Lx
n =0
∞
v n ( x, t ) −
∞
n=0
n =0
∞
∞
un ( x , t ) −
n=0
An (3.6)
Bn
.
n =0
Following Adomian analysis, the nonlinear system (3.1) is transformed into a set of recursive relations given by u0 (x, t ) = f1 (x) un+1 (x, t ) = J α (Φ1 − Lx vn − An ),
n≥0
(3.7)
n ≥ 0.
(3.8)
v0 (x, t ) = f2 (x) vn+1 (x, t ) = J α (Φ2 − Lx un − Bn ),
It is an essential feature of the decomposition method that the zeroth components u0 (x, t ) and v0 (x, t ) are defined always by all terms that arise from initial data and from integrating the inhomogeneous terms. Having defined these the zeroth pair (u0 , v0 ) can be determined recurrently by using (3.7) and (3.8). The remaining pairs (un , vn ), n ≥ 1, can be easily determined in a parallel manner. Additional pairs for the decomposition series normally account for higher accuracy. Having determined the components of u(x, t ) and v(x, t ), the solution (u, v) of the system follows immediately in the form of a power series expansion upon using (3.4). The series obtained can be summed up in many cases to give a closed form solution. For concrete problems, the n term approximants can be used for numerical purposes. Comparing the scheme presented above with existing techniques such as the characteristics method and Riemann invariants, it is clear that the decomposition method introduces a fundamentally qualitative difference in approach, because no assumptions are made. The approach is straightforward and
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271
rapid convergence is guaranteed. To give a clear overview of the content of this work, several illustrative examples have been selected to demonstrate the efficiency of the method. In this paper the nonlinear terms by an infinite series of polynomials take the forms u|u|2 =
∞
An
(3.9)
Bn
(3.10)
n=0
u|v|2 =
∞ n =0
v2 u =
v|v|2 =
Cn
∞
(3.11)
Dn
(3.12)
En
(3.13)
n =0
v|u|2 =
∞ n =0
∞
u2 v =
Fn
(3.14)
Gn
(3.15)
n =0
∞
vvt =
n=0
where An , Bn , Cn , Dn , En , Fn and Gn are the Adomian polynomials defined as follows. The Adomian polynomials for the nonlinear term u|u|2 are given by A0 = |u0 |2 u0 A1 = 2|u0 |2 u1 + u20 u¯ 1 2
u0 u21
A2 = 2|u0 | u2 + ¯
(3.16) 2
+ | u1 | u0 +
¯ .
u20 u2
2
Those for the nonlinear term u|v| are given by B0 = |v0 |2 u0 B1 = |u0 |2 u1 + u0 v1 u¯ 0 + u0 v0 v¯ 1
(3.17)
B2 = u0 v0 v¯ 2 + u0 |v1 | + u1 (v0 v¯ 1 + v1 v¯ 0 ) + u2 |v0 | . 2
2
Those for the nonlinear term v 2 u are given by C0 = v02 u0 C1 = 2v0 v1 u0 + v02 u1 C2 =
(3.18)
v + 2u1 v0 v1 + 2u0 v0 v2 +
u2 02
v .
u0 12
Those for the nonlinear term v|v| are given by 2
D0 = |v0 |2 v0 D1 = 2|v0 |2 v1 + v02 v¯ 1
(3.19)
D2 = 2|v0 | v2 + v¯ v + |v1 |2 v0 + v02 v¯ 2 . 2
2 0 1
Those for the nonlinear term v|u|2 are given by E0 = |u0 |2 v0 E1 = |v0 |2 v1 + v0 u1 v¯ 0 + v0 u0 u¯ 1
(3.20)
E2 = v0 u0 u¯ 2 + v0 |u1 |2 + v1 (u0 u¯ 1 + u1 u¯ 0 ) + v2 |u0 |2 . Those for the nonlinear term u2 v are given by F0 = u20 v0 F1 = 2u0 u1 v0 + u20 v1 F2 = v
2 2 u0
+ 2v1 u0 u1 + 2v0 u0 u2 + v
(3.21) 2 0 u1
.
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A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
And those for the nonlinear term vvt are given by G0 = v0 v0t G1 = v0 v1t + v1 v0t
(3.22)
G2 = v0 v2t + v1 v1t + v2 v0t .
4. Solution of weakly birefringent optical fibers with time-fractional order Let us consider a single-mode fiber with weak birefringence. The fiber is assumed to be straight (no twists) so as to avoid any effects arising from optical activity [21]. If u is the field envelope of the polarization along the x axis of the birefringent fiber and v the envelope along the y axis, then u and v obey the following pair of coupled evolution equations of fractional order:
∂ α u β ∂ 2 u w 0 n2 i α − + ∂t 2 ∂ x2 2c i
∂αv β ∂ 2v w0 n2 − + α ∂t 2 ∂ x2 2c
2
2
|u| + |v| 3
2
u+
1 3
v ue 2
−4iρ t
=0
(4.1)
2 1 |v|2 + |u|2 v + v 2 u e4iρ t = 0 3
(4.2)
3
where w0 = 2π c /λ0 is the carrier angular frequency of the light field, λ0 is its wave length, n2 is the nonlinear Kerr coefficient of glass, β is the fiber dispersive coefficient evaluated at w0 , and ρ is related to the birefringence of the fiber. Consider the generalized coupled evolution equations in an operator form: iDαt u − α
iDt v −
β 2
β 2
uxx +
vxx +
w0 n2 2c
2 1 |u|2 + |v|2 u + v 2 u e−4iρ t = 0
(4.3)
2 1 2 4iρ t 2 2 |v| + |u| v + v u e =0
(4.4)
3
w 0 n2 2c
3
3
3
0 < α ≤ 1, t > 0 with the initial data u(x, 0) = a0 sech
v(x, 0) = b0 tanh
x
(4.5)
x0
x
.
x0
(4.6)
The exact solution, for the special case α → 1, is given by u(x, t ) = a0 sech
v(x, t ) = b0 tanh
x
ei(σ −ρ)t
(4.7)
ei(σ +ρ)t .
(4.8)
x0
x x0
We may follow the same analysis as applied before. Operating with J α on (4.3) and (4.4) we obtain u(x, t ) = u(x, 0) + iJ α
v(x, t ) = v(x, 0) + iJ
α
β 2
β 2
uxx −
vxx −
w0 n2 2c
w 0 n2 2c
|u|2 +
2
2 3 2
1 |v|2 u + v 2 u e−4iρ t 3
2
|v| + |u| 3
1
v + v ue 3
2
4iρ t
.
(4.9)
(4.10)
The decomposition method represents the linear terms u(x, t ) and v(x, t ) by the decomposition series u(x, t ) =
∞
un ( x , t )
(4.11)
vn (x, t ).
(4.12)
n=0
v(x, t ) =
∞ n=0
A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
273
Substituting the nonlinear terms by an infinite series of polynomials ((3.9)–(3.14)), and ((4.11)–(4.12)) into ((4.9)–(4.10)) yields ∞
un (x, t ) = u(x, 0) + iJ
α
∞ w0 n2
∞
i
6c
un ( x , t )
−
n=0
xx
w0 n2
∞
2c
An +
n=0
∞ 2
3 n =0
Bn
Cn J α ( e−4iρ t )
(4.13)
n=0
vn (x, t ) = v(x, 0) + iJ
α
∞ w0 n2
i
6c
β
2
n =0
−
∞
2
n =0
−
β
∞
v n ( x, t )
−
n=0
xx
w0 n2
2c
∞
Dn +
n =0
∞ 2
3 n =0
En
Fn J α ( e4iρ t ).
(4.14)
n=0
To accelerate the convergence of the solution, the decomposition method will be employed. The decomposition form, although it introduces a slightly different approach in the definition of the recursive relation, in fact results in a qualitative tool in minimizing the size of the calculations. The decomposition method defines the recursive relations in the form u0 (x, t ) = u0 (x, 0) un+1 (x, t ) = iJ
α
β 2
(unxx ) −
w0 n2
An +
2c
2 3
−
Bn
w0 n2
iCn J α ( e−4iρ t ),
6c
n≥0
(4.15)
and
v0 (x, t ) = v0 (x, 0) w 0 n2 2 w 0 n2 α β vn+1 (x, t ) = iJ (vnxx ) − D n + En − iFn J α ( e4iρ t ), 2
2c
3
6c
n ≥ 0.
(4.16)
By using Eqs. ((3.16)–(3.21)), and Eqs. ((4.15)–(4.16)), we can calculate some of the terms of the decomposition series (4.11) and (4.12) as u0 = f (x)
u1 = i f1 (x)
Γ (α + 1)
u2 = i f3 (x)
tα
t
− f2 (x)
∞ k =0
2α
− f4 (x)
Γ (2α + 1)
(−4iρ)k t α+k Γ (α + k + 1)
(−4iρ)k t α+k Γ (α + k + 1)
∞ k=0
and
v0 = g (x) v 1 = i g 1 ( x)
Γ (α + 1)
v 2 = i g 3 ( x)
tα
t
− g2 (x)
∞ k=0
2α
Γ (2α + 1)
− g4 (x)
(4iρ)k t α+k Γ (α + k + 1)
∞ k=0
(4iρ)k t α+k Γ (α + k + 1)
,
where f (x) = a0 sech f 1 ( x) = f 2 ( x) = f 3 ( x) = f 4 ( x) =
β 2
6c 2
(2)
f1
w0 n2 6c
x
x0
f (2) −
w 0 n2 β
w 0 n2
2c
2
|f |2 f + |g |2 f
3
2
g f
−
w 0 n2 2c
2
2|f | f1 + f f1 + (|g | f1 + f (g g¯1 + g1 g¯ ))
(2gg1 f + g 2 f1 )
2
2
3
2
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A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
and g (x) = b0 tanh g1 (x) = g2 (x) = g3 (x) =
β 2
x0
w 0 n2
2c
2
|g |2 g + |f |2 g
3
2
f g
6c 2
x
g (2) −
w 0 n2 β
(2)
g1
−
w 0 n2
2c
2
2|g | g1 + g g¯1 + (|f | g1 + g (f f¯1 + f1 f¯ )) 2
2
2
3
w0 n2
(2ff1 f + f 2 g1 ) 6c and so on. Substituting u0 , u1 , u2 , u3 , . . . and v0 , v1 , v2 , v3 , . . . into (4.11) and (4.12) gives the solutions u(x, t ) and v(x, t ) in a series form by g4 (x) =
u(x, t ) = u0 + u1 + u2 + u3 + · · ·
(4.17)
v(x, t ) = v0 + v1 + v2 + v3 + · · · . 5. Solution of the coupled soliton and a forward pump wave in a fiber with time-fractional order
With the assumption of uniform depletion, the pump wave is assumed to be a c w wave along the fiber. Chi and Wen [22] shows the effects caused by a small relative group velocity between a forward pump wave and the soliton. We consider the following coupled dimensionless wave equations:
∂ α u 1 ∂ 2u + as 2 + ηs (|u|2 + 2|v|2 )u = −iΓs u + iG |v|2 u (5.1) ∂tα 2 ∂x α ∂ v ∂u 1 ∂ 2v λs i −V − ap 2 + ηp (|v|2 + 2|u|2 )v = −iΓp v − i G |u|2 v (5.2) α ∂t ∂x 2 ∂x λf where u is the soliton and v is a forward pump wave; these are due to the Kerr effect. For Raman scattering, λs and λf are the wavelengths of the soliton and the pump wave, V is the relative velocity, as , ap , ηs , ηp , Γs , Γp , and G are arbitrary constants. i
Consider the generalized coupled dimensionless wave equations in an operator form: Dαt u = i
1 2
α
as D2xx u + ηs |u|2 u + (2ηs + G) |v|2 u − Γs u
Dt v = i −
1 2
ap D2xx
(5.3)
λs 2 v + vvt + ηp |v| v + 2ηp − G |u| v − Γp v λf 2
(5.4)
0 < α ≤ 1, t > 0. The initial conditions are taken from [22] as u(x, 0) =
as
ηs
sech(x)
(5.5)
v(x, 0) = b0
where
as
ηs
(5.6)
and b0 are the amplitudes of the soliton and pump wave.
We may follow the same analysis as applied before. Operating with J α on (5.3) and (5.4), we obtain u(x, t ) = u(x, 0) + iJ α
1 2
as D2xx u + ηs |u|2 u + (2ηs + G) |v|2 u − Γs u
λs 2 v(x, t ) = v(x, 0) + iJ − v + vvt + ηp |v| v + 2ηp − G |u| v − Γp v . 2 λf The decomposition method represents the linear terms u(x, t ) and v(x, t ) by the decomposition series α
u(x, t ) =
∞
∞ n=0
1
ap D2xx
2
(5.8)
un ( x , t )
n=0
v(x, t ) =
(5.7)
(5.9)
vn (x, t ).
A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
275
Substituting the nonlinear terms by an infinite series of polynomials ((3.9)–(3.12) and (3.14)), and (5.9) into ((5.7)–(5.8)) yields ∞
un (x, t ) = u(x, 0) + iJ α
2
n =0
∞
vn (x, t ) = v(x, 0) + iJ
n =0
α
1
as 1
− ap 2
∞
un (x, t )
+ ηs
n =0
xx
∞
An + (2ηs + G)
n =0
xx
∞
Bn − Γs u
(5.10)
n=0
∞ λs + Gn + ηp Dn + 2ηp − G En − Γp v . (5.11) λf n =0 n =0 n=0 ∞
v n ( x, t )
n=0
∞
∞
To accelerate the convergence of the solution, the decomposition method will be employed. The decomposition form, although it introduces a slightly different approach in the definition of the recursive relation, in fact results in a qualitative tool in minimizing the size of the calculations. The decomposition method defines the recursive relations in the form u0 (x, t ) = u(x, 0) un+1 (x, t ) = iJ α
1 2
as unτ τ + ηs An + (2ηs + G) Bn − Γs u ,
n≥0
(5.12)
and
v0 (x, t ) = v(x, 0) 1 λs α vn+1 (x, t ) = iJ − ap vnτ τ + Gn + ηp Dn + 2ηp − G En − Γp v , 2 λf
n ≥ 0.
(5.13)
By using Eqs. ((3.16)–(3.17), (3.19)–(3.20), and (3.22)), and Eqs. ((5.12)–(5.13)) we can calculate some of the terms of the decomposition series (5.9) as u0 = f (x) u1 = if1 (x) u2 = if2 (x)
tα
Γ (α + 1) t 2α
Γ (2α + 1)
and
v0 = g (x) v1 = ig1 (x) v2 = ig2 (x)
tα
Γ (α + 1) t 2α
Γ (2α + 1)
where f (x) = u(x, 0) = f 1 ( x) = f 2 ( x) =
1 2 1 2
as /ηs sech(x)
as f (2) + ηs |f |2 f + (2ηs + G) |g |2 f − Γs f (2)
as f 1
+ ηs 2 |f |2 f1 + f 2 f1 + (2ηs + G) |g |2 f1 + f (g g¯1 + g1 g¯ )
and g (x) = v(x, 0) = b0
λs G |f |2 g λf 1 λs (2) (1) (1) 2 2 g2 (x) = − ap g1 + gg1 + g1 g + ηp 2 |g | g1 + g g¯1 + 2ηp − G (|f |2 g1 + g (f f¯1 + f1 f¯ )) 2 λf
1 g1 (x) = − ap g (2) + gg (1) + ηp |g |2 g + 2
2ηp −
and so on. Substituting u0 , u1 , u2 , u3 , . . . and v0 , v1 , v2 , v3 , . . . into Eq. (5.9) gives the solutions u(x, t ) and v(x, t ) in a series form by u(x, t ) = u0 + u1 + u2 + u3 + · · ·
v(x, t ) = v0 + v1 + v2 + v3 + · · · .
(5.14)
276
A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
6. Conclusions The basic goal of this work has been to employ the Adomian decomposition method form as a reasonable basis for studying the solution of the nonlinear generalized coupled evolution equations. The goal has been achieved by obtaining analytical approximations with a high degree of accuracy. Figs. 1 and 2 show the computational results obtained for approximation solutions for α → 1 of system (4.1)–(4.2) by using the ADM in comparison with the exact solution. Fig. 3 shows the evolution results for u(x, t ) and v(x, t ) at α = 0.99. Figs. 4 and 5 show the computational results obtained for the approximation solution for α → 1 of the system (5.1)–(5.2) and the evolution results for u(x, t ) and v(x, t ) at α = 0.99. We find that the ADM provides an efficient numerical solution with high accuracy.
a
b
4 × 109 5 × 109 | u3 | 3 × 109 2.5 × 109 2 × 109 -10
4 × 109 5 × 109 | u3 | 3 × 109 2.5 × 109 2 × 109 -10
0.1 0.075 0.05t -5 x
5
0.05t -5
0.025
0
0.1 0.075
x
0.025
0 5
10 0
10 0
Fig. 1. The numerical results for u(x, t ) at α = 1 (a), and comparison with the exact solution (b). as = 23.7, ap = −36.2, ηs = 6.1 × 10−10 , ηp = 6.4 × 10−10 , Γs = 3, Γp = 5, G = 0.05, V = 590, λs = 1.3, λf = 1.23, b0 = 3.
a
| v7 |
b
1×1010 5×109 5×109 2.5×109 0 -10
0.1
| v7 |
0.075 0.05t -5 x
1×1010 5×109 5×109 2.5×109 0 -10
5
0.075 0.05t -5
0.025
0
0.1
x
0.025
0 5
10 0
10
Fig. 2. The numerical results for v(x, t ) at α = 1 (a), and comparison with the exact solution (b).
a
|
u3 |
b
4×109 5×109
0.1
3×109
2.5×109
0.75
2×109 -10
0.05t -5
0.025
0 x
5
10 0
1×1010 | v7 | 5×109 5×109 2.5×109 0 -10
0.1 0.75 0.05t -5
0.025
0 x
Fig. 3. The numerical results for u(x, t ) and v(x, t ) at α = 0.99.
5
10 0
A.A.M. Arafa, S.Z. Rida / Mathematical and Computer Modelling 56 (2012) 268–277
a
277
b
2×108 | v | 1.5×108 1×108 5×107 0
3 2
400000 300000 | u | 200000 100000 0
t
-10
2
0 10
t
-10
1 x
3
1 x
0 10
0
0
Fig. 4. The numerical results for u(x, t ) and v(x, t ) at α → 1. λ0 = 0.62, β = 5.4, n2 = 1.2 × 10−22 , ρ = 2.35 × 10−2 , σ = 0.08, x0 = 7.6, a0 = 3.9 × 109 , b0 = 1.3 × 1010 .
a
b
400000 | u | 300000 200000 100000 0
2×108 3 2 1
3
1×108 2 t
0 t
-10
|v|
-10
0 x
1
0 10
x 0
10
0
Fig. 5. The numerical results for u(x, t ) and v(x, t ) at α = 0.99.
The computational size has been reduced and rapid convergence has been guaranteed. The decomposition method introduces a significant improvement in this field over existing techniques. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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