Schr dinger Equation
Schr dinger Equation
In partial fulfillment
PhD (Mathematics)
Schr dinger Equation
A Post Graduate Thesis submitted to the Department of Mathematics as partial
fulfillment of the requirement for the award of Degree of PhD in Mathematics.
Name
Supervisor Dr. Kashif Ali Associate Professor Department of Mathematics COMSATS University Islamabad, Lahore Campus. Co-Supervisor Dr. Syed Tahir Raza Rizvi Assistant Professor Department of Mathematics COMSATS University Islamabad, Lahore Campus.
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And
Children
ACKNOWLEDGEMENT
In the name of Allah, the Most Gracious and the Most Merciful
Alhamdulillah, all praises to Allah for the strengths and His blessing in completing this
thesis. I feel much pleasure in expressing my heartiest gratitude to my ever affectionate
supervisor Dr. Kashif Ali and co-supervisor Dr. Syed Tahir Raza Rizvi for their dynamic
and friendly supervision and with their inspiring attitudes made it very easy to undertake
this project. Secondly, I express my heartiest and sincerest gratitude to my institution and a
special thanks to the Head of Department of Mathematics for providing us with very good
research facilities, required for the accomplishment of our goals. My deepest appreciations
are extended to my beloved Parents (late), sisters and children, for my success is really the
fruit of their devoted prayers.
Badar Nawaz
Equation
A soliton which is also known as the traveling wave solution has a unique property,” the
collision of two solitons produce the waves whose permanentstructure remain the same”.
These wave solutions are very stable and having a key role in the mathematical physics,
especially in fiber technology. ”A soliton does not change its amplitude, shape and speed
for a long distance in a non-local and nonlinear optical media”. The applications of soliton
solutions are also in various branches of physics. The soliton solutions can be calculated
by using the different types of nonlinearities. The opticalsolitons are very important in the
study of nonlinear optical fibers.
In this thesis firstly, we determine different types of soliton solutions of dimensionless
form of Quintic Complex Ginzburg-Landau (CGLQ) equation byusing modified extended
tanh− function method (METFM) and the extended trial equation method (ETEM). Sec-
ondly, we obtain bright, singular and Jacobi elliptic soliton solutions for the time fractional
perturbed NLSE (TFPNLSE) by using ETEM with Kerr, power and log law nonlineari-
ties. Thirdly, we find the combo and dipole soliton solutionsfor CGLQ model with differ-
ent ansatz methods. Next, we construct different soliton solutions for the paraxial NLSE
(PNLSE) in Kerr media by ETEM. Then, we get the bright and darksolitons for non-
Kerr law NLSE with third order (3OD) and fourth order (4OD) dispersions by Sine-cosine
method (SCM)and Bernoullis equation method (BEM) with nonlinearities. In the last, by
using Hirota bilinear method (HBM), we obtain the multiple solitons for the nonlinear
Telegraph equation (NLTE) and the nonlinear PHI-four equation (NLPFE).
x
1.1.1 Spatial Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Temporal Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Bright Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Dark Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Singular Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Classification of Solitons on the Base of Shape. . . . . . . . . . . . . 6
1.5.1 Bell Solitons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.2 Kink Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.3 Breath Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Literature Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Refractive Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Phase Velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Self Phase Modulation (SPM). . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Group Velocity Dispersion. . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Nonlinear Schrodinger Equation. . . . . . . . . . . . . . . . . . . . . . 11
2.5.1 Quintic Complex Ginzburg-Landau Equation. . . . . . . . . . 12 2.5.2 Time Fractional Perturbed Nonlinear Shrodinger Equation. . . 12 2.5.3 The Paraxial Nonlinear Shrodinger Equation . . . . . . . . . . 12 2.5.4 The Nonlinear Shrodinger Equation with Third and Fourth Order Dispersions 2.5.5 Nonlinear Telegraph Equation. . . . . . . . . . . . . . . . . . . 13 2.5.6 Nonlinear PHI-Four Equation. . . . . . . . . . . . . . . . . . . 13
2.6 Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6.1 Kerr Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 13 2.6.2 Power Law Nonlinearity. . . . . . . . . . . . . . . . . . . . . . 14 2.6.3 Dual-Power Law Nonlinearity. . . . . . . . . . . . . . . . . . . 14 2.6.4 Parabolic Law Nonlinearity. . . . . . . . . . . . . . . . . . . . 14 2.6.5 Higher Order Polynomial Law Nonlinearity. . . . . . . . . . . 15 2.6.6 Triple-Power Law Nonlinearity. . . . . . . . . . . . . . . . . . 15 2.6.7 Anti-Cubic Law Nonlinearity . . . . . . . . . . . . . . . . . . . 15 2.6.8 Saturation Law Nonlinearity. . . . . . . . . . . . . . . . . . . . 15 2.6.9 Log Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Jacobi Elliptic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Extended Trial Equation Method. . . . . . . . . . . . . . . . . . . . . . 16 2.9 Modified extended tanh-function method. . . . . . . . . . . . . . . . . 17
xi
2.10 Sine-Cosine Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.11 Bernoulli’s Equation Approach. . . . . . . . . . . . . . . . . . . . . . 20 2.12 Hirota’s Bilinear Method. . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Soliton Solutions for Quintic Complex Ginzburg-Landau Model . . . . 23 3.1 CGLQ Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Extended Trial Equation Method. . . . . . . . . . . . . . . . . . . . . . 24 3.3 Modified Extended Tanh-Function Method. . . . . . . . . . . . . . . . 29
3.3.1 Case. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.3.2 Case. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.3 Case. 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.4 Case. 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Optical Soliton for Perturbed Nonlinear Fractional Schrodinger Equation 35 4.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Kerr Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 Power Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 Log Law Nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5 Solitary Wave Solutions for Quintic Complex Ginzburg-Landau Model 54 5.1 Extraction of Combo Solitons. . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Dark-in-the-Bright Soliton. . . . . . . . . . . . . . . . . . . . . . . . . 59
6 Optical Solitons for Paraxial Wave Equation in Kerr Media . . . . . . . 62 6.1 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Extended Trial Equation Method. . . . . . . . . . . . . . . . . . . . . 64
6.2.1 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7 Optical Solitons for Non-Kerr Law Nonlinear Schrodinger Equation with Higher Order 7.1 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2 Kerr Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 7.3 Power Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7.4 Anti-Cubic Law Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 77 7.5 Parabolic Law Nonlinearity. . . . . . . . . . . . . . . . . . . . . . . . . 79 7.6 Cubic Quintic Law Nonlinearity. . . . . . . . . . . . . . . . . . . . . . 82
8 Multiple Complex Soliton for Nonlinear Telegraph Equation . . . . . . 84 8.1 Nonlinear Telegraph Equation. . . . . . . . . . . . . . . . . . . . . . . 85 8.2 One Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 8.3 Two Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.4 Three Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.5 N Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9 Multiple Complex Soliton for PHI-Four Equation . . . . . . . . . . . . . 91 9.1 Nonlinear PHI-Four Equation. . . . . . . . . . . . . . . . . . . . . . . 92 9.2 One Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
xii
9.3 Two Soliton Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.4 Three Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.5 N Soliton Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
10 Conclusions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . 98 10.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 10.2 Open Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100
LIST OF FIGURES
Fig 3.1 δ = 5.0,ε = 0.05,α1 = 0.09,a= 12.5,β = 0.02,ψ0 = 10.0. . . . . . . . . 32 Fig 3.2 δ = 1.05,ε = 0.05,α1 = 10.4,a= 1.5,β = 2.03,ψ0 = 1.0. . . . . . . . . . 32 Fig 3.3 δ = 1.0,ε = 5.0, α1 = 1.09,a= 2.5,β = 0.09,ψ0 = 1.0. . . . . . . . . . . 33
Fig 4.1 R= 0.09, T = 0.9, Y = 2.0,K = 5.05,α = 9.0,β = 1.5,n= 1. . . . . . . . 46 Fig 4.2 R= 1.05, T = 1.05,Y = 2.05,K = 2.05,α = 5.0,β = 5.5,n= 1. . . . . . . 46 Fig 4.3 U = 1.05,V1 = 0.5, R= 2.0, P= 2.0,K = 1.05,α = 5.0,β = 5.5,n= 1. . . 47 Fig 4.4 U = 5.05,V1 = 0.5, R= 2.0, P= 2.0,K = 1.05,α = 5.0,β = 5.5,n= 1. . . 47
Fig 5.1 λ = 5.05,ε = 2.5,ω = 3.5D = 5.0,β = 0.05. . . . . . . . . . . . . . . . . 57 Fig 5.2 λ = 5.05,A= 2.5,η = 3.5δ = 5.0,µ = 0.05. . . . . . . . . . . . . . . . . 59 Fig 5.3 λ = 5.05,κ = 2.5,ω = 3.5,ν = 2.5. . . . . . . . . . . . . . . . . . . . . . 61
Fig 6.1 δ1 = 1.5,δ2 = 2.5, α1 = 2.0,γo = 1.5,β4 = 0.5,β = 0.5. . . . . . . . . . . 69 Fig 6.2 δ1 = 2.5,δ2 = 1.05,α1 = 2.0,γo = 1.5,β4 =−5.5,β = 0.5. . . . . . . . . 69 Fig 6.3 δ1 = 1.5,δ2 = 2.5,α1 = 2.0,γo = 1.5,β4 = 0.5,β = 0.5. . . . . . . . . . . 70 Fig 6.4 δ1 = 1.5,δ2 = 2.5,α1 = 2.0,γo = 1.5,β4 = 0.5,β = 0.5. . . . . . . . . . . 71
Fig 7.1 ω = 2,λ = 2,a= 2,b= 1.3,κ = 1.1,θ = 1. . . . . . . . . . . . . . . . . 75 Fig 7.2 ω = 2,λ = 2,a= 2,b=−1.3,κ = 0.3,η = 3,θ = 1,δ = 1. . . . . . . . . 79 Fig 7.3 ω = 0.2,A1 = 1,a= 2,b= 1.3,κ = 0.001,η = 0.01,Ao = 99,θ = 1. . . . 81 Fig 7.4 ω = 2,A= 2,a= 2,b= 1.3,κ = 0.01,η = 2,θ = 1,δ = 1. . . . . . . . . . 83
Fig 8.1 p= 0.5+ i,β = 1,α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Fig 8.2 p= 0.5+ i, r = 0.15+ i,β = 1,α = 1. . . . . . . . . . . . . . . . . . . . . 88 Fig 8.3 p= 0.5+ i, r = 0.15+ i,q=−0.2+ i,β = 1,α = 1. . . . . . . . . . . . . 89
Fig 9.1 λ=1.0,m=1.0,p= 0.05+ i . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Fig 9.2 λ=1.0,m=1.0,p= 0.05+ i, r = 0.05+ i . . . . . . . . . . . . . . . . . . . 95 Fig 9.3 λ=1.0,m=1.0,p= 0.05+ i, q= 0.05+ i, r = 0.05+ i . . . . . . . . . . . . 96
xiv
Differential equation (DE) is the equation involving derivatives or differential coefficients.
The natural phenomenas are modelled by DEs and they are widely use to form mathemati-
cal models in engineering, business, chemistry, physics, cosmology, ecology and biological
sciences. The DEs are mainly of two types: ordinary differential equations (ODE) with one
dependent and one independent variable while the partial differential equations (PDE) with
one dependent and more than one independent variables. The linear DEs contain the de-
pendent variable and/or its derivatives linear in each term, otherwise it is nonlinear DE. The
general form of linear ODE of ordern is
a0(x) +a1(x) d dx
+a2(x) d2 dx2 + · · ·+an(x)
dn dxn = F(x) (1.0.1)
Where, = f (x)
The general form of linear PDE of two variables of order 2 is
b0(x,y) ∂ 2 ∂x2 +b1(x,y)
Here, some examples of nonlinear PDEs (NLPDEs) are given as
∂ 2 ∂x2 +
t −xxt+ax(1−t) = 0 (1.0.4)
Where, the amplitude of the wave is represented by(x, t) anda is nonzero real constant.
The NLSE is
2
Where(x, t) is complex, the constantsα andβ are the coefficients of dispersion and non-
linear term respectively.
Solitons are the non-dispersive waves which were first discovered in 1834 by John Scott
Russell [51]. In the narrow channel of Edinburgh, he observed some solitary waves in wa-
ter. The important properties of these waves are that they behave like particles in dispersive
nonlinear media. Solitons do not change their shape and speed after the collisions with the
other solitons. In 1874, Rayleigh also mentioned the possibilities of such kind of waves.
KdV equation was derived by Korteweg and de Vries, which shows the existence of such
solitons [26].
dµ dt
dµ dx
= 0 (1.0.6)
in the early 1970’s, a lot of work was done to obtain the soliton solutions of the Sine-
Gordon equation (SGE). It is used to study the differential geometry of hyperbolic curved
surface.
− d2µ dx2 +sin(µ) = 0 (1.0.7)
Soltions have so many applicatons in different fields of physics like optics, solid state
physics, particle physics, quantum and plasma physics. Thesoliton concept is actually
based on integrability of nonlinear differential equation(NLDE). One of its main class is
NLSE which was integrated by Zakharov [72].
1.1 Optical Solitons
An optical soliton is a light pulse that do not disperse or diffract during its travel in some
host medium. In optical fibers solitons have been suggested by Haegawa and Tappert in
1973. Optical solitons are generated by balancing self modulation and the group velocity
dispersion (GVD). We have studied the several forms of solitons in optics, like spatial (that
propagate several diffraction lengths but not broadening), temporal, spatio-temporal [69],
discrete [18] and bragg soliton [20]. Soliton plays a vital role in all optical transmission
3
system. In the last few years, one of the most manifest development in optical transmission
system was the introduction of dispersion management. The major types of solitons are
spatial and temporal.
1.1.1 Spatial Solitons
”Spatial solitons are the result of perfect balancing between nonlinear self focusing and
linear diffraction”. It has the natural tendency to broadenduring the propagation in a dis-
persive linear medium. For pulse in space, the broadening iscaused by diffraction. We get
the spatial solitons due the propagation a pulse (optical beam) in a nonlinear media which
does not diffract. During the propagation its diameter remain unchanged. Spatial soliton
based on the Kerr effect by which a self phase modulation (SPM) is produced.
1.1.2 Temporal Solitons
Temporal solitons are, ”the non-spreading optical pulse formed when the GVD is totally
counteracted by nonlinear SPM effect”. These pulses cover the distance of several hundreds
of kilometers with the same temporal width in optical fibers.
1.2 Bright Solitons
Bright solitons are present in form of pulses and these pulses are on zero background. In
dispersive area, refractive index depends on it and it decreases when frequency increases.
In this region temporal bright solitons occur [25], while the dark solitons appear in normal
dispersion region. In optical fibers, solitons are formed due to the nonlinear effect. GVD
has two different signs, it is affiliated with dark and brightsolitons. In optical fibers, it is
irregular and the wave have constant amplitude. Due to instability, these waves are unstable
and breakdown into pulses. These pulses are known as bright solitons. After colliding with
each other during propagation in local dispersion, these pulses maintain its shape and speed.
These waves are known as non-topological waves.
E(x, t) = αsechpφ (1.2.1)
φ = β (x−νt) (1.2.2)
α, β and ν represent the amplitude and the inverse width and the velocity of solitons
respectively .
1.3 Dark Solitons
Dark solitons are intensity dips that have being on the background of the continues waves.
”The dark solitons exist for the self-defocusing nonlinearities and the spatial bright solitons
exist for the self-focusing nonlinearities”. In fibers, bright solitons do not exist in GVD,
then the waves with constant amplitude form the dark solitons.
Dark solitons are wave packets that carries energy and propagate through the mediums such
as solid, liquid, gas and plasma. Traveling speed of these waves is much higher than the
ordinary waves. Due to the local change in phase and amplitude change of the background
waves, dark solitons is less straight forward than the bright solitons. These waves are
known as topological waves.
E(x, t) = γtanhpφ (1.3.1)
φ = δ (x−νt), p> 0 (1.3.2)
γ andδ represent the amplitude and the inverse width of soliton respectively. In non kerr
media dark solitons are unstable while black dark solitons are stable. Dark solitons have
zero powers at their center.
5
E(x, t) = ρcschpφ (1.4.1)
φ = κ(x−νt), p> 0 (1.4.2)
ρ , κ andν stand for the amplitude, the inverse width and the velocity of soliton respectively.
1.5 Classification of Solitons on the Base of Shape
Some solitons are classified on base of shape.
1.5.1 Bell Solitons
From the KdV equation we find the bell shape and low frequency solitons solution. These
solitons are also called non-topological solitons. The soliton solutions of NLSE also have a
bell shape and those soliton solutions are independent of the amplitude and high frequency
solution.
1.5.2 Kink Solitons
”The Kink solitons ( also known as topological solitons) canbe found by Sine-G equation
and their velocity does not depend on the wave amplitude”. The Bloch wall between two
magnatic domains is a good physical example of kink-soliton.
1.5.3 Breath Solitons
”A soliton solution that had non-dispersive standing modescalled breather solitons”. They
occur in the networks of oscillators and time periodic spatially localized solution.
6
1.6 Literature Review
Biswas [7] studied the 1-soliton solution by using the dual power law nonlinearity for the
(1+2)− dimensional NLSE. Lu et al. [34] solved the combined Kdv and mKdv equtaion
for the jacobi elliptic function solution. The generalizedNLSE was solved by Biswas et al
[14] for the dark and bright soliton solutions. Similar soliton solution with the time depen-
dent coefficient in non-Kerr media observed by the Biswas et.al [10]. Abdou [1] discussed
in detail the generalized auxiliary equation method and itsapplications. In birefringent
fiber with Hamiltonian perturbations, Biswas [8] found the dark and bright solitons with
the Kerr law nonlinearity. The different nonlinear problems were studied by Fermi. et al
[22]. Zhou et al. [73] studied the Biswas-Milovic model (BMM) by using the generalized
Kudryashov method in the nonlinear optics. In [74], Zhou et al. got the soliton solutions
for the non-Kerr nonlinear negative index materials. With the help of log law, Crutcher et
al. [17] gave the soliton solutions for the BMM. Mirzazadeh et al. [38] used the power
law nonlinearty to get different soliton solutions of generalized resonant dispersive NLSE.
Rizvi et al. [49] gave the optical solitons which are defined in the physical model and they
found out the exact solitons through the parabolic law nonlinearity. With the help of inter
model dispersion, Rizvi et al. [50] explained the optical 1-soliton solution in dual core
fiber. Solitary wave solutions of CGLQ were found by Rizvi et al. [48]. Eslami et al.
[20] obtained the optical solitons by using first integral method for the resonant NLSE with
time-dependent coefficients. With the help of Kerr and powerlaw nonlinearities, Biswas et
al. [12] retrived the Cubic-quartic optical solitons solution of the NLSE.
With the spatio-temporal dispersion and coupled Hirota equation, optical soliton were ob-
served by Savescu et al. in [53]. To find the optical solitons in birefringent, they [54] used
the Kerr law nonlinearity for the four wave mixing. Triki et al. [61] used dual-power and
parabolic nonlinearities to obtain the dark optical solitons and conservation laws. Ali et
al. [2] observed the optical solitons for paraxial wave equation in Kerr media. Andersen
et al. [3] observed the dark spatial solitons for the direct measurement of their transverse
velocity. Azzouzi et al. [4] got the dipole solitons for a high-order NLSE with non-Kerr
terms and also [5], Azzouzi et al. solved the high dispersivecubic-quintic NLSE for the
7
soliton solutions. By using SCM, Bibi et al. [6] solved the KdV equations and obtained the
different solitons. Milovic and Biswas [37] solved the NLSEfor the dark and bright soli-
tons in optics with the parabolic law nonlinearity. Schr ¨odinger-Hirota equation was solved
by Biswas et al. [11] for optical solitons and complexitons.Cevikel et al. [15] used ansatz
method and got the dark-bright solitons for the evolution equations. Choudhuri et al. [16]
solved the higher-order NLSE with non-Kerr terms and got thedark-in-the-bright solitons.
Ekici et al. [19] solved the Kundu-Eckhaus equation, hence got the dark and singular opti-
cal solitons by using the extendedG′/G-expansion scheme and the ETEM respectively. Fan
et al. [21] studied the analytic solitons for the high-orderNLSE in the fiber lasers. Gardner
et al. solved the KdV equation [24] and obtained its soliton solutions. Li et al. [27] solved
the high-order NLSE for its solitons and in fiber lasers by using the symbolic computations
they [28] also obtained stable propagation of optical solitons. Liu et al. [29] generated
the controlled multiple solitons by introducing the parameters, they also solved the Hirota
equation with variable-coefficients and observed the effects on the obtained solitons [30].
They also got the mixed lump stripe solutions and lump soliton for (3+1)-dimensional
soliton equation by using HBM [31]. Wazwaz solved the Sawada-Kotera-Ito seventh-order
equation by the help of HBM and the tanh-coth method in [62] toget the multiple solitons,
solved two-mode mKdV equation [63] for the multiple solitons. Yang et al. solved the
fifth-order variable-coefficient NLSE [67] for the soliton solutions. Zhou et al. observed
the optical metamaterials for the combo-solitons by using the cubic-quintic nonlinearity
[75].
8
9
In this chapter, we present some basic definitions, that willhelp in understanding our re-
sults.
2.1 Refractive Index
Refractive index (index of refraction) is a dimensionless number, describe the bending of
rays of light when passes from one medium to another.
n= c υ
(2.1.1)
Where,υ andc are the phase velocity of light in the medium and the speed of the light in
space respectively. The variations of Kerr effect (which isalso called refractive index) in
the field produce SPM and self focusing.
2.1.1 Phase Velocity
The phase velocityνp of a travelling wave is defined as
νp = λ T
2.2 Self Phase Modulation (SPM)
First time in history (in 1978) Stolen and Lin observed SPM which is the nonlinear optical
effect. A light pulse produces a nonlinear optical effect due to its own intensity by the Kerr
effect. A nonlinear change in refractive index is describedas
n= αI (2.2.1)
Whereα andI are the nonlinear index and the optical intensity respectively. SPM is very
useful for many devices and system application solitons formation, demultiplexing, all op-
tical regeneration, optical switching, spectral pulse compression and etc.
10
2.3 Dispersion
Dispersion is the process in which phase velocity relate on frequency. Dispersive media
is that media which have this common property. One of the mostwell known example of
dispersion is ’rainbow’, in which white light scatter into different wavelengths (colors).
2.4 Group Velocity Dispersion
”The phenomena in a transparent medium the group velocity oflight depend on the wave-
length or optical frequency is group velocity dispersion (GVD)”. It is given as
GDV = ∂
) = ∂ 2K ∂ψ2 (2.4.1)
WhereK is the wave number and unit of GVD iss 2
m. The cause of ultrashort pulse is GVD.
In optical fibers, the effect of nonlinearities strongly based on GVD.
2.5 Nonlinear Schrodinger Equation
Over the last four decades, the research work in the field of optical fibers have brought a
revolutionary success. NLSE is one of the special kind of equation on which lot of work
have been done by mathematicians and physicists. In opticalfibers, this equation helped
us to understand the the dynamics of optical solitons. The mathematical model of NLSE is
given below
it +αxx+βL(||) = 0 (2.5.1)
where(x, t) is complex, the first term and the second term in Eq. (2.5.1)represent the
evolution and GVD respectively. Some well known NLSEs are given below
11
Consider the dimensionless form of CGLQ equation [40] is
i ∂ ∂ t
∂ 2 ∂x2 + iε||2+ iµ||4 (2.5.2)
where, the normalized envelope of the electric field is denoted by(x, t), D =−sgnβ2 and
β = go w2
g|β2| , whereβ2 is the second order GVD andgo is the linear gain.δ , ε andµ are the
nonlinearity coefficients.
The time fractional perturbed nonlinear Shrodinger equation (TFPNLSE) [43] is given as
i ∂ αu ∂ tα +
∂x + iγ3
(F(|u|2))u= 0 (2.5.3)
where,t > 0 and 0< α ≤ 1, γ1 is third order dispersion,γ2 andγ3 are versions of nonlinear
dispersions.
2.5.3 The Paraxial Nonlinear Shrodinger Equation
The paraxial nonlinear Shrodinger equation (PNLSE) [2] inKerr media is given by
iuz+ α 2
utt + β 2
uyy+ γ|u|2u= 0 (2.5.4)
where,u is the complex wave envelope function. Dispersion, diffraction and Kerr nonlin-
earity are represented byα,β andγ respectively. Ifαβ > 0, then Eq. (2.5.4) becomes ellip-
tic NLSE (ENLSE) and Ifαβ < 0, then Eq. (2.5.4) becomes hyperbolic NLSE (HNLSE).
2.5.4 The Nonlinear Shrodinger Equation with Third and Fourth Order Dis-
persions
The NLSE with 3 OD and 4 OD order dispersion is given by [39]
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+F(|u(x, t)|2)u(x, t) = 0 (2.5.5)
12
The values of 3 OD and 4 OD are controlled by real parametersa andb respectively.F is
a function that describes a general form of the intensity dependent refractive index.
2.5.5 Nonlinear Telegraph Equation
The mathematical model for the nonlinear telegraph equation (NLTE) [42] is given by:
utt −uxx+ut +αu+βu3 = 0 (2.5.6)
Whereα andβ are constants
2.5.6 Nonlinear PHI-Four Equation
utt(x, t)−uxx(x, t)+m2u(x, t)+λu3(x, t) = 0 (2.5.7)
Wherem andλ are constants.
2.6 Nonlinearities
The linear process is a process in which the response to the process and the stimulus given to
the process are proportional otherwise the process is nonlinear. The nonlinear effects arise
in optical fibers due to the increase in the data rates, transmission lengths and number of
wavelengths [3]. In fibers, fiber nonlinearity arise due to two mechanisms, one is refractive
index present in the glass and second one is scattering. There are many definite forms of
nonlinearities in optical fibers which arise due to the propagation of the solitons. Some
different nonlinearities are given as following.
2.6.1 Kerr Law Nonlinearity
L(s) = s
13
and is also known as the cubic nonlinearity. Most of the optical fibers follow the Kerr
nonlinearity.
L(s) = sq
To avoid the collapse, in power law it is necessary takeq in the the interval 0< q < 2.
Similarly, to avoid self-focussing singularity, takeq 6= 2. In plasma, the nonlinearity of
power law is also present. This nonlinearity is also noticeable in optics and semiconductors.
2.6.3 Dual-Power Law Nonlinearity
L(s) = sq+ηs2q
The saturation of the nonlinear refractive index produces dual-power law nonlinearity”. By
using this form of nonlinearity, many other optical nonlinearities are formed. The solitons
become unstable for 1< q< 2. The solitons solution collapse with each other, ifq≥ 2.
2.6.4 Parabolic Law Nonlinearity
L(s) = s+δs2
and ”the nonlinear interaction between Langmuir waves and electrons produces parabolic
nonlinearity is also known as the cubic-quintic nonlinearity” [37].
14
”The higher order polynomial law nonlinearity is defined as
L(s) = s+δs2+υs3
The higher order polynomial law nonlinearity is an extendedform of the parabolic law”.
2.6.6 Triple-Power Law Nonlinearity
L(s) = sq+δs2q+υs3q
and the extension of the dual-power law nonlinearity and thegeneralized form of the higher
order polynomial law nonlinearity produce the triple-power law nonlinearity”.
2.6.7 Anti-Cubic Law Nonlinearity
The anti-cubic law nonlinearity is
L(s) = γ s2 +ζs+ϑs2
This nonlinearity is used to find the exact solitary wave solution. It is very important
nonlinearity used in optical metamaterials.
2.6.8 Saturation Law Nonlinearity
L(s) = ρs
1+ρs
To describe the variation of the dielectrics constant present in the gas vapors, forρ > 0
saturation law is used [36]. The saturation nonlinearity isused for modeling the soliton
propagation, in case of semiconductor-doped fibers inspiteof the Kerr law.
15
L(s) = α ln(β 2s)”
This is an important nonlinearity which appears in many different fields of physics [13] and
use to find 1−soliton solution.
2.7 Jacobi Elliptic Function
”The auxiliary theta functions and the basic elliptic functions are the Jacobi elliptic function
(JEF). The JEFs are found in the description of motion of pendulum as well as in the design
of electronic elliptic function”. Carl Gustav Jakob Jacobifound the JEFs first time in 1829.
There are twelve JEFs. Kink-type soliton solution are generated by JEFs [64].
2.8 Extended Trial Equation Method
First, we explain the ETEM [19, 40, 43, 45, 46]. Consider
P(,Dα t ,x,xx,xxx, ...) = 0 (2.8.1)
and by using the transformation like
(x, t) = u(κ)eiψ , κ = x−bt, ψ = cx−at+ψ0 (2.8.2)
we have a nonlinear ODE
N(u,u′,u′′, ...) = 0 (2.8.3)
u= ν
∑ n=0
τnyn (2.8.4)
γσ yσ + γσ−1yσ−1+ ...+ γ1y+ γ0 (2.8.5)
From Eq. (2.8.4) and Eq. (2.8.5) we can write,
(u ′ )2 =
(2.8.7)
Here f (y) and g(y) are the polynomials. From the Eq. (2.8.5), we got the elementary
integral
dy (2.8.8)
By using the balance procedure, we have a relation amongθ , ν andσ . By computing some
values ofθ , ν andσ and a system of polynomials to get the roots off (y), we solved Eq.
(2.8.8) and find the exact solutions for Eq. (2.8.3). In the last, we get the soliton solutions
for Eq. (2.8.1).
2.9 Modified extended tanh-function method
For using the METFM [40] to solve Eq. (2.8.1), we consider thefollowing traveling wave
solution
17
Then Eq. (2.8.1) becomes ODE as given in Eq. (2.8.3) and by introducing a new variable
ω = ω(κ) the solution is expressed as
u(κ) = m
∑ i=0
whereω ′ = dω dκ
Here, m is positive integer which can be find by the balance procedure. From the Eq.
(2.8.3), Eq. (2.9.2) and Eq. (2.9.3), we get a system of equations and then from these
obtained equations, we get the values ofk,ρ ,α0,α1, ...,αm,β0,β1, ...,βm
Finally, we get the general solutions from the Riccati Eq. (2.9.3) as follows
ω(κ) =− √ −ktanh(
Therefore, obtain the exact soliton solutions of Eq. (2.8.1).
18
Then, Eq. (2.10.1) becomes ODE
Q(φ ,φ ′,φ ′′,φ ′′, ...) = 0 (2.10.3)
Solution of Eq. (2.10.3) is
φ(κ) = asinb(µκ), |κ | ≤ π 2µ
(2.10.4)
or
(2.10.5)
φ ′′(κ) = ab(b−1)µ2sinb−2(µκ)−ab2µ2sinb(µκ) (2.10.7)
or
19
φ ′′(κ) = ab(b−1)µ2cosb−2(µκ)−ab2µ2cosb(µκ) (2.10.9)
By using φ(κ) and its derivatives in Eq. (2.10.3) and by balancing the coefficients of
sin(µκ) or cos(µκ) for the same powers, then we get the values ofa,b,µ from the resulting
equations and finally, we get the required solutions [6, 39].
2.11 Bernoulli’s Equation Approach
Consider the nonlinear PDE
Then, Eq. (2.11.1) becomes ODE
S(ξ ,ξ ′,ξ ′′,ξ ′′, ...) = 0 (2.11.3)
The exact solution for Eq. (2.11.3) is
ξ (σ) = m
Bi( f (σ))i, i = 0,1,2, ...,m, Bm 6= 0 (2.11.4)
Set
ξ ′(σ) = η f (σ)− ( f (σ))2 (2.11.6)
20
By using Eq. (2.11.3) and Eq. (2.11.4), we get the positive integerm. By using theξ
and its required high derivatives in Eq. (2.11.3), we get a polynomial equation inf (σ).
Then equating powers off (σ) to zero, we obtained a system of equations which yields the
values ofBi ,η andρ . Finally, the required solutions of Eq. (2.11.1) obtained [6, 39].
2.12 Hirota’s Bilinear Method
To apply the HBM [2, 41, 62], convert the given PDE in ODE, given as
N(u,ut ,ux,uxt,uxx,uxxx, ...) = 0 (2.12.1)
in quadratic form by using any suitable transformation, then by using the depended variable
transformation
(2.12.2)
and derivatives off andg expressed in Hirota’sD−operator defined by
Dm x Dn
n]g(x, t) f (x′, t ′)|x′=x,t ′=t (2.12.3)
Then we will get some polynomial equations in Hirota’sD−operator. To obtain the soliton
solutions, we setg and f as
g= ∞
with a small parameterε. For one soliton solution, we use
g= εg1, f = 1+ ε f1 (2.12.6)
21
g1 = eη1, η1 = a1x+b1t +c1 (2.12.7)
Wherea1,b1 are complex parameters andc1 is a real constant. For two soliton solution, we
use
g= εg1+ ε2g2, f = 1+ ε f1+ ε2 f2 (2.12.8)
and suppose that
g2 = eη2, η2 = a2x+b2t +c2 (2.12.9)
Wherea2,b2 are complex parameters andc2 is a real constant. So forn soliton solution,
we use
g= εg1+ ε2g2+ ...+ εngn, f = 1+ ε f1+ ε2 f2+ ...+ εn fn (2.12.10)
and suppose that
Wherean,bn are complex parameters andcn is a real constant.
22
Model
23
i ∂ ∂ t
∂ 2 ∂x2 + iε||2+ iµ||4 (3.1.1)
Consider transformation for soliton structure
(x, t) = u(κ)eiψ , κ = x−bt, ψ = cx−at+ψ0 (3.1.2)
Whereu(κ), b, ψ(x, t), a andψ0 are the amplitude, the speed, the phase component, the
frequency, the wave number and the phase constant of the soliton respectively. We get the
following ODE by using Eq. (3.1.2) into Eq. (3.1.1)
(Dq+A)u ′′ +(pq+B)u+2(q+C)u3+Eu5 = 0 (3.1.3)
where,
p= 2a−Da2, q= b−Dc, A=−4β 2,B=−4β (δ −β ),C=−2βε,E =−4β µ
Now for the CGLQ equation, the ETEM is given in the following section.
3.2 Extended Trial Equation Method
First, we explain the ETEM [19, 40, 43, 45, 46] and then apply it to the dimensionless
CGLQ equation. To get the solutions for Eq. (3.1.3) we use thefollowing assumption for
soliton structure
u= ν
∑ n=0
τnyn (3.2.1)
γσ yσ + γσ−1yσ−1+ ...+ γ1y+ γ0 (3.2.2)
From Eq. (3.2.1) and Eq. (3.2.2) we can write,
(u ′ )2 =
(3.2.4)
Here f (y) andg(y) are the polynomials iny and the elementary integral from Eq. (3.2.2) is
±(κ −κ0) = ∫
dy√ (y)
θ = 4ν +σ +2 (3.2.6)
Takeσ = 0 , ν = 1, andθ = 6 so the result is given as,
u= τ0+ τ1y (3.2.7)
2γ0 (3.2.8)
By substituting Eq. (3.2.7) and (3.2.8) into Eq. (3.1.3) andtaking the free parameters,
c= c,τ0 = τ0,τ1 = τ1,ξ0 = ξ0,ξ1 = ξ1,ξ2 = ξ2,ξ3 = ξ3
25
we get the set of some equations which yields the soliton solutions,
a= FDc2−B−H
2F , b= Dc+F, γ0 = G, ξ4 =−G(F +C+5Eτ2
0)τ 2 1
±(κ −κ0) = H ∫
ρ6 (3.2.10)
Integrating Eq. (3.2.10), we have the following solutions,For one root(y) = (y−ρ1) 6
±(κ −κ0) =− H 2(y−ρ1)2 (3.2.11)
For two roots(y) = (y−ρ1) 3(y−ρ2)
3, ρ1 > ρ2
±(κ −κ0) = 2H
For two roots(y) = (y−ρ1) 3(y−ρ2)
3, ρ2 > ρ1
±(κ −κ0) = H
2(y−ρ3) 2, ρ1 > ρ2 > ρ3
±(κ −κ0) = H
[ (ρ1−ρ2) ln |y−ρ3|+
(ρ2−ρ3) ln |y−ρ1|+(ρ3−ρ1) ln |y−ρ2| ] , ρ1 > ρ2 > ρ3
(3.2.14)
Substituting the solutions from Eq. (3.2.11)- Eq. (3.2.14)in Eq. (3.1.2),
(x, t) = [
9(x− Bt−κ0)2− D
) −
)]
A= H 2 , B= D+F, C=
FDc2−B−H 2F
27
By substitutingτ0 =−τ1ρ1, κ0 = 0 andψ0 = 0 in Eq. (3.2.15), we get the rational function
solution
) (3.2.19)
By substitutingτ0 = κ0 = ψ0 = 0 in Eq. (3.2.16), we get solitary wave solution
(x, t) = [
1 3 1 +1)2
1 3 1 +1)2
√ 9(x− Bt)2− D
D
By substitutingτ0 =−τ1ρ1, κ0 = 0 andψ0 = 0 in Eq. (3.2.17), we get confluent hyperge-
ometric solution
) (3.2.21)
By substitutingτ0 = κ0 = ρ1 = ρ2 = 0 andτ1 = ρ3 = 1 in Eq. (3.2.18), we get the singular
soliton solution
]
(3.2.22)
WhereB andC are amplitudes,B andC are the inverse widths andc is the velocity of
solitons.
28
3.3 Modified Extended Tanh-Function Method
Now, we will solve Eq. (3.1.3) with the aid of METFM. In this method, we consider the
solution [21]
u(κ) = m
∑ i=0
By balancingu ′′
andu3 we get the parameterm= 1. Thus the solution from Eq. (3.3.1)
takes the form
u(κ) = α0+α1λ +β0+β1λ−1 (3.3.2)
Now, we putu and its derivatives from Eq. (3.3.2) into Eq. (3.1.3) , we geta system of
various equations which produces the following solutions.
3.3.1 Case. 1
a= a, b= Dc, c= c, d = c(βc2−δ )
2β , α0 =−β0, α1 = α1, β0 = β0, β1 = 0
Ford > 0, we get new solitary wave and antikink soliton solutions
(x, t) = [
29
Ford < 0, the dark and singular soliton solutions are given below
(x, t) =− [
)
wherec, a and ψ0 are frequency, the wave number and the phase constant of solitons
respectively.
D−α2 1
, d = 2c(D−α2
(Dcε −2β )α2 1
a= a, c= c, α0 =−β0, α1 = α1, β0 = β0, β1 = 0
Ford > 0, we obtain new solitary wave and antikink soliton solutions
(x, t) = [
(Dcε −2β )α2 1
(Dcε −2β )α2 1
1)
(Dcε −2β )α2 1
(Dcε −2β )α2 1
1)
(x, t) =− [
(Dcε −2β )α2 1
(Dcε −2β )α2 1
1)
(Dcε −2β )α2 1
(Dcε −2β )α2 1
1)
2β ε , c=
Ford > 0, we obtain new solitary wave and antikink solitons
(x, t) = [
0.0 0.5
and
32
(x, t) =− [
and
where 2β εα2
1 , a andψ0 are the frequency, the wave number and the phase constant of solitons
respectively.
α0 =−β0, α1 = α1, β0 = β0, β1 = 0
33
Ford > 0, we obtain new solitary wave and antikink solitons
(x, t) = [
Ford < 0, the dark solitons and the singular solitons are
(x, t) =− [
)
wherec is frequency,βc(1−c)+δ ε is the wave number andψ0 is phase constant of the solitons.
The results of this chapter have been published in [40].
34
Schrodinger Equation
i ∂ αu ∂ tα +
∂x + iγ3
(F(|u|2))u= 0 (4.1.1)
where,t > 0 and 0< α ≤ 1, γ1 is third order dispersion,γ2 andγ3 are versions of nonlinear
dispersions. By using fractional complex wave transformation
u(x, t) = u(ξ ), ξ = kx+ lt α
Γ(α +1) (4.1.2)
l andk are constants. Eq. (4.1.1) becomes the following nonlinearcomplex ODE
ilu′+k2u′′+ γ(F(|u|2))u+ iγ1k3u′′′+ iγ2k(F(|u|2))u′+ iγ3k(F(|u|2))′u= 0 (4.1.3)
4.2 Kerr Law Nonlinearity
F(u) = u (4.2.1)
Eq. (4.1.3) reduces to
ilu′+k2u′′+ γ(|u|2)u+ iγ1k3u′′′+ iγ2k(|u|2)u′+ iγ3k(|u|2)′u= 0 (4.2.2)
By using the complex transformation
u(ξ ) = (ξ )eiβξ (4.2.3)
in Eq. (4.2.2), the real part is
(−lβ −k2β 2+k3β 3γ1) +(γ −kβγ2)3+k2(1−3kβγ1) ′′ = 0 (4.2.4)
and imaginary part is
(l +2k2β −3k3β 2γ1) ′+k3γ1 ′′′+k(γ2+2γ3)2 ′ = 0 (4.2.5)
By substitutingγ1 = 0 andk(γ2+2γ3) = 0 in Eq. (4.2.5), we get
l =−2k2β (4.2.6)
k2β 2 +(γ −kβγ2)3+k2 ′′ = 0 (4.2.7)
Now, we choose the following assumption to get the soliton solution for Eq. (4.2.7)
= p
∑ i=0
cryr +cr−1yr−1+ ...+c1y+c0 (4.2.9)
From Eq. (4.2.8) and Eq. (4.2.9) we can write,
( ′ )2 =
) (4.2.11)
37
Here f (y) andg(y) are the polynomials and Eq. (4.2.9) reduced to the elementary integral
±(ξ −ξ0) = ∫
dy√ Φ(y)
q= 2p+ r +2 (4.2.13)
Taker = 0 , p= 1, andq= 4 so the result is given as,
= a0+a1y (4.2.14)
2c0 (4.2.15)
By substituting Eq. (4.2.14) and Eq. (4.2.15) into Eq. (4.2.7) and taking the free parame-
ters,
a0 = a0,b0 = b0,c0 = c0,a1 = a1,b4 = b4
we obtained the set of algebraic equations which give the soliton solutions,
β = γ
) +
±(ξ −ξ0) = A ∫
WhenΦ(y) = (y−d1) 4
u(x, t) =
[ a0+a1d1+
4A2(d2−d1)a1
1c0γ) a2
a2 1c0γ0Γ(α+1)
u(x, t) =
[ a0+a1d2+
(d2−d1)a1
a2 1c0γ0Γ(α+1)
u(x, t) =
[ a0+a1d1−
cosh
WhenΦ(y) = (y−d1)(y−d2)(y−d3)(y−d4); d1 > d2 > d3 > d4
u(x, t) =
[ a0+a1d1+
(4.2.26)
Seta0 = −a1d1 andξ0 = 0, the solutions Eq. (4.2.21) - Eq. (4.2.23) reduced to rational
function solutions
) (4.2.29)
Seta0 =−a1d2 andξ0 = 0 in Eq. (4.2.24), we get the soliton solution
u(x, t) = F
) (4.2.30)
By settinga0 =−a1d1 andξ0 = 0 in Eq. (4.2.25) and Eq. (4.2.26) reduced to bright soliton
u(x, t) = H
( iC(x− Atα)
u(x, t) = L
( iC(x− Atα)
G= k(d1−d2)
J = d3−d2, K =
A , L =−a1(d1−d2)(d4−d2)
M = d4−d2, N = d1−d4, O= k √ (d1−d3)(d2−d4)
2A
41
Here,F andH are amplitude of soliton andG, K are inverse width of solitons.
Remark-1 We also get second form of singular optical soliton solutions, when modulus
P→ 1
( iC(x− Atα)
) (4.2.33)
Remark-2 We get periodic singular wave soliton solutions, when modulusP→ 0
u(x, t) = L
( iC(x− Atα)
The power law nonlinearity is
F(|u|) = |u|n (4.3.1)
For avoiding the self-focusing singularity, we take the power law nonlinearity for 0< n< 2
and in particularn 6= 2 . By using Eq. (4.3.1) in Eq. (4.1.3), we get
ilu′+k2u′′+ γ|u|2nu+ iγ1k3u′′′+ iγ2k|u|2nu′+ iγ3k(|u|2n)′u= 0 (4.3.2)
By using Eq. (4.2.3) in Eq. (4.3.2), the imaginary and real parts are
(l +2k2β −3k3β 2γ1) ′+k3γ1 ′′′+k(γ2+2nγ3)2n ′ = 0 (4.3.3)
and
(−lβ −k2β 2+k3β 3γ1) +(γ −kβγ2)2n+1+k2(1−3kβγ1) ′′ = 0 (4.3.4)
42
By substitutingγ1 = 0 andk(γ2+2nγ3) = 0 in Eq. (4.3.3), we get
l =−2k2β (4.3.5)
By using = ψ 1 2n in Eq. (4.3.6), we get
4n2k2β 2ψ2+4n2(γ −kβγ2)ψ3+(1−2n)k2(ψ ′)2+2nk2ψψ ′′ = 0 (4.3.7)
Now replacing by ψ and then by using Eq. (4.2.8) - Eq. (4.2.11) in Eq. (4.3.7) andafter
balancing the highest order nonlinear terms
q= p+ r +2 (4.3.8)
Taker = 0, p= 1 andq= 3 so the result is given as,
ψ = a0+a1y (4.3.9)
2c0 (4.3.10)
Put Eq. (4.3.9) and (4.3.10) into Eq. (4.3.7) and by taking the free parameters,
a0 = a0,c0 = c0,a1 = a1,b3 = b3
Then, we have some equations which gives the solutions,
β = γ
±(ξ −ξ0) = A ∫
WhenΠ(y) = (y−e1) 3
u(x, t) =
u(x, t) =
u(x, t) =
[ a0+a1e3+
( (a1(e2−e3))
(4.3.19)
Let δ0 = −δ1ξ1 andτ0 = 0, then the solution in Eq. (4.3.16) reduces to rational function
solution
u(x, t) = R
exp
 Q¤2
Figure 4.1:R= 0.09, T = 0.9, Y = 2.0,K = 5.05,α = 9.0,β = 1.5,n= 1.
Eq. (4.3.18) reduced to singular soliton solution
u(x, t) = S
exp
Figure 4.2:R= 1.05, T = 1.05,Y = 2.05,K = 2.05,α = 5.0,β = 5.5,n= 1.
Let δ0 = −δ1ξ3 andτ0 = 0, then the solution in Eq. (4.3.19) reduces to Jacobi elliptic
function solution
( Vi(x− Rtα), X
46
The solutions exist forδ1 > 0 while R, S are amplitudes of soliton andT is the inverse
width of the solitons.
Remark-1 The second form of singular optical soliton for the modulusX → 1 is
u(x, t) = Utanh 1 n
( Vi(x− Rtα)
0.0
 Q¤2
Figure 4.3:U = 1.05,V1 = 0.5, R= 2.0, P= 2.0,K = 1.05,α = 5.0,β = 5.5,n= 1.
Remark-2 The periodic singular wave soliton solutions are obtained,when modulusX → 0
u(x, t) = Usin 1 n
( Vi(x− Rtα)
0.00.51.0 t
4  Q¤2
Figure 4.4:U = 5.05,V1 = 0.5, R= 2.0, P= 2.0,K = 1.05,α = 5.0,β = 5.5,n= 1.
47
By using Eq. (4.4.1) in Eq. (4.1.3), we get
ilu′+k2u′′+ γuln |u|2+ iγ1k3u′′′+ iγ2ku′ ln |u|2+ iγ3ku(ln |u|2)′ = 0 (4.4.2)
By using Eq. (4.2.3) in Eq. (4.4.2), the imaginary and real parts are
(l +2k2β −3kβ 2γ1+2kγ3) ′+k3γ1 ′′′+2kγ2 ′ ln = 0 (4.4.3)
and
(−lβ −k2β 2+k3β 3γ1) +(k2−3γ1k3β ) ′′+(2γ −2kγ2β ) ln = 0 (4.4.4)
By substitutingγ1 = 0 andγ2 = 0 in Eq. (4.4.3), we get
l =−2k(kβ + γ3) (4.4.5)
Now, Eq. (4.4.4) implies
By using
in Eq. (4.4.6), we get
kβ (kβ −2γ3)ψ4+2γψ3−k2ψ2ψ ′′+2k2ψ(ψ ′)2+k2(ψ ′)2 = 0 (4.4.8)
48
By replacing by ψ and then by using Eq. (4.2.8) - Eq. (4.2.11) in Eq.(4.4.8) andafter
balancing the highest order nonlinear terms
q= p+ r +2 (4.4.9)
Taker = 0, p= 1 andq= 3 so the result is given as,
ψ = a0+a1y (4.4.10)
Put Eq. (4.4.10) into Eq. (4.4.8) and by taking the free parameters,
a0 = a0,c0 = c0,a1 = a1,b3 = b3
we have various equations from which, the following solitonsolutions are obtained,
β = γ3
k2 (4.4.14)
Thus by takingb4 = 1, the elementary integral in Eq. (4.2.20) becomes,
±(ξ −ξ0) = A1
c0
When(y) = (y−δ1) 4
u(x, t) = exp
u(x, t) = exp
u(x, t) = exp
When(y) = (y−δ1) 2(y−δ2)(y−δ3); δ1 > δ2 > δ3
u(x, t) = exp
cosh
) tα )] (4.4.20)
When(y) = (y−δ1)(y−δ2)(y−δ3)(y−δ4); δ1 > δ2 > δ3 > δ4
u(x, t) = exp
( δ4−δ2+(δ1−δ4)
) tα )]
(4.4.21)
The rational soliton solutions fora0 =−a1δ1 andξ0 = 0, from Eq. (4.4.16) - Eq. (4.4.18)
are
Here we get bright soliton solution after solving Eq. (4.4.20)
u(x, t) = exp
Eq. (4.4.21) reduces to the following Jacobi elliptic soliton solution
u(x, t) = exp
]−1
exp
√ γ2 3 +2γ
E = 4A2 1(δ2−δ1)a1, F = k(δ1−δ2), G= (δ1−δ2)a1, H =
k(δ1−δ2)
A1 , K =−2a1(δ1−δ2)(δ1−δ3)
L = 2δ1−δ2−δ3, M = δ3−δ2, N = k √
(δ1−δ2)(δ1−δ3)
O=−a1(δ1−δ2)(δ4−δ2), P= δ4−δ2, Q= δ1−δ4
R= k √ (δ1−δ3)(δ2−δ4)
2A1
Remark-1 In limiting case as modulusS→ 1, the second form of singular optical soliton
is
] (4.4.28)
52
Remark-2 The periodic singular wave solitons obtained for the modulus S→ 0
u(x, t) = exp
] (4.4.29)
The above results of this chapter have been published in [43].
53
Ginzburg-Landau Model
5.1 Extraction of Combo Solitons
In this section, we find the soliton solutions [48] for CGLQ model. The dimensionless
CGLQ model is
iuz+ D 2
utt +u|u|2 = iδu+ iAutt + iεu|u|2+ iµu|u|4 (5.1.1)
whereu(z, t) is a normalized envelope for the given electric field,D = −sgnβ2 andA =
go w2
g|β2| , whereβ2 is the second order GVD andgo is the linear gain.δ , ε and µ are the
coefficients of nonlinearity, consider the wave solution [69].
u(z, t) = q(z, t)eiφ(kz−ωt). (5.1.2)
whereφ(z, t) = kz−ωt is the nonlinear phase shift andq(z, t) is the shape of the pulse.
Put u(z, t) and its required derivatives from Eq. (5.1.2) in Eq. (5.1.1), then the real and
imaginary parts are given as
(−k−ω2D 2 )q−2ωAqt +
D 2
and
qz+(ω2A−δ )q−ωDqt −Aqtt − ε)|q|2q−µ|q|4q= 0 (5.1.4)
Now we consider the ansatz method of Li [27]
q(z, t) = iβ +λ tanh(ηξ )+ iρ sech(ηξ ) (5.1.5)
whereξ = t − χz andη is the pulse width,χ is the shift of the group velocity and the
nonlinear phase shift for solitary wave solution with|t| → ∞ is
φ(z, t) = arctan
( β +ρ sech(ηξ )
The amplitude is
|q(z, t)|= {(λ 2+β 2)+2βρ sech(ηξ )+(ρ2−λ 2)sech2(ηξ )}1/2 (5.1.7)
where the parametersη,χ ,ω andk are all real butλ ,β andρ may be real or complex. Now
substituting Eq. (5.1.5) into Eq. (5.1.3) and Eq. (5.1.4), we get set of algebraic equations.
(λ 2+β 2−k−ω2D 2 )λ = 0 (5.1.8)
(ρ2−λ 2−Dη2)λ = 0 (5.1.9)
(λ 2+3β 2−k−ω2D 2 +
D 2
−(χ +ωD)λη = 0 (5.1.11)
{ω2A−δ − (λ 2+β 2)ε −µ(λ 2+β 2)2}λ = 0 (5.1.12)
2εβλ{ε −2µ(λ 2+β 2)}= 0 (5.1.13)
{(ω2A−δ )−Aη2−2εβ 2− ε(λ 2+β 2)−4µβ 2(λ 2+β 2)−µ(λ 2+β 2)2}ρ = 0(5.1.14)
{−εβ (ρ2−λ 2)−2εβρ2−4µβ 3ρ2−2µβ (λ 2+β 2)(ρ2−λ 2)−4µρ2β (λ 2+β 2)}ρ = 0(5.1.15)
Consider the following cases.
56
In this case, forρ = 0 andλ 6= β 6= 0 the solution is
q(z, t) = iβ +λ tanh[η(t−χz)] (5.1.16)
Thus we get different parameter as
η2 = −4β 2
χ = ωD (5.1.19)
The intensity function is available forβ = ±λ 2 . The solution describes the dark soliton
extracted from Eq. (5.1.16). This solution depends on the value of λ andβ . The band
width has a connection withD depending on the sign ofD. To get positive value ofη we
must chooseD < 0. Thus by using Eq. (5.1.18) and Eq. (5.1.19) we get the following
envelope function:
u(z, t) =
)] (5.1.20)
The solution in Eq. (5.1.20) tells about dark solitary wave having amplitude just depending
on time.
-10 -5
0 5
Case. 2
57
In this case, we takeλ = 0 andρ 6= 0, thus we get the solution
q(z, t) = iβ + iρλ sech[η(t −χz)] (5.1.21)
The solitary wave intensity is
|q(z, t)|2 = β 2 (
1+4sech2[η(t −χz)]
ω =
(5.1.25)
The solution of Eq. (5.1.21) represents the bright solitarywaves depending on the value of
β 2 = ε 2µ . The band width depends onβ 2, ε andA, therefore we must takeεA < 0. The
phase and the amplitude of bright solitons from Eq. (5.1.21)(by lettingχ = 0) depends on
time and coordinate in propagation direction. Now the envelope function is
u(z, t) =
[ iβ + i
58
5.2 Dark-in-the-Bright Soliton
In this section, we will find dark-in-the-bright soliton solutions for CGLQ model by apply-
ing the ansatz method of Choudhuri [16] to Eq. (5.1.3) and Eq.(5.1.4)
q(z, t) = iζ +λ sech(νξ ) tanh(σξ ) (5.2.1)
whereξ = t −χz, σ andν are the pulse widths andχ is the shift of the inverse GVD. For
ζ = ν = 0, Eq. (5.2.1) gives the dark solitons. The dipole soliton solutions are obtained by
using an Ansatz method, whenλ , ζ , σ andν are observed in Eq.(5.2.1). One notice that
ν, χ , k, andω are all real where asλ , ζ , andσ can be real or complex depending on the
parametersD, A, ε, µ andδ . From Eq. (5.2.1), the amplitude functionq(z, t) is
|q|= [ζ 2+{λ sech(νξ ) tanh(σξ )}2] 1 2 (5.2.2)
With its corresponding phase shiftψ(z, t)
ψ(z, t) = arctan
) (5.2.3)
By substituting equation Eq. (5.2.1) and its derivatives into Eq.(5.1.3) and Eq. (5.1.4), we
will get the following equations
59
−Dλ 2
(−k−ω2D 2 )ζ +ζ 3 = 0 (5.2.7)
χλσ = 0 (5.2.8)
−λ 5µ = 0 (5.2.11)
(ε +2µζ 2)ζ λ 2 = 0 (5.2.14)
60
Now to get solitary wave solutions we have to implant some conditions on the solution of
Eq. (5.2.4)-Eq. (5.2.14), there would be two casesζ = 0 andλ 6= 0 or ζ 6= 0 andλ = 0.
In caseλ = 0, we can not obtain our required results. It is noted that, the occurrence of
the parameterζ in Eq. (5.2.1) provides soliton solutions with the time variable approaches
infinity. Therefore, the solution ofq(z, t) from Eq. (5.2.1) reduces to the following form by
puttingζ = 0,
under the following parametric conditions,ε = 0,µ = 0,Aω2−δ = 0.
Thus we get the envelope function
u(z, t) = λ sech(νξ ) tanh(σξ )ei(kz−ωt) (5.2.16)
Whereχ = 0, ω = √
2A and amplitude depend onλ , ν andσ and these constants
can be determined by initial pulse condition. The soliton solutions are the combination of
bright and dark solitary wave. This result shows the condition where GVD and TOD pro-
duce the dipole solitons. Thus, the dipole solitons exist innon-kerr media with high order
dispersions.
These results have been published in [48].
61
62
The paraxial NLSE [2] in Kerr media is given by
iuz+ α 2
utt + β 2
uyy+ γ|u|2u= 0 (6.1.1)
where,u is the complex wave envelope function. Dispersion, diffraction and Kerr nonlin-
earity are represented byα,β andγ respectively. Ifαβ > 0, then Eq. (6.1.1) becomes
elliptic nonlinear Schrodinger equation (ENLSE) and Ifαβ < 0, then Eq. (6.1.1) becomes
hyperbolic nonlinear Schrodinger equation (HNLSE). Consider the following wave trans-
formations
where
ξ = y+z−ct (6.1.3)
Using Eq. (6.1.2) and Eq. (6.1.3) into Eq. (6.1.1), we get thereal and imaginary parts
(β +αc2) ′′− (βa2+2a+αa2c2) + γ3 = 0 (6.1.4)
and
βa=−1−αac2 (6.1.6)
′′−a2 + γa3 = 0 (6.1.7)
In the following section, we will analyze Eq. (6.1.7) with the aid of ETEM [19, 40, 43, 45,
46].
6.2 Extended Trial Equation Method
To get the exact solutions for Eq. (6.1.7) we start with the following assumption
= r
∑ i=0
γqxq+ γq−1xq−1+ ...+ γ1x+ γ0 (6.2.2)
From Eq. (6.2.1) and Eq. (6.2.2) we can write,
( ′)2 = g(x) h(x)
) (6.2.4)
Hereg(x) andh(x) are the polynomials ofx. We can reduce Eq. (6.2.2) to the elementary
integral
∫ √ h(x) g(x)
dx (6.2.5)
Substituting Eq. (6.2.1), Eq. (6.2.3) and Eq. (6.2.4) into Eq. (6.1.7) and with the help of
homogenous balance, we get the following relation,
64
If we takeq= 0 , r = 1, ands= 4, then
= α0+α1x (6.2.7)
3 = α3 0 +3α2
0α1x+3α0α2 1x2+α3
2γ0 (6.2.9)
By substituting Eq. (6.2.7) - Eq. (6.2.9) into Eq. (6.1.7), we get the set of the required
equations which gives the solutions,
α0 = α0,α1 = α1,β0 = β0,β4 = β4,γ0 = γ0
β1 = 4β4α0
α3 1
±(ξ −ξ0) = A ∫ (
±(ξ −ξ0) =− A x−δ1
(6.2.14)
±(ξ −ξ0) = A
δ1−δ2 ln
, δ1 > δ2 > δ3
±(ξ −ξ0) = 2A√
(δ1−δ3)(δ2−δ4) H(θ ,h), δ1 > δ2 > δ3 > δ4 (6.2.18)
where
x4+ β3
β4 x3+
u(y,z, t) =
[ α0+α1δ1±
cosh
(6.2.25)
After settingα0 = −α1δ1 and ξ0 = 0, the Eq. (6.2.20) - Eq. (6.2.22) give the rational
function solitons
exp
[ ia
[ ia
)] (6.2.28)
After settingα0 =−α1δ2 andξ0 = 0, in Eq. (6.2.23), we get the required soliton solution
u(y,z, t) = F
[ ia
Figure 6.1:δ1 = 1.5,δ2 = 2.5, α1 = 2.0,γo = 1.5,β4 = 0.5,β = 0.5.
By settingα0 =−α1δ1 andξ0 = 0 Eq. (6.2.24), we get bright soliton solution
u(y,z, t) = H
[ ia
Figure 6.2:δ1 = 2.5,δ2 = 1.05,α1 = 2.0,γo = 1.5,β4 =−5.5,β = 0.5.
Also by settingα0 =−α1δ1 andξ0 = 0 in Eq. (6.2.25), we get the Jacobi elliptic function
solution
[ ia
D = 4A2(δ2−δ1)α1, E = (δ1−δ2) 2, F = (δ1−δ2)α1
G= δ1−δ2
J = δ3−δ2, K =
M = δ4−δ2, N = δ1−δ4, O=
√ (δ1−δ3)(δ2−δ4)
(δ1−δ3)(δ2−δ4)
HereF andH are amplitude of soliton andG, K are inverse width of solitons.
6.2.1 Remarks
In the limiting case, as the modulusP → 1 the Jacobi elliptic function solution in Eq.
(6.2.31) reduces to solitary wave solutions
u(y,z, t) = L
[ ia
Figure 6.3:δ1 = 1.5,δ2 = 2.5,α1 = 2.0,γo = 1.5,β4 = 0.5,β = 0.5.
In the limiting case, as the modulusP → 0 the Jacobi elliptic function solution in Eq.
(6.2.31) reduces to the periodic singular wave soliton solutions
u(y,z, t) = L
[ ia
The results of this chapter have been published in [2].
71
Equation with Higher Order Dispersions
72
7.1 Mathematical Model
The NLSE with 3rd order and 4th order dispersion is given by [39]
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+F(|u(x, t)|2)u(x, t) = 0 (7.1.1)
The values of 3rd order and 4th order dispersions are controlled by real parametersa and
b.
s= x−υt, φ(x, t) =−κx+ωt +θ (7.1.3)
with κ , ω, ν andθ represent the frequency, the wave number, the speed and the phase
constant of soliton respectively. Now using Eq. (7.1.2) andEq. (7.1.3) into Eq. (7.1.1), we
get real and imaginary parts
bg′′′′+(3aκ −6bκ2)g′′+(bκ4−ω −aκ3)g+F(|g|2)g= 0 (7.1.4)
and
From imaginary part, we get
a= 4bκ (7.1.6)
ν = 4bκ3−3κ2a (7.1.7)
7.2 Kerr Law Nonlinearity
The Kerr law nonlinearity [12] isF(p) = cp. Thus Eq. (7.1.1) reduces to,
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+c(|u(x, t)|2)u(x, t) = 0 (7.2.1)
Thus Eq. (7.1.4) becomes,
Taking the assumption for sine-cosine method [39]:
g= λcosβ (µs) (7.2.3)
Using Eq. (7.2.3) along with its derivatives into Eq. (7.2.2), we get set of following alge-
braic equations
16bλ µ4−4P2λ µ2+P1λ = 0 (7.2.6)
74
By solving the obtained equations, we get the results given below
β =−2,λ = (3aκ −6bκ2)
µ =
25b (7.2.8)
wherea, b, c andκ are the constants. Now substituting these values into Eq. (7.2.1) to get
bright soliton solution,
q(x, t) = λsech2 [√
6bκ2−3aκ 20b
( x− (4bκ3−3κ2a)t
)] ei(−κx+ωt+θ ) (7.2.9)
The 3D and 2D plots for Eq. (7.2.9) are given as
-10 -5
0 5
10 x
0.0 0.5
Figure 7.1:ω = 2,λ = 2,a= 2,b= 1.3,κ = 1.1,θ = 1.
In the following section, we use power law to find soliton solutions with the aid of sin-
cosine method.
7.3 Power Law Nonlinearity
The power law [12] isF(p) = cpn. Thus Eq. (7.1.1) reduces to,
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+c(|u(x, t)|2n)u(x, t) = 0 (7.3.1)
and Eq. (7.1.4) becomes,
bg′′′′+(3aκ −6bκ2)g′′+(bκ4−ω −aκ3)+cg2n+1 = 0 (7.3.2)
Now by using Eq. (7.2.3) along with its derivatives into Eq. (7.3.2), we get the following
equations
bλβ (β −1)(β −2)(β −3)µ4cosβ−4(µs)−bλβ (β −1)(β −2)(β −3)µ4cosβ−2(µs)
−bλβ (β −1)(β −2)µ4cosβ−2(µs)−bλβ 3(β −1)µ4cosβ−2(µs)+bλβ 3(β −1)µ4cosβ (µs)
+bλβ 3µ4cosβ (µs)+(3aκ −6bκ2)λβ (β −1)µ2cosβ−2(µs)
−(3aκ −6bκ2)λβ 2µ2cosβ (µs)+(bκ4−ω −aκ3)λcosβ (µs)+cλ 2n+1cos(2n+1)β (µs) = 0
Now by equating the coefficients and exponents ofcosk(.), we obtain the algebraic equa-
tions which yields the solutions withβ = −2 n ,
bλ (12n3+44n2+48n+16)µ4+cn4λ 2n+1 = 0 (7.3.3)
−bλ (8n3+34n2+64n+34)µ4+P2λ (2n3+4n2)µ2 = 0 (7.3.4)
16bλ µ4−4P2n2λ µ2+P1n4λ = 0 (7.3.5)
76
λ =
b(n2+2n+2)2 (7.3.8)
Wherea, b, c, n andκ are the constants. Thus we obtain bright soliton solution for Eq.
(7.3.1).
[√ (6bκ2−3aκ)n2
] ei(−κx+ωt+θ ) (7.3.9)
7.4 Anti-Cubic Law Nonlinearity
The anti-cubic law nonlinearity isF(p) = ξ p−2+η p+δ p2 . Thus Eq. (7.1.1) reduces to,
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+(ξ |u(x, t)|−4+η|u(x, t)|2+δ |u(x, t)|4)q= 0(7.4.1)
and the real part Eq. (7.1.4) becomes,
bg′′′′+(3aκ −6bκ2)g′′+(bκ4−ω −aκ3)g+ξg−3+ηg3+δg5 = 0 (7.4.2)
Now by using Eq. (7.2.3) along with its derivatives into Eq. (7.4.2), we get the following
equations
77
bλβ (β −1)(β −2)(β −3)µ4cosβ−4(µs)−bλβ (β −1)(β −2)(β −3)µ4cosβ−2(µs)
−bλβ (β −1)(β −2)µ4cosβ−2(µs)−bλβ 3(β −1)µ4cosβ−2(µs)+bλβ 3(β −1)µ4cosβ (µs)
+bλβ 3µ4cosβ (µs)+(3aκ −6bκ2)λβ (β −1)µ2cosβ−2(µs)− (3aκ −6bκ2)λβ 2µ2cosβ (µs)
+(bκ4−ω −aκ3)λcosβ (µs)+ξ λ−3cos−3β (µs)+ηλ 3cos3β (µs)+δλ 5cos5β (µs) = 0
By equating the coefficients and exponent ofcosk(.), so we obtain some equations which
yields the solitons
bλ µ4−P2λ µ2+P1λ = 0 (7.4.5)
Eq. (7.4.3), Eq. (7.4.4) and Eq. (7.4.5) yield the followingsolutions.
λ =
√ −24bη
√ η
ω = bκ4−aκ3+ 24bη2((3aκ −6bκ2)−b) (20b−2(3aκ −6bκ2))2 (7.4.7)
wherea,b,η,κ ,δ ,ξ are the constants.
78
Hence, bright soliton solution for Eq. (7.4.1) is given by
q(x, t) = λsech
ei(−κx+ωt+θ )
The 3D and 2D plots for Eq. (7.4.8) are given as
-5 0 5x
-4 -2 2 4
Figure 7.2:ω = 2,λ = 2,a= 2,b=−1.3,κ = 0.3,η = 3,θ = 1,δ = 1.
7.5 Parabolic Law Nonlinearity
The parabolic law nonlinearity isF(p) = ξ p+η p2. Thus Eq. (7.1.1) reduces to,
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+(ξ |u(x, t)|2+η|u(x, t)|4)u(x, t) = 0 (7.5.1)
So Eq. (7.1.4) becomes,
bg′′′′+(3aκ −6bκ2)g′′+(bκ4−ω −aκ3)g+ξg3+ηg5 = 0 (7.5.2)
By balancing we getn= 1. For Bernoulli’s equation approach, we use the following sub-
stitution
U ′(s) = δU(s)−U2(s) (7.5.5)
By using Eq. (7.5.3)- Eq. (7.5.5) along with its derivativesinto Eq. (7.5.2) and equating
the coefficientsU and its powers to zero, we obtain the following equations,
24bA1+ηA5 1 = 0 (7.5.6)
−60bA1δ +5ηA0A4 1 = 0 (7.5.7)
50bA1δ 2+6aκA1−12bκ2A1+10ηA2 0A3
1+ξA3 1 = 0 (7.5.8)
−15bA1δ 3+3ξA0A2 1+10ηA3
0A2 1−9aκA1δ +18bκ2A1δ = 0 (7.5.9)
bA1δ 4+bκ4A1−aκ3A1+3ξA2 0A1+5ηA4
0A1 (7.5.10)
−ωA0+ξA3 0+ηA5
The above equations possess following solutions,
δ = ηA0A3
80
A0 =
(7.5.13)
whereκ , A0, A1 are arbitrary constants andbη < 0. By using values ofδ andν into Eq.
(7.5.4) we get,
q(x, t) =
ei(−κx+ωt+θ )
The 3D and 2D plots for Eq. (7.5.14) are given as
-10 -5
0 5
Figure 7.3:ω = 0.2,A1 = 1,a= 2,b= 1.3,κ = 0.001,η = 0.01,Ao = 99,θ = 1.
81
7.6 Cubic Quintic Law Nonlinearity
In Cubic Quintic law nonlinearity, we assume that,F(p) = cp2+dp3. Thus Eq. (7.1.1)
reduces to,
iu(x, t)t + iauxxx(x, t)+buxxxx(x, t)+(c|u(x, t)|4+d|u(x, t)|6)q= 0 (7.6.1)
So Eq. (7.1.44) becomes,
bg′′′′+(3aκ −6bκ2)g′′+(bκ4−ω −aκ3)g+cg5+dg7 = 0 (7.6.2)
Using homogenous balance method we get,N = 2/3. Assume the solution of Eq. (7.6.2)
as
where,U satisfies
U ′(s) = δU(s)−U2(s) (7.6.5)
By using Eq. (7.6.3)- Eq. (7.6.5) into Eq. (7.6.6) and equating the coefficients of powers
of U to zero, we obtain the equations which yields the required soliton solutions,
δ =
82
whereκ , A are arbitrary constants. By using value ofδ andν, U(s) becomes,
U(s) = 1 2
q(x, t) = A
ei(−κx+ωt+θ )
The 2D and 3D plots for Eq. (7.6.8) are given as follows
-1.0 -0.5
0.0 0.5
0.1735 0.1740 0.1745 0.1750 0.1755 0.1760 0.1765
Figure 7.4:ω = 2,A= 2,a= 2,b= 1.3,κ = 0.01,η = 2,θ = 1,δ = 1.
The above stated chapter have been published in [39].
83
Equation
84
utt −uxx+ut +αu+βu3 = 0 (8.1.1)
Whereα,β are constants and we use the transformation
u(x, t) = g f , g= g(x, t), f = f (x, t) (8.1.2)
By using Eq. (8.1.2) into Eq. (8.1.1), we get
(D2 x−D2
(D2 t −D2
Dtg. f = 0 (8.1.5)
Dm t Dn
ng(x, t) f (x, t)|t ′=t,x′=x (8.1.6)
By considering the series expansion ofg and f with a small parameterε
g= ∞
85
g= εg1, f = 1+ ε f1 (8.2.1)
By using Eq. (8.2.1) in Eq. (8.1.3) and considering the coefficients ofε, we get
g1xx−g1tt = 0 (8.2.2)
g1 = epx+qt (8.2.3)
From Eq. (8.2.2) and Eq. (8.2.3) implies the dispersion relation
q=±p (8.2.4)
By using Eq. (8.2.1) in Eq. (8.1.4) and by comparing the coefficients ofε2, we have
2 f1 f1tt −2 f1t f1t −2 f1 f1xx+2 f1x f1x = α f 2 1 +βg2
1 (8.2.5)
which implies
f1 =±i
√ β α
whereβα > 0. Thus we obtained one soliton solution withε = 1
u(x, t) = ep(x±t)
(8.2.7)
The 3 dimensional and 2 dimensional plots witht = 0 are as follows
86
1
2
3
8.3 Two Soliton Solution
g= εg1+ ε2g2, f = 1+ ε f1+ ε2 f2 (8.3.1)
By using Eq. (8.3.1) in Eq. (8.1.3) and considering the coefficients ofε2, we get
g2xx+ f1g1xx−2 f1xg1x+ f1xxg1−g2tt − f1g1tt +2 f1tg1t − f1ttg1 = 0 (8.3.2)
Here, we take
s=±r (8.3.4)
By using Eq. (8.3.1) in Eq. (8.1.4) and by comparing the coefficients ofε4, we have
2 f2 f2tt −2 f2t f2t −2 f2 f2xx+2 f2x f2x = α f 2 2 +βg2
2 (8.3.5)
which implies
f2 =±i
√ β α
87
whereβα > 0. Hence, the two soliton solution withε = 1 is given by
u(x, t) = ep(x±t)+er(x±t)
1± i √
√ β α er(x±t)
(8.3.7)
The 3 dimensional and 2 dimensional plots for Eq. (8.3.7) with t = 0 are as follows
-8 -4 0 4 8 x
1
2
3
8.4 Three Soliton Solution
For three soliton solution we consider
g= εg1+ ε2g2+ ε3g3, f = 1+ ε f1+ ε2 f2+ ε3 f3 (8.4.1)
By using Eq. (8.4.1) in Eq. (8.1.3) and considering the coefficients ofε3, we get
g3xx+ f1g2xx+ f2g1xx−2 f1xg2x−2 f2xg1x+ f1xxg2+ f2xxg1
−g3tt − f1g2tt − f2g1tt +2 f1tg2t +2 f2tg1t − f1ttg2− f2ttg1 = 0
(8.4.2)
v=±q (8.4.4)
By using Eq. (8.4.4) in Eq. (8.1.4) and by comparing the coefficients ofε4, we have
2 f3 f3tt −2 f3t f3t −2 f3 f3xx+2 f3x f3x = α f 2 3 +βg2
3 (8.4.5)
which implies
f3 =±i
√ β α
whereβα > 0. Thus the three soliton solution withε = 1 is
u(x, t) = ep(x±t)+er(x±t)+eq(x±t)
1± i √
√ β α eq(x±t)
(8.4.7)
The 3-D and 2-D plots for Eq. (8.4.7) witht = 0 are shown as
-10 -5 0 5 10 x
1
2
3
u2
Figure 8.3:p= 0.5+ i, r = 0.15+ i,q=−0.2+ i,β = 1,α = 1.
8.5 N Soliton Solution
u(x, t) = ep1(x±t)
1± i √
Three soliton solution
u(x, t) = ∑3
1± i √
N soliton solution
u(x, t) = ∑N
1± i √
The results of this chapter have been submitted [42].
90
91
utt(x, t)−uxx(x, t)+m2u(x, t)+λu3(x, t) = 0 (9.1.1)
we use the following transformation
u(x, t) = g f , g= g(x, t), f = f (x, t) (9.1.2)
By using Eq. (9.1.2) into Eq. (9.1.1), we get
(D2 x−D2
(D2 t −D2
m2 f 2+λg2 = 0 (9.1.5)
where, Hirota bilinear operatorD is defined as
Dm x Dn
n]g(x, t) f (x′, t ′)|x′=x,t ′=t (9.1.6)
By considering the series expansion ofg and f with a small parameterε
g= ∞
92
g= εg1, f = 1+ ε f1 (9.2.1)
By using Eq. (9.2.1) in Eq. (9.1.3) and considering the coefficients ofε, we get
g1xx−g1tt = 0 (9.2.2)
g1 = epx+qt (9.2.3)
From Eq. (9.2.2) and Eq. (9.2.3) implies the dispersion relation
q=±p (9.2.4)
By using Eq. (9.2.1) in Eq. (9.1.5) and by comparing the coefficients ofε2, we have
m2 f 2 1 +λg2
1 = 0 (9.2.5)
u(x, t) = ep(x±t)
(9.2.7)
The fig. given below shows the 3-D and 2-D images of one-soliton.
93
0.5
1.0
1.5
9.3 Two Soliton Solution
g= εg1+ ε2g2, f = 1+ ε f1+ ε2 f2 (9.3.1)
By using Eq. (9.3.1) in Eq. (9.1.3) and considering the coefficients ofε2, we get
g2tt + f1g1tt −2 f1tg1t + f1ttg1−g2xx− f1g1xx+2 f1xg1x− f1xxg1 = 0 (9.3.2)
Here, we take
s=±r (9.3.4)
By using Eq. (9.3.1) in Eq. (9.1.5) and by comparing the coefficients ofε4, we have
m2 f 2 2 +λg2
2 = 0 (9.3.5)
u(x, t) = ep(x±t)+er(x±t)
1± i √
√ λ
m er(x±t) (9.3.7)
The figure given below shows the 3-D and 2-D images of two-soliton interaction.
-10 -5
0.5
1.0
1.5
9.4 Three Soliton Solution
For three soliton solution take
g= εg1+ ε2g2+ ε3g3, f = 1+ ε f1+ ε2 f2+ ε3 f3 (9.4.1)
By using Eq. (9.4.1) in Eq. (9.1.3) and considering the coefficients ofε3, we get
g3tt + f1g2tt + f2g1tt −2 f1tg2t −2 f2tg1t + f1ttg2+ f2ttg1−g3xx
− f1g2xx− f2g1xx+2 f1xg2x+2 f2xg1x− f1xxg2− f2xxg1 = 0
(9.4.2)
v=±q (9.4.4)
By using Eq. (9.4.1) in Eq. (9.1.5) and by comparing the coefficients ofε6, we have
m2 f 2 3 +λg2
3 = 0 (9.4.5)
1± i √
√ λ
(9.4.7)
The fig. given below shows the 3-D and 2-D images of three-soliton interaction.
-10 -5
0.5
1.0
1.5
Figure 9.3:λ=1.0,m=1.0,p= 0.05+ i, q= 0.05+ i, r = 0.05+ i
9.5 N Soliton Solution
u(x, t) = ep1(x±t)
1± i √
Three soliton solution
u(x, t) = ∑3
1± i √
N soliton solution
u(x, t) = ∑N
1± i √
These results have been submitted [41].
97
In this dissertation, we firstly obtained rational function, confluent hypergeometric and soli-
tary wave solutions for CGLQ model by using ETEM and antikinkand some new solitary
wave solutions by using METFM. The results are new and very encouraging for future re-
search. Secondly, we obtained different kinds of soliton solutions for the time fractional
perturbed NLSE by using ETEM with different nonlinearitieslike Kerr, power and log law.
Thirdly, we got the combo soliton and dipole soliton for CGLQmodel by using ansatz
method of Li [27] and ansatz method of Choudhuri [16] respectively. Next, we obtained
the Jacobi elliptic, periodic bright and singular soliton by using the ETEM with Kerr law
nonlinearity. We obtained bright, dark and kink solitons for cubic-quartic NLSE. We used
two methods to obtain bright soliton by the aid of SCM under Kerr law, power law and anti-
cubic law nonlinearities and we used Bernoulli equation approach to get dark soliton with
parabolic and cubic quintic law nonlinearities. In the last, we obtained multiple solitons
solutions for NLTE and for PFE by using HBM. All the obtained results are new, novel and
these results may be used in telecommunication and optical fiber industry.
10.2 Open Problems
Open Problem: 1
Find the interaction properties and conservation laws of CGLQ model with the help of
HBM and multiplier approach respectively.
Open Problem: 2
Find the interaction properties and conservation laws for time fractional perturbed NLSE
with the help of HBM and undetermined coefficients approach respectively.
Open Problem: 3
Can we find the conservation laws for paraxial NLSE? Can we obtain the one-soliton trans-
formation and two soliton interaction for normalized NLSE?
99
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