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The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schrödinger equation and its solutions
Accepted Manuscript Title: The generalized nonlinear higher order of KdVequations from the higher ordernonlinear Schr¨odinger equation and its solutions Author: A.R. Seadawy PII: DOI: Reference:
S0030-4026(17)30352-2 http://dx.doi.org/doi:10.1016/j.ijleo.2017.03.086 IJLEO 59009
To appear in: Received date: Accepted date:
19-12-2016 20-3-2017
Please cite this article as: A.R. Seadawy, The generalized nonlinear higher order of KdVequations from the higher ordernonlinear Schrddotodinger equation and its solutions, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.03.086 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schr¨ odinger equation and its solutions A. R. Seadawy
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Mathematics Department, Faculty of science, Taibah University, Al-Ula, Saudi Arabia. Mathematics Department, Faculty of Science, Beni-Suef University, Egypt. E-mail: [email protected]
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Abstract The higher order nonlinear Schr¨odinger (NLS) equation describes ultrashort pluse propagation in optical fibres. The generalized nonlinear fifth-order of
us
KdV equations derived from the higher order NLS equation by using multiple scales methods. We obtained the traveling wave solutions for some different kinds of the
an
generalized nonlinear fifth-order of KdV (fifth-order Lax; fifth-order Ito; fifth-order Sawada-Kotera; fifth-order Kaup-Kupershmidt; fifth-order Caudrey-Dodd-Gibbon)
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equations by applying the auxiliary equation of the direct algebraic method. These solutions for the generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and
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the movement role of the waves by making the graphs of the exact solutions are
Ac ce pt e
analyzed. All solutions are exact and stable. Keywords. Higher order NLS equation; Generalized higher order of KdV equation; Traveling wave solutions; Mathematical Physics methods. 2000 MR Subject Classification 35G20 · 35Q53 · 37K10 · 49S05 · 76A60
1
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1
Derivation of the generalized fifth order KdV equation from higher order nonlinear Schr¨ odinger equation
Nonlinear wave phenomena exist in many fields, such as fluid mechanics, plasma
ip t
physics, biology, hydrodynamics, solid state physics and optical fibers, etc. In order to better understand these nonlinear phenomena, it is important to seek their exact solutions. They can help to analyze the stability of these solutions and the movement
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role of the wave by making the graphs of the exact solutions [1-11].
Optical solitons have been the subjects of extensive research in nonlinear optics
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due to their potential applications in telecommunication and ultra fast signal processing systems [12-14]. Optical solitons arise from the balance between group velocity
an
dispersion effect and nonlinear effect arising due to nonlinear change in the refractive index [13]. A higher-order Schrodinger equation containing parameters, which
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is used to describe pulse propagation in optical fibres, is shown to admit an infinitedimensional prolongation structure for exactly four combinations of the parameters, besides the classical NLS equation [15-23].
d
Consider one of the integrable cases of higher order nonlinear Schr¨odinger (NLS)
Ac ce pt e
equation which is known as the Kaup-Newell equation [23] 1 iqτ = i(a3 qq ∗ qζ + a3 q 2 q ∗ ) − (a1 qζζ + a2 q 2 q ∗ ) 2
By separating the phase and amplitude in the form as p p q(ζ, τ ) = eiθ(ζ,τ ) N(ζ, τ ), q ∗ (ζ, τ ) = e−iθ(ζ,τ ) N(ζ, τ ).
(1)
(2)
By substituting from equation (2) into Kaup-Newell equation (1) and grouping the real and imaginary parts, the following real differential equations can be obtained as Nτ =
Vτ =
1 6a3 Nζ N 3 − 8a1 Nζ N 2 V − 8a1 Vζ N 3 , 2 4N
1 (2a1 Nζζζ N 2 − 4a1 Nζζ Nζ N + 8a1 Nζ3 + 2a3 Nζ N 3 V + 4a2 Nζ N 3 − 4N 3 8a1 Vζ V N 3 + 2a3 Vζ N 4 ),
(3)
2
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where θ(ζ, τ ) = V (ζ, τ ). The following series expansions for solutions as N =1+
∞ X
2n
ǫ Nn (x, t1 , t2,...,tn ),
V =
∞ X
ǫ2n Vn (x, t1 , t2,...,tn ),
(4)
n=1
n=1
where ζ and τ are slow variables with respect to scaling parameters ǫ > 0 respectively
ip t
as follows t1 = ǫ3 τ,
x = ǫ(ζ + 2τ ),
t2 = ǫ5 τ, ....tn = ǫ2n+1 τ.
(5)
cr
By substituting from series expansions (4) and (5) into a system of real differential The coefficients ǫ3 as 1 (3a3 − 4)N1 , 4a1
a2 =
1 [2a2 N1xxx + (18a3 − 3a23 − 24)N1 N1x ]. 4(a3 − 2) 1 a3 = 5, then
N1t1 =
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Choosing a1 = 1,
1 (−3a23 + 16a3 − 16)N1 , 8a1
an
V1 =
us
equations (3) and collect the coefficients of the powers of ǫ equal to zero separately.
Ac ce pt e
d
9 1 N1t1 = (N1xxx − N1 N1x ). 6 2 Making the following transformation t1 →
1 t1 , 6
(6)
4 N1 → − u 3
Inserting (6) into series expansion at the coefficient ǫ7 , we obtain
N1t2 =
1 N2t +24N1xxxxx +56N1 N1xxx +528N1x N1xx +11682N2N1x +1038N12N1x − 4752 1 5544V2 N1x − 216V2xxx − 1296V2x N1 + 1188V2t1 .
(7)
For equation (7), take N2 , V2 as following N2 = K11 N1xx + K11 N12
V2 = K21 N1xx + K22 N12 ,
By choosing the coefficients K11 = −
7 6
K12 =
95 48
K21 = −
79 24
K22 =
949 , 192
3
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we obtain N1t2 =
1 N1xxxxx + 10N1 N1xxx + 20N1x N1xx + 30N12 N1x . 216
(8)
Now by using the following transformation 1 t2 , , 216
4 N1 → − u 3
ip t
t2 →
cr
The fifth-order lax’s KdV equation can be derived as
us
ut2 = uxxxxx + 10uuxxx + 20ux uxx + 30u2 ux . By choosing 337 216
K12 =
755 288
K21 = −
3779 864
an
K11 = −
K22 =
(9)
7549 , 1152
The fifth-order Sawada-Kotera’s equation can be obtained as ut2 = uxxxxx + 5uuxxx + 5ux uxx + 5u2 ux . By supposing 173 144
K12 =
Ac ce pt e
K11 = −
d
M
(10)
145 72
K21 = −
1951 576
K22 =
1451 , 288
The fifth-order Kaup-Kupershmidt’s KdV equation can be got as ut2 = uxxxxx + 10uuxxx + 25ux uxx + 20u2 ux .
(11)
Consider the generalized fifth-order KdV equation as [24-29] ut + αu2ux + βux uxx + γuuxxx + λuxxxxx = 0 .
(12)
Where α, β, γ, are real parameters and λ = ±1. The generalized fifth-order KdV
equation (12) describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice [1] and it is an important mathematical model with wide applications in quantum mechanics, nonlinear optics, solid state physics, 4
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plasma physics, fluid physics and quantum field theory. Some important particular cases of equation (12) are Case I: When α = 30, β = 20, γ = 10, λ = 1, equation (12) called the fifth-order Lax equation as
ip t
ut + 30u2ux + 20ux uxx + 10uuxxx + uxxxxx = 0 .
(13)
cr
Case II: When α = 2, β = 6, γ = 3, λ = 1, equation (12) named the fifth-order Ito equation as
(14)
us
ut + 2u2ux + 6ux uxx + 3uuxxx + uxxxxx = 0 .
Case III: When α = 45, β = 15, γ = 15, λ = 1, equation (12) called the
an
fifth-order Sawada-Kotera equation as
ut + 45u2ux + 15ux uxx + 15uuxxx + uxxxxx = 0 .
(15)
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Case IV: When α = 20, β = 25, γ = 10, λ = 1, equation (12) named the fifth-order Kaup-Kupershmidt equation as
(16)
d
ut + 20u2ux + 25ux uxx + 10uuxxx + uxxxxx = 0 .
Ac ce pt e
Case V: When α = 180, β = 30, γ = 30, λ = 1, equation (12) called the
fifth-order Caudrey-DoddGibbon equation as ut + 180u2ux + 30ux uxx + 30uuxxx + uxxxxx = 0 .
2
(17)
Travelling wave solutions of the generalized fifth order KdV equation
Suppose that the travelling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 +
n X i ai φ (η) + bi φ−i (η) ,
(18)
i=1
and ′
φ =
p c2 φ 2 + c3 φ 3 + c4 φ 4 ,
′
φ = 5
p
c2 φ 2 + c4 φ 4 + c6 φ 6
(19)
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where ξ = kx − ωt and k, ω, a1 , ......... an , b1 , ......... bn are arbitrary constants. Consider the traveling wave solutions (18) and (19), then equation (12) becomes −ωu′ + αku2 u′ + βk 3 u′ u′′ + γk 3 uu(3) + λk 5 u(5) = 0 . Families I:
ip t
2.1
(20)
Balancing the nonlinear term uu(3) and the highest order derivative u(5) gives n = 2.
cr
Then the traveling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 + a1 φ(ξ) + a2 φ2 (ξ) + b1 φ−1 (ξ) + b2 φ−2 (ξ),
us
(21)
where ξ = kx − ωt and k, ω, a0, a1 , a2 , b1 , b2 are arbitrary constants. Substituting
an
from equations (21) into equation (20) and collecting coefficients, then setting coefficients equal to zero, we obtain a system of algebraic equations. By solving this
q 2 2 4 2 −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) ,
c23 , 4c4
q 2 2 4 2 −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) ,
Ac ce pt e
3 a1 = 2α
c2 =
d
c3 a0 = 16αc4
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system, the parameters k, ω, a0 , a1 , a2 , b1 , b2 can be determined as
q 3c4 2 2 4 2 a2 = −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) , αc3 q k 3 c33 2 2 4 2 k c3 (12α − β(β + 2γ)) ± β c3 k ((β + 2γ) − 40α) ; ω= 128αc24
(22)
Case I: The coefficients of the fifth-order Lax equation can be obtained as
k2 β 2 a0 = − , 8γ
2
a1 = −3k β ,
2
a2 = −6k γ ,
b1 = b2 = 0,
β2 α= , 4γ
7k 5 β 4 ω= ; 32γ 2 (23)
Case II: The coefficients of the fifth-order Ito equation can be determined as √ √ √ (−9 ± 2 19)k 2 β 2 , a1 = 3(−9 ± 2 19)k 2 β , a2 = 6(−9 ± 2 19)k 2 γ , a0 = 8γ 6
Page 6 of 22
FigureH1 - aL
FigureH1 - bL 0.1
-10
-0.2
-5
ip t
5
-5
5
10
-0.1
cr
-0.4
-0.2
us
-0.6
-0.8
-0.3
M
an
Bistable bright solitary wave solutions of the fifth-order Lax equation (13) with various different shapes plotted in Figures (1a-1b).
2
-2
4
-10
Ac ce pt e
-4
FigureH2 - aL
d
FigureH1 - cL
5
-5
10
-10
-0.5
-20
-1.0
-30
-1.5
-40
-2.0
-50
Bistable dark solitary wave solution of the fifth-order Lax equation (13) with various different shape plotted in Figure (1c). Velocity potential in the fluid solution (28) with coefficients (24) of the fifth-order Ito equation plotted in Figure (2a).
7
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FigureH2 - bL
FigureH2 - cL 5
-5
5
-5
ip t
-2
-4
-0.1
cr
-6
-0.2
-8
us
-10
-0.3
-12
-14
M
an
Velocity potentials in the fluid solutions (29-30) with coefficients (24) of the fifth-order Ito equation plotted in Figures (2b-2c).
Ac ce pt e
0.16
-10
-5
FigureH3 - bL
d
FigureH3 - aL
0.16
0.14
0.14
0.12
0.12
0.10
0.10
5
10
-10
-5
5
10
Periodic solitary wave solutions of the fifth-order Sawada-Kotera equation (15) with coefficients (25) plotted in Figures (3a-3b).
8
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√ 3(4 ± 19)k 5 β 4 β2 , ω= ; (24) b1 = b2 = 0, α = 4γ 16γ 2 Case III: The coefficients of the fifth-order Sawada-Kotera equation can be derived as p
5(k 6 α2 + 4kω) k2 β 2 β2 2 , a1 = −k β , a2 = − , b1 = b2 = 0, γ = 30k 2α 4α (25)
ip t
a0 =
−5αk 3 ±
cr
Case IV: The coefficients of the fifth-order Kaup-Kupershmidt equation can be obtained as
us
v u p u 4 3 2 5 2 t 3 5αγk ± 5k γ (k α + 4ω) −5αγk 4 ± 5k 3 γ 2 (k 5 α2 + 4ω) a0 = , a = ± , 1 20γk 2 10 v u u 2 5αγk 4 ± p5k 3 γ 2 (k 5 α2 + 4ω) t 2 3 b1 = b2 = 0, β = ± 2 ; (26) a2 = − γk 2 , 2 k 15 Case V: The coefficients of the fifth-order Caudrey-Dodd-Gibbon equation can be derived as p
√ a1 = ±k 2 αγ,
d
5k(k 5 α2 + 4ω) , 60k
a2 = ±γk 2 ,
b1 = b2 = 0,
√ β = ±2 αγ. (27)
Ac ce pt e
a0 =
−5αk 3 ±
M
an
p
By substituting from equation (22) into (21), the following solution of Generalized
nonlinear fifth-order KdV equation (12) can be obtained as p k2 2 − 40α −β − 2γ ± (β + 2γ) c43 − 24c2 c4 c23 u1 (x, t) = 16αc4c23 √ √ 2 ! c2 c2 1 + tanh (kx − ωt) + ξ0 + 48c22 c24 1 + tanh (kx − ωt) + ξ0 2 2 (28) p 2 2 k −β − 2γ ± (β + 2γ) − 40α √ 2 2 u2 (x, t) = √ 2 (ρ + cosh [ c2 (kx − ωt) + ξ0 ]) c3 16αc4 ρ + cosh c2 (kx − ωt) + ξ0 √ √ +12 ρ2 + 2ρ cosh [ c2 (kx − ωt) + ξ0 ] + cosh [2 c2 (kx − ωt) + ξ0 ] √ √ +2ρ sinh [ c2 (kx − ωt) + ξ0 ] + sinh [2 c2 (kx − ωt) + ξ0 ]) c2 c4 9
Page 9 of 22
FigureH3 - cL
FigureH4 - aL 5
-5
-10
5
-5
10
-0.5
ip t
-0.2
-1.0
-0.4
cr
-1.5
-0.6
-2.0
us
-2.5
-3.0
-0.8
-1.0
FigureH4 - bL
d
M
an
Velocity potential in the fluid solution (30) with coefficients (25) of the fifth-order Sawada-Kotera equation (15) plotted in Figure (3c). Bistable solitary wave solution of the fifth-order KaupKupershmidt equation (16) with coefficients (26) plotted in Figure (4a).
FigureH4 - cL
5
-4
Ac ce pt e
-5
2
-2
4
-2
-20
-4
-40
-6
-60
-8
-10
-80
-12
-100
Bistable solitary wave solution of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (26) plotted in Figures (4b-4c).
10
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FigureH5 - aL -10
FigureH5 - bL 5
-5
10
5
-5 -0.05
ip t
-0.1
-0.10
-0.2
cr
-0.15
-0.3
-0.20
-0.4
us
-0.25
-0.5
-0.30
-0.35
FigureH5 - cL
2
-2
4
Ac ce pt e
-4
d
M
an
Bistable solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (27) plotted in Figures (5a-5b).
-2
-4
-6
-8
Velocity potential in the fluid solution (30) with coefficients (27) of the fifth-order Caudrey-DoddGibbon equation (17) plotted in Figure (5c).
11
Page 11 of 22
√ √ −6 1 + 2ρ2 + 4ρ cosh [ c2 (kx − ωt) + ξ0 ] + cosh [2 c2 (kx − ωt) + ξ0 ] (29)
ip t
√ √ √ +2ρ sinh [ c2 (kx − ωt) + ξ0 ] + sinh [2 c2 (kx − ωt) + ξ0 ]) c3 c2 c4 ) c4 p k2 3 2 −β − 2γ ± (β + 2γ) − 40α − 24c2 c23 u3 (x, t) = 2 16αc3 c4 √ √ ! ρ 1 + λ2 + cosh c2 (kx − ωt) + ξ0 √ 1+ λ + sinh c2 (kx − ωt) + ξ0
cr
(30)
us
+48c22 c4
√ √ !2 ρ 1 + λ2 + cosh c2 (kx − ωt) + ξ0 √ 1+ λ + sinh c2 (kx − ωt) + ξ0
Similarly, by substituting from equation (23) into equation (13), we obtained the solutions of the fifth-order Lax equation and their figures (1a-1c). From equation (24), we
an
derived the travelling wave solutions of the fifth-order Ito equation and these solutions plotted in figures (2a-2c). Figures (3a-3c) represented the solutions of the fifth-order
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Sawada-Kotera equation (15) by substituting from (25) in equation (15). From equation (26), we obtained the solutions of the fifth-order Kaup-Kupershmidt equation (16) in form travelling wave solutions and their figures (4a-4c). Finally, figures (5a-5c)
Ac ce pt e
d
represented the solutions of the fifth-order Caudrey-Dodd-Gibbon equation (17). 2.2
Families II:
Balancing the nonlinear term uu(3) and the highest order derivative u(5) gives n = 4. The traveling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 + a1 φ(ξ) + a2 φ2 (ξ) + a3 φ3 (ξ) + a4 φ4 (ξ),
(31)
where ξ = kx−ωt and k, ω, a0, a1 , a2 , a3 , a4 are arbitrary constants. Substituting from equations (31) into equation (20) and collecting coefficients, then setting coefficients equal to zero, we obtain a system of algebraic equations. By solving this system, the parameters k, ω, a0, a1 , a2 , a3 , a4 can be determined as p c2 k 2 2 −(β + 2γ) ± (β + 2γ) − 40α , a0 = α
a1 = a3 = 0 ,
12
Page 12 of 22
12k 2 a2 = − α
r
p 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ)2 − 40α ,
p 12c6 k 2 −β − 2γ ± (β + 2γ)2 − 40α , α p r p −β − 2γ ± (β + 2γ)2 − 40α 2 √ c2 c6 −20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α , c4 = 10α 2 p 2k 5 c22 −12α + β β + 2γ ± (β + 2γ)2 − 40α (32) ω= α
ip t
a4 =
a1 = a3 = 0,
a4 = −24k 2 γ,
us
√ a2 = ±24k 2 αγ,
a0 = −2k 2 α,
cr
Case I: The coefficients of the fifth-order Lax equation can be obtained as √ β = ± αγ,
ω = −56k 5 α2 ; (33)
2
a2 = −60k β ,
a1 = a3 = 0 ,
2
a4 = −120k γ,
β2 α= , 4γ
6k 5 β 4 ω=− 2 ; γ (34)
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5k 2 β 2 , a0 = − 2γ
an
Case II: The coefficients of the fifth-order Ito equation can be determined as
Case III: The coefficients of the fifth-order Sawada-Kotera equation can be derived as p
√ a2 = 8k 2 αγ,
d
5(4k 6 α2 + kω) , 15k
a1 = a3 = 0,
a4 = −8k 2 γ,
√ β = −2 αγ; (35)
Ac ce pt e
a0 =
−10αk 3 +
Case IV: The coefficients of the fifth-order Kaup-Kupershmidt equation can be
obtained as a0 =
p k 5 γ 2 (45k 5 β 4 + 64ωγ 2 ) , a2 = −3k 2 β, a1 = a3 = 0, 16γ 2 k 3 p 2 5 15γβ k ± k 5 γ 2 (45k 5 β 4 + 64ωγ 2) a4 = −6k 2 γ, α = ; (36) 32γ 2 k 5
−9γβ 2 k 5 ±
Case V: The coefficients of the fifth-order Caudrey-Dodd-Gibbon equation can
be derived as a0 =
−10αk 3 ±
p 5k(4k 5 α2 + ω) , 30k
√ a2 = 4k 2 αγ,
a1 = a3 = 0,
√ a4 = −4γk 2 , β = −2 αγ. (37)
13
Page 13 of 22
FigureH6 - aL
FigureH6 - bL -6 5
-5
-7
ip t
-100
-8 -9
-200
cr
-10
-300
us
-11
-400
-4
-12
2
-2
4
M
an
Bistable and kink solitary wave solution of the fifth-order Lax equation (13) with coefficients (33) plotted in Figures (6a-6b).
-4
2
-2
4
Ac ce pt e
-6
FigureH7 - aL
d
FigureH6 - cL
6 5
-5
-2
-5
-4
-10
-6
-8
-15
Periodic solitary wave solution of the fifth-order Lax equation (13) with coefficients (33) plotted in Figure (6c). Bright solitary wave solution of the fifth-order Ito equation (14) with coefficients (34) plotted in Figure (7a).
14
Page 14 of 22
FigureH7 - bL
FigureH7 - cL
-2 3
ip t
2
-4
1
cr
-6 -10
5
-5
10
us
-1
-8
-10
5
-5
10
-2
FigureH8 - aL -4
2
-2
4
FigureH8 - bL 2 6
Ac ce pt e
-6
d
M
an
Figures (7b-7c) represented periodic and kink solitary wave solution of the fifth-order Ito equation (14) with coefficients (34).
-50
1
-100
-3
-150
-2
1
-1
2
3
-1
-200
-250
-2
-300
-3
-350
Bistable and kink solitary wave solutions of the fifth-order Sawada-Kotera equation (15) with coefficients (37) plotted in Figures (8a-8b).
15
Page 15 of 22
We obtained the solutions of the generalized fifth-order KdV equation (12) by
p √ 2 (38) −(β + 2γ) ± (β + 2γ) − 40α (1 + tanh [ c2 (kx − ωt) + θ0 ])
us
+c2 c6 k
2
cr
ip t
substituting from equation (32) into (31) as following 12 p √ c2 u1 (x, t) = k 2 −(β + 2γ) ± (β + 2γ)2 − 40α + 2 (1 + tanh [ c2 (kx − ωt) + θ0 ]) α c4 r p 2 2 c4 k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α
p c2 k 2 −(β + 2γ) ± (β + 2γ)2 − 40α α p 48k 2 c6 c22 β + 2γ ± (β + 2γ)2 − 40α − p 2 √ α c24 − 4c2 c6 cosh 2 c2 (kx − ωt) + θ0 − c4 r p 2 24c2 k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ)2 − 40α p + √ αc4 − α c24 − 4c2 c6 cosh 2 c2 (kx − ωt) + θ0
(39)
d
M
an
u2 (x, t) =
p k2 −(β + 2γ) ± (β + 2γ)2 − 40α α p 48k 2 c6 c2 β + 2γ ± (β + 2γ)2 − 40α + 2 p √ 2 α c4 − c4 − 4c2 c6 sin [2 −c2 (kx − ωt) + θ0 ] r p 2 2 24k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α p − √ αc4 − α c24 − 4c2 c6 sin [2 −c2 (kx − ωt) + θ0 ]
Ac ce pt e
u3 (x, t) = c2
(40)
Figures (2a-2b), represent the both dark and bright solitary wave solutions (15) of
the Boussinesq equation (10), with k = 1 , ω = 2 , β = 4 and α = 0 ; k = −1 , ω = 2 and α = 1 in the interval [−10, 10] . Similarly, we derived the bistable, kink and periodic solitary wave solutions of the fifth-order Lax equation by substituting from equation (33) into equation (13); we 16
Page 16 of 22
FigureH8 - cL -4
FigureH9 - aL 2
-2
4
-4
2
-2
4
-50
-100 -100
-150
ip t
-200
-200
-300
cr
-250
-300
us
-400
-350
an
Figure (8c) represented periodic solitary wave solution of the fifth-order Sawada-Kotera equation (15) with coefficients (35). Figure (9a) represented bistable solitary wave solution of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (36).
represented these solutions in figures (6a-6c). From equation (34), we obtained the
M
bright, kink periodic and kink solitary wave solutions of the fifth-order Ito equation and these solutions plotted in figures (7a-6c). Figures (8a-8c) represented the bistable, kink and periodic solitary wave solutions of the fifth-order Sawada-Kotera equation
d
(15) by substituting from (35) in equation (15). From equation (36), we obtained
Ac ce pt e
the solutions of the fifth-order Kaup-Kupershmidt equation (16) in form bistable, kink and periodic solitary wave solutions and their figures (9a-9c). Finally, figures (10a-10c) represented the bistable, kink and periodic solitary wave solutions of the fifth-order Caudrey-Dodd-Gibbon equation (17).
17
Page 17 of 22
FigureH9 - bL
FigureH9 - cL
-19.40 0.5
-0.5
1.0
ip t
-1.0 -19.45
-50
cr
-19.50
-100
-4
2
-2
us
-19.55
4
-150
M
an
Kink and periodic solitary wave solutions of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (36) plotted in Figures (9b-9c).
FigureH10 - bL
d
FigureH10 - aL
5
-1.0
Ac ce pt e
-5 -100
-1.5
-200
-2.0
-300
-2.5
-400
-3.0
-3.5
-500
-4
-600
-2
2
4
Bistable and kink solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (37) plotted in Figures (10a-10b).
18
Page 18 of 22
FigureH10 - cL 2
-3
-2
1
-1
2
3
ip t
-2
-4
cr
-6
-8
us
-10
an
Periodic solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (37) plotted in Figure (10c).
References
M
[1] Whitham G. B., Linear and Nonlinear Waves, New York: Wiley (1972). [2] Choi, W. and Camassa, R., Weakly nonlinear internal waves in a two uid system,
d
J. Fluid Mech 313 (1996) 83-103.
Ac ce pt e
[3] Seadawy, A.R., Dianchen L.: Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov-Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results in Phys. 6, 590 (2016)
[4] Seadawy, A.R.: Stability analysis of traveling wave solutions for generalized coupled nonlinear KdV equations. Appl. Math. Inf. Sci. 10, 209 (2016)
[5] Seadawy, A.R.: New exact solutions for the KdV equation with higher order nonlinearity by using the variational method. Comp. and Math. Appl. 62, 3741 (2011)
[6] Joseph, R.I.: Solitary waves in finite depth fluid. J. Phys. A: Math. Gen. 10, L225 (1977)
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[7] Helal, M.A., Seadawy, A.R.: Benjamin-Feir instability in nonlinear dispersive waves. Computers and Mathematics with Appl. 64, 3557 (2012) [9] Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries-Zakharov-Kuznetsov equation in a magnetized electron-
ip t
positron plasma. Physica A: Statistical Mechanics and its Appl. 455, 44 (2016)
cr
[10] Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Soviet Physics Doklady 15, 539 (1970)
us
[11] Seadawy A. R.: Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. Computers and Mathe-
an
matics with Appl. 71, 201 (2016)
[12] Aly R. Seadawy, Travelling wave solutions of a weakly nonlinear two-dimensional
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higher order Kadomtsev-Petviashvili dynamical equation for dispersive shallow water waves, Eur. Phys. J. Plus (2017) 132: 29.
d
[13] Hasegawa, A.: An historical review of application of optical solitons for high
Ac ce pt e
speed communication. Chaos 10, 475 (2000). [14] Hasegawa A, Tappert DF. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers I: anomalous dispersion regime. Appl Phys Lett 1973;23(3):1424.
[15] Manirupa Saha, Amarendra K. Sarma, Solitary wave solutions and modulation instability analysis of the nonlinear Schrodinger equation with higher order dispersion and nonlinear terms, Commun Nonlinear Sci Numer Simulat 18 (2013) 24202425.
[16] Nijhoft, J. H. B. and Roelofs, G. H. M., Prolongation structures of a higher-order nonlinear SchrZidinger equation, J. Phys. A: Math. Gen. 25 (1992) 2403-2416.
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[17] Min Li, Tao Xu and Lei Wang, Dynamical behaviors and soliton solutions of a generalized higher-order nonlinear Schrodinger equation in optical fibers, Nonlinear Dyn (2015) 80:14511461. [18] Seadawy A. R., Approximation solutions of derivative nonlinear Schrodinger
ip t
equation with computational applications by variational method, the European
cr
Phys. J. Plus (2015) 130: 182.
[19] M.A. Helal and A.R. Seadawy, Exact soliton solutions of an D-dimensional non-
us
linear Schr¨odinger equation with damping and diffusive terms, Z. Angew. Math. Phys. 62 (2011), 839-847.
an
[20] A.H. Khater, M.A. Helal and A.R. Seadawy, General soliton solutions of ndimensional nonlinear Schr¨odinger equation, IL Nuovo Cimento 115B, (2000)
M
1303-1312.
[21] A.R. Seadawy, Exact Solutions of a two-dimensional nonlinear Schr¨odinger e-
d
quation, Applied Mathematics Letters 25 (2012) 687-691. [22] M.A. Helal and A.R. Seadawy, Variational method for the derivative nonlin-
Ac ce pt e
ear Schr¨odinger equation with computational applications, Physica Scripta, 80, (2009) 350-360.
[23] Ozer, M. N. and Doken, F. T., Derivation of integrable nonlinear evolution equations from the higher order NLS equation, J. Phys. A: Math. Gen. 36 (2003) 2319-2323.
[24] Changfu Liu and Zhengde Dai, Exact soliton solutions for the fifth-order SawadaKotera equation, Applied Mathematics and Computation 206 (2008) 272275.
[25] Kaya D. An explicit and numerical solutions of some fifth-order KdV equation by decomposition method. Applied mathematics and computation, 2003, 144: 353-363. 21
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[26] K. Sawada and T. Kotera, A Method for Finding N-Soliton Solutions of the K.d.V. Equation and K.d.V.-Like Equation. Prog. Theor. Phys. 51 (1974), 1355-1367. [27] P. J. Caudrey, R. K. Dodd, and J. D. Gibbon, A New Hierarchy of Korteweg-De
ip t
Vries Equations Proc. R. Soc. London, Ser. A 351 (1976), 407-422.
cr
[28] D. J. Kaup, On the Inverse Scattering Problem for Cubic Eigenvalue Problems of the Class ψxxx + 6Qψx + 6Rψ = λψ. Stud. Appl. Math. 62 (1980), 189-216.
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[29] B. A. Kupershmidt, A super Korteweg-de Vries equation: An integrable system.
Ac ce pt e
d
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an
Phys. Lett. A 102 (1984), 213-215.
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S0030-4026(17)30352-2 http://dx.doi.org/doi:10.1016/j.ijleo.2017.03.086 IJLEO 59009
To appear in: Received date: Accepted date:
19-12-2016 20-3-2017
Please cite this article as: A.R. Seadawy, The generalized nonlinear higher order of KdVequations from the higher ordernonlinear Schrddotodinger equation and its solutions, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.03.086 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
The generalized nonlinear higher order of KdV equations from the higher order nonlinear Schr¨ odinger equation and its solutions A. R. Seadawy
ip t
Mathematics Department, Faculty of science, Taibah University, Al-Ula, Saudi Arabia. Mathematics Department, Faculty of Science, Beni-Suef University, Egypt. E-mail: [email protected]
cr
Abstract The higher order nonlinear Schr¨odinger (NLS) equation describes ultrashort pluse propagation in optical fibres. The generalized nonlinear fifth-order of
us
KdV equations derived from the higher order NLS equation by using multiple scales methods. We obtained the traveling wave solutions for some different kinds of the
an
generalized nonlinear fifth-order of KdV (fifth-order Lax; fifth-order Ito; fifth-order Sawada-Kotera; fifth-order Kaup-Kupershmidt; fifth-order Caudrey-Dodd-Gibbon)
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equations by applying the auxiliary equation of the direct algebraic method. These solutions for the generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and
d
the movement role of the waves by making the graphs of the exact solutions are
Ac ce pt e
analyzed. All solutions are exact and stable. Keywords. Higher order NLS equation; Generalized higher order of KdV equation; Traveling wave solutions; Mathematical Physics methods. 2000 MR Subject Classification 35G20 · 35Q53 · 37K10 · 49S05 · 76A60
1
Page 1 of 22
1
Derivation of the generalized fifth order KdV equation from higher order nonlinear Schr¨ odinger equation
Nonlinear wave phenomena exist in many fields, such as fluid mechanics, plasma
ip t
physics, biology, hydrodynamics, solid state physics and optical fibers, etc. In order to better understand these nonlinear phenomena, it is important to seek their exact solutions. They can help to analyze the stability of these solutions and the movement
cr
role of the wave by making the graphs of the exact solutions [1-11].
Optical solitons have been the subjects of extensive research in nonlinear optics
us
due to their potential applications in telecommunication and ultra fast signal processing systems [12-14]. Optical solitons arise from the balance between group velocity
an
dispersion effect and nonlinear effect arising due to nonlinear change in the refractive index [13]. A higher-order Schrodinger equation containing parameters, which
M
is used to describe pulse propagation in optical fibres, is shown to admit an infinitedimensional prolongation structure for exactly four combinations of the parameters, besides the classical NLS equation [15-23].
d
Consider one of the integrable cases of higher order nonlinear Schr¨odinger (NLS)
Ac ce pt e
equation which is known as the Kaup-Newell equation [23] 1 iqτ = i(a3 qq ∗ qζ + a3 q 2 q ∗ ) − (a1 qζζ + a2 q 2 q ∗ ) 2
By separating the phase and amplitude in the form as p p q(ζ, τ ) = eiθ(ζ,τ ) N(ζ, τ ), q ∗ (ζ, τ ) = e−iθ(ζ,τ ) N(ζ, τ ).
(1)
(2)
By substituting from equation (2) into Kaup-Newell equation (1) and grouping the real and imaginary parts, the following real differential equations can be obtained as Nτ =
Vτ =
1 6a3 Nζ N 3 − 8a1 Nζ N 2 V − 8a1 Vζ N 3 , 2 4N
1 (2a1 Nζζζ N 2 − 4a1 Nζζ Nζ N + 8a1 Nζ3 + 2a3 Nζ N 3 V + 4a2 Nζ N 3 − 4N 3 8a1 Vζ V N 3 + 2a3 Vζ N 4 ),
(3)
2
Page 2 of 22
where θ(ζ, τ ) = V (ζ, τ ). The following series expansions for solutions as N =1+
∞ X
2n
ǫ Nn (x, t1 , t2,...,tn ),
V =
∞ X
ǫ2n Vn (x, t1 , t2,...,tn ),
(4)
n=1
n=1
where ζ and τ are slow variables with respect to scaling parameters ǫ > 0 respectively
ip t
as follows t1 = ǫ3 τ,
x = ǫ(ζ + 2τ ),
t2 = ǫ5 τ, ....tn = ǫ2n+1 τ.
(5)
cr
By substituting from series expansions (4) and (5) into a system of real differential The coefficients ǫ3 as 1 (3a3 − 4)N1 , 4a1
a2 =
1 [2a2 N1xxx + (18a3 − 3a23 − 24)N1 N1x ]. 4(a3 − 2) 1 a3 = 5, then
N1t1 =
M
Choosing a1 = 1,
1 (−3a23 + 16a3 − 16)N1 , 8a1
an
V1 =
us
equations (3) and collect the coefficients of the powers of ǫ equal to zero separately.
Ac ce pt e
d
9 1 N1t1 = (N1xxx − N1 N1x ). 6 2 Making the following transformation t1 →
1 t1 , 6
(6)
4 N1 → − u 3
Inserting (6) into series expansion at the coefficient ǫ7 , we obtain
N1t2 =
1 N2t +24N1xxxxx +56N1 N1xxx +528N1x N1xx +11682N2N1x +1038N12N1x − 4752 1 5544V2 N1x − 216V2xxx − 1296V2x N1 + 1188V2t1 .
(7)
For equation (7), take N2 , V2 as following N2 = K11 N1xx + K11 N12
V2 = K21 N1xx + K22 N12 ,
By choosing the coefficients K11 = −
7 6
K12 =
95 48
K21 = −
79 24
K22 =
949 , 192
3
Page 3 of 22
we obtain N1t2 =
1 N1xxxxx + 10N1 N1xxx + 20N1x N1xx + 30N12 N1x . 216
(8)
Now by using the following transformation 1 t2 , , 216
4 N1 → − u 3
ip t
t2 →
cr
The fifth-order lax’s KdV equation can be derived as
us
ut2 = uxxxxx + 10uuxxx + 20ux uxx + 30u2 ux . By choosing 337 216
K12 =
755 288
K21 = −
3779 864
an
K11 = −
K22 =
(9)
7549 , 1152
The fifth-order Sawada-Kotera’s equation can be obtained as ut2 = uxxxxx + 5uuxxx + 5ux uxx + 5u2 ux . By supposing 173 144
K12 =
Ac ce pt e
K11 = −
d
M
(10)
145 72
K21 = −
1951 576
K22 =
1451 , 288
The fifth-order Kaup-Kupershmidt’s KdV equation can be got as ut2 = uxxxxx + 10uuxxx + 25ux uxx + 20u2 ux .
(11)
Consider the generalized fifth-order KdV equation as [24-29] ut + αu2ux + βux uxx + γuuxxx + λuxxxxx = 0 .
(12)
Where α, β, γ, are real parameters and λ = ±1. The generalized fifth-order KdV
equation (12) describes motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice [1] and it is an important mathematical model with wide applications in quantum mechanics, nonlinear optics, solid state physics, 4
Page 4 of 22
plasma physics, fluid physics and quantum field theory. Some important particular cases of equation (12) are Case I: When α = 30, β = 20, γ = 10, λ = 1, equation (12) called the fifth-order Lax equation as
ip t
ut + 30u2ux + 20ux uxx + 10uuxxx + uxxxxx = 0 .
(13)
cr
Case II: When α = 2, β = 6, γ = 3, λ = 1, equation (12) named the fifth-order Ito equation as
(14)
us
ut + 2u2ux + 6ux uxx + 3uuxxx + uxxxxx = 0 .
Case III: When α = 45, β = 15, γ = 15, λ = 1, equation (12) called the
an
fifth-order Sawada-Kotera equation as
ut + 45u2ux + 15ux uxx + 15uuxxx + uxxxxx = 0 .
(15)
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Case IV: When α = 20, β = 25, γ = 10, λ = 1, equation (12) named the fifth-order Kaup-Kupershmidt equation as
(16)
d
ut + 20u2ux + 25ux uxx + 10uuxxx + uxxxxx = 0 .
Ac ce pt e
Case V: When α = 180, β = 30, γ = 30, λ = 1, equation (12) called the
fifth-order Caudrey-DoddGibbon equation as ut + 180u2ux + 30ux uxx + 30uuxxx + uxxxxx = 0 .
2
(17)
Travelling wave solutions of the generalized fifth order KdV equation
Suppose that the travelling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 +
n X i ai φ (η) + bi φ−i (η) ,
(18)
i=1
and ′
φ =
p c2 φ 2 + c3 φ 3 + c4 φ 4 ,
′
φ = 5
p
c2 φ 2 + c4 φ 4 + c6 φ 6
(19)
Page 5 of 22
where ξ = kx − ωt and k, ω, a1 , ......... an , b1 , ......... bn are arbitrary constants. Consider the traveling wave solutions (18) and (19), then equation (12) becomes −ωu′ + αku2 u′ + βk 3 u′ u′′ + γk 3 uu(3) + λk 5 u(5) = 0 . Families I:
ip t
2.1
(20)
Balancing the nonlinear term uu(3) and the highest order derivative u(5) gives n = 2.
cr
Then the traveling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 + a1 φ(ξ) + a2 φ2 (ξ) + b1 φ−1 (ξ) + b2 φ−2 (ξ),
us
(21)
where ξ = kx − ωt and k, ω, a0, a1 , a2 , b1 , b2 are arbitrary constants. Substituting
an
from equations (21) into equation (20) and collecting coefficients, then setting coefficients equal to zero, we obtain a system of algebraic equations. By solving this
q 2 2 4 2 −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) ,
c23 , 4c4
q 2 2 4 2 −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) ,
Ac ce pt e
3 a1 = 2α
c2 =
d
c3 a0 = 16αc4
M
system, the parameters k, ω, a0 , a1 , a2 , b1 , b2 can be determined as
q 3c4 2 2 4 2 a2 = −k c3 (β + 2γ) ± c3 k ((β + 2γ) − 40α) , αc3 q k 3 c33 2 2 4 2 k c3 (12α − β(β + 2γ)) ± β c3 k ((β + 2γ) − 40α) ; ω= 128αc24
(22)
Case I: The coefficients of the fifth-order Lax equation can be obtained as
k2 β 2 a0 = − , 8γ
2
a1 = −3k β ,
2
a2 = −6k γ ,
b1 = b2 = 0,
β2 α= , 4γ
7k 5 β 4 ω= ; 32γ 2 (23)
Case II: The coefficients of the fifth-order Ito equation can be determined as √ √ √ (−9 ± 2 19)k 2 β 2 , a1 = 3(−9 ± 2 19)k 2 β , a2 = 6(−9 ± 2 19)k 2 γ , a0 = 8γ 6
Page 6 of 22
FigureH1 - aL
FigureH1 - bL 0.1
-10
-0.2
-5
ip t
5
-5
5
10
-0.1
cr
-0.4
-0.2
us
-0.6
-0.8
-0.3
M
an
Bistable bright solitary wave solutions of the fifth-order Lax equation (13) with various different shapes plotted in Figures (1a-1b).
2
-2
4
-10
Ac ce pt e
-4
FigureH2 - aL
d
FigureH1 - cL
5
-5
10
-10
-0.5
-20
-1.0
-30
-1.5
-40
-2.0
-50
Bistable dark solitary wave solution of the fifth-order Lax equation (13) with various different shape plotted in Figure (1c). Velocity potential in the fluid solution (28) with coefficients (24) of the fifth-order Ito equation plotted in Figure (2a).
7
Page 7 of 22
FigureH2 - bL
FigureH2 - cL 5
-5
5
-5
ip t
-2
-4
-0.1
cr
-6
-0.2
-8
us
-10
-0.3
-12
-14
M
an
Velocity potentials in the fluid solutions (29-30) with coefficients (24) of the fifth-order Ito equation plotted in Figures (2b-2c).
Ac ce pt e
0.16
-10
-5
FigureH3 - bL
d
FigureH3 - aL
0.16
0.14
0.14
0.12
0.12
0.10
0.10
5
10
-10
-5
5
10
Periodic solitary wave solutions of the fifth-order Sawada-Kotera equation (15) with coefficients (25) plotted in Figures (3a-3b).
8
Page 8 of 22
√ 3(4 ± 19)k 5 β 4 β2 , ω= ; (24) b1 = b2 = 0, α = 4γ 16γ 2 Case III: The coefficients of the fifth-order Sawada-Kotera equation can be derived as p
5(k 6 α2 + 4kω) k2 β 2 β2 2 , a1 = −k β , a2 = − , b1 = b2 = 0, γ = 30k 2α 4α (25)
ip t
a0 =
−5αk 3 ±
cr
Case IV: The coefficients of the fifth-order Kaup-Kupershmidt equation can be obtained as
us
v u p u 4 3 2 5 2 t 3 5αγk ± 5k γ (k α + 4ω) −5αγk 4 ± 5k 3 γ 2 (k 5 α2 + 4ω) a0 = , a = ± , 1 20γk 2 10 v u u 2 5αγk 4 ± p5k 3 γ 2 (k 5 α2 + 4ω) t 2 3 b1 = b2 = 0, β = ± 2 ; (26) a2 = − γk 2 , 2 k 15 Case V: The coefficients of the fifth-order Caudrey-Dodd-Gibbon equation can be derived as p
√ a1 = ±k 2 αγ,
d
5k(k 5 α2 + 4ω) , 60k
a2 = ±γk 2 ,
b1 = b2 = 0,
√ β = ±2 αγ. (27)
Ac ce pt e
a0 =
−5αk 3 ±
M
an
p
By substituting from equation (22) into (21), the following solution of Generalized
nonlinear fifth-order KdV equation (12) can be obtained as p k2 2 − 40α −β − 2γ ± (β + 2γ) c43 − 24c2 c4 c23 u1 (x, t) = 16αc4c23 √ √ 2 ! c2 c2 1 + tanh (kx − ωt) + ξ0 + 48c22 c24 1 + tanh (kx − ωt) + ξ0 2 2 (28) p 2 2 k −β − 2γ ± (β + 2γ) − 40α √ 2 2 u2 (x, t) = √ 2 (ρ + cosh [ c2 (kx − ωt) + ξ0 ]) c3 16αc4 ρ + cosh c2 (kx − ωt) + ξ0 √ √ +12 ρ2 + 2ρ cosh [ c2 (kx − ωt) + ξ0 ] + cosh [2 c2 (kx − ωt) + ξ0 ] √ √ +2ρ sinh [ c2 (kx − ωt) + ξ0 ] + sinh [2 c2 (kx − ωt) + ξ0 ]) c2 c4 9
Page 9 of 22
FigureH3 - cL
FigureH4 - aL 5
-5
-10
5
-5
10
-0.5
ip t
-0.2
-1.0
-0.4
cr
-1.5
-0.6
-2.0
us
-2.5
-3.0
-0.8
-1.0
FigureH4 - bL
d
M
an
Velocity potential in the fluid solution (30) with coefficients (25) of the fifth-order Sawada-Kotera equation (15) plotted in Figure (3c). Bistable solitary wave solution of the fifth-order KaupKupershmidt equation (16) with coefficients (26) plotted in Figure (4a).
FigureH4 - cL
5
-4
Ac ce pt e
-5
2
-2
4
-2
-20
-4
-40
-6
-60
-8
-10
-80
-12
-100
Bistable solitary wave solution of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (26) plotted in Figures (4b-4c).
10
Page 10 of 22
FigureH5 - aL -10
FigureH5 - bL 5
-5
10
5
-5 -0.05
ip t
-0.1
-0.10
-0.2
cr
-0.15
-0.3
-0.20
-0.4
us
-0.25
-0.5
-0.30
-0.35
FigureH5 - cL
2
-2
4
Ac ce pt e
-4
d
M
an
Bistable solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (27) plotted in Figures (5a-5b).
-2
-4
-6
-8
Velocity potential in the fluid solution (30) with coefficients (27) of the fifth-order Caudrey-DoddGibbon equation (17) plotted in Figure (5c).
11
Page 11 of 22
√ √ −6 1 + 2ρ2 + 4ρ cosh [ c2 (kx − ωt) + ξ0 ] + cosh [2 c2 (kx − ωt) + ξ0 ] (29)
ip t
√ √ √ +2ρ sinh [ c2 (kx − ωt) + ξ0 ] + sinh [2 c2 (kx − ωt) + ξ0 ]) c3 c2 c4 ) c4 p k2 3 2 −β − 2γ ± (β + 2γ) − 40α − 24c2 c23 u3 (x, t) = 2 16αc3 c4 √ √ ! ρ 1 + λ2 + cosh c2 (kx − ωt) + ξ0 √ 1+ λ + sinh c2 (kx − ωt) + ξ0
cr
(30)
us
+48c22 c4
√ √ !2 ρ 1 + λ2 + cosh c2 (kx − ωt) + ξ0 √ 1+ λ + sinh c2 (kx − ωt) + ξ0
Similarly, by substituting from equation (23) into equation (13), we obtained the solutions of the fifth-order Lax equation and their figures (1a-1c). From equation (24), we
an
derived the travelling wave solutions of the fifth-order Ito equation and these solutions plotted in figures (2a-2c). Figures (3a-3c) represented the solutions of the fifth-order
M
Sawada-Kotera equation (15) by substituting from (25) in equation (15). From equation (26), we obtained the solutions of the fifth-order Kaup-Kupershmidt equation (16) in form travelling wave solutions and their figures (4a-4c). Finally, figures (5a-5c)
Ac ce pt e
d
represented the solutions of the fifth-order Caudrey-Dodd-Gibbon equation (17). 2.2
Families II:
Balancing the nonlinear term uu(3) and the highest order derivative u(5) gives n = 4. The traveling wave solutions of equation (12) has the following form u(x, t) = u(ξ) = a0 + a1 φ(ξ) + a2 φ2 (ξ) + a3 φ3 (ξ) + a4 φ4 (ξ),
(31)
where ξ = kx−ωt and k, ω, a0, a1 , a2 , a3 , a4 are arbitrary constants. Substituting from equations (31) into equation (20) and collecting coefficients, then setting coefficients equal to zero, we obtain a system of algebraic equations. By solving this system, the parameters k, ω, a0, a1 , a2 , a3 , a4 can be determined as p c2 k 2 2 −(β + 2γ) ± (β + 2γ) − 40α , a0 = α
a1 = a3 = 0 ,
12
Page 12 of 22
12k 2 a2 = − α
r
p 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ)2 − 40α ,
p 12c6 k 2 −β − 2γ ± (β + 2γ)2 − 40α , α p r p −β − 2γ ± (β + 2γ)2 − 40α 2 √ c2 c6 −20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α , c4 = 10α 2 p 2k 5 c22 −12α + β β + 2γ ± (β + 2γ)2 − 40α (32) ω= α
ip t
a4 =
a1 = a3 = 0,
a4 = −24k 2 γ,
us
√ a2 = ±24k 2 αγ,
a0 = −2k 2 α,
cr
Case I: The coefficients of the fifth-order Lax equation can be obtained as √ β = ± αγ,
ω = −56k 5 α2 ; (33)
2
a2 = −60k β ,
a1 = a3 = 0 ,
2
a4 = −120k γ,
β2 α= , 4γ
6k 5 β 4 ω=− 2 ; γ (34)
M
5k 2 β 2 , a0 = − 2γ
an
Case II: The coefficients of the fifth-order Ito equation can be determined as
Case III: The coefficients of the fifth-order Sawada-Kotera equation can be derived as p
√ a2 = 8k 2 αγ,
d
5(4k 6 α2 + kω) , 15k
a1 = a3 = 0,
a4 = −8k 2 γ,
√ β = −2 αγ; (35)
Ac ce pt e
a0 =
−10αk 3 +
Case IV: The coefficients of the fifth-order Kaup-Kupershmidt equation can be
obtained as a0 =
p k 5 γ 2 (45k 5 β 4 + 64ωγ 2 ) , a2 = −3k 2 β, a1 = a3 = 0, 16γ 2 k 3 p 2 5 15γβ k ± k 5 γ 2 (45k 5 β 4 + 64ωγ 2) a4 = −6k 2 γ, α = ; (36) 32γ 2 k 5
−9γβ 2 k 5 ±
Case V: The coefficients of the fifth-order Caudrey-Dodd-Gibbon equation can
be derived as a0 =
−10αk 3 ±
p 5k(4k 5 α2 + ω) , 30k
√ a2 = 4k 2 αγ,
a1 = a3 = 0,
√ a4 = −4γk 2 , β = −2 αγ. (37)
13
Page 13 of 22
FigureH6 - aL
FigureH6 - bL -6 5
-5
-7
ip t
-100
-8 -9
-200
cr
-10
-300
us
-11
-400
-4
-12
2
-2
4
M
an
Bistable and kink solitary wave solution of the fifth-order Lax equation (13) with coefficients (33) plotted in Figures (6a-6b).
-4
2
-2
4
Ac ce pt e
-6
FigureH7 - aL
d
FigureH6 - cL
6 5
-5
-2
-5
-4
-10
-6
-8
-15
Periodic solitary wave solution of the fifth-order Lax equation (13) with coefficients (33) plotted in Figure (6c). Bright solitary wave solution of the fifth-order Ito equation (14) with coefficients (34) plotted in Figure (7a).
14
Page 14 of 22
FigureH7 - bL
FigureH7 - cL
-2 3
ip t
2
-4
1
cr
-6 -10
5
-5
10
us
-1
-8
-10
5
-5
10
-2
FigureH8 - aL -4
2
-2
4
FigureH8 - bL 2 6
Ac ce pt e
-6
d
M
an
Figures (7b-7c) represented periodic and kink solitary wave solution of the fifth-order Ito equation (14) with coefficients (34).
-50
1
-100
-3
-150
-2
1
-1
2
3
-1
-200
-250
-2
-300
-3
-350
Bistable and kink solitary wave solutions of the fifth-order Sawada-Kotera equation (15) with coefficients (37) plotted in Figures (8a-8b).
15
Page 15 of 22
We obtained the solutions of the generalized fifth-order KdV equation (12) by
p √ 2 (38) −(β + 2γ) ± (β + 2γ) − 40α (1 + tanh [ c2 (kx − ωt) + θ0 ])
us
+c2 c6 k
2
cr
ip t
substituting from equation (32) into (31) as following 12 p √ c2 u1 (x, t) = k 2 −(β + 2γ) ± (β + 2γ)2 − 40α + 2 (1 + tanh [ c2 (kx − ωt) + θ0 ]) α c4 r p 2 2 c4 k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α
p c2 k 2 −(β + 2γ) ± (β + 2γ)2 − 40α α p 48k 2 c6 c22 β + 2γ ± (β + 2γ)2 − 40α − p 2 √ α c24 − 4c2 c6 cosh 2 c2 (kx − ωt) + θ0 − c4 r p 2 24c2 k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ)2 − 40α p + √ αc4 − α c24 − 4c2 c6 cosh 2 c2 (kx − ωt) + θ0
(39)
d
M
an
u2 (x, t) =
p k2 −(β + 2γ) ± (β + 2γ)2 − 40α α p 48k 2 c6 c2 β + 2γ ± (β + 2γ)2 − 40α + 2 p √ 2 α c4 − c4 − 4c2 c6 sin [2 −c2 (kx − ωt) + θ0 ] r p 2 2 24k 2c2 c6 20α + (β + 2γ) β + 2γ ± (β + 2γ) − 40α p − √ αc4 − α c24 − 4c2 c6 sin [2 −c2 (kx − ωt) + θ0 ]
Ac ce pt e
u3 (x, t) = c2
(40)
Figures (2a-2b), represent the both dark and bright solitary wave solutions (15) of
the Boussinesq equation (10), with k = 1 , ω = 2 , β = 4 and α = 0 ; k = −1 , ω = 2 and α = 1 in the interval [−10, 10] . Similarly, we derived the bistable, kink and periodic solitary wave solutions of the fifth-order Lax equation by substituting from equation (33) into equation (13); we 16
Page 16 of 22
FigureH8 - cL -4
FigureH9 - aL 2
-2
4
-4
2
-2
4
-50
-100 -100
-150
ip t
-200
-200
-300
cr
-250
-300
us
-400
-350
an
Figure (8c) represented periodic solitary wave solution of the fifth-order Sawada-Kotera equation (15) with coefficients (35). Figure (9a) represented bistable solitary wave solution of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (36).
represented these solutions in figures (6a-6c). From equation (34), we obtained the
M
bright, kink periodic and kink solitary wave solutions of the fifth-order Ito equation and these solutions plotted in figures (7a-6c). Figures (8a-8c) represented the bistable, kink and periodic solitary wave solutions of the fifth-order Sawada-Kotera equation
d
(15) by substituting from (35) in equation (15). From equation (36), we obtained
Ac ce pt e
the solutions of the fifth-order Kaup-Kupershmidt equation (16) in form bistable, kink and periodic solitary wave solutions and their figures (9a-9c). Finally, figures (10a-10c) represented the bistable, kink and periodic solitary wave solutions of the fifth-order Caudrey-Dodd-Gibbon equation (17).
17
Page 17 of 22
FigureH9 - bL
FigureH9 - cL
-19.40 0.5
-0.5
1.0
ip t
-1.0 -19.45
-50
cr
-19.50
-100
-4
2
-2
us
-19.55
4
-150
M
an
Kink and periodic solitary wave solutions of the fifth-order Kaup-Kupershmidt equation (16) with coefficients (36) plotted in Figures (9b-9c).
FigureH10 - bL
d
FigureH10 - aL
5
-1.0
Ac ce pt e
-5 -100
-1.5
-200
-2.0
-300
-2.5
-400
-3.0
-3.5
-500
-4
-600
-2
2
4
Bistable and kink solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (37) plotted in Figures (10a-10b).
18
Page 18 of 22
FigureH10 - cL 2
-3
-2
1
-1
2
3
ip t
-2
-4
cr
-6
-8
us
-10
an
Periodic solitary wave solution of the fifth-order Caudrey-Dodd-Gibbon equation (17) with coefficients (37) plotted in Figure (10c).
References
M
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cr
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