Numerical Solution of Hirota Equation

Kingdom
of Saudi Arabia
Ministry of Higher Education
Umm AL-Qura University
Faculty of Applied Science for Girls
Department of Mathematics
Numerical Solution of Hirota Equation
A Thesis
Submitted In Partial Fulfillment of the Requirements
For The Master’s Degree in Science
Pure Mathematics (Numerical Analysis)
By
Weam Ghazi Al.Harbi
Supervised By
Dr. Mohammad Said Hammoudeh Ismail
Associate Professor of Numerical Analysis
Department of Mathematics
Faculty of Science
King Abdulaziz University
1430-2009

المملكة العربية السعودية
وزارة التعليم العالي
أم القرى جامعة
العلوم التطبيقية فرع الطالبات كلية
الرايضيات
قسم
العليا ادلراسات
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Numerical Solution of Hirota Equation
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أستاذ التحليل العددي المشارك بجامعة المكل عبد العزيز بجده
كلية العلوم قسم الرايضيات
٢٠٠٩/١٤٣٠

ABSTRACT
Numerical Solution of Hirota
Equation
The aim of this thesis is to solve numerically the Hirota equation using
finite difference method.
This dissertation consists of three chapters and a list of references.
In Chapter 1: we present in details, this equation and the exact
solution, also we study its conserved quantities. The solution of the block
penta-diagonal system and block septa-diagonal system are derived. We
described the fixed point method and Newton’s method for solving the
nonlinear system.
In Chapter 2: we solve the Hirota equation numerically by using
Crank-Nicolson method. The accuracy of the resulting scheme is second
order in space and time and is unconditionally stable. Newton’s and fixed
point methods are used for solving the nonlinear system. Also we use a
linearized implicit technique.
We give the some numerical examples to show that this method is
conserving the conserved quantities. We give some experiments, like,
single soliton and collision of two solitons.
In Chapter 3: we use the same approach as we did in the previous
chapter, but this time we are approximating the space derivatives in the
Hirota equation by a fourth order replacement. The accuracy of the
resulting scheme is fourth order in space and second order in time and is
unconditionally stable. Newton’s and fixed point methods are used for
solving the nonlinear system. Also we use a linearized implicit technique.
We give the some numerical examples to show that this method is
conserving quantities. We give some experiments like the one mentioned
in chapter 2.
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.
:
:
.
(Block Penta-diagonal System)
.(Block Septa-diagonal System)
.
:
.
t,x
.
:
x
.
t
x
.
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Introduction
In this work, we aim to solve numerically the Hirota equation [11]
∂ u
+
∂t
3
2 ∂u
∂ u
α u + γ = 0
(1)
∂x
∂x
3
3
where u is a complex valued function of the spatial coordinate x and the time t, α
and γ are positive real constants. This equation is an integrable equation which has a
number of physical applications, such as the propagation of optical pluses in nematic
liquid crystal waveguides. The Hirota equation is closely related to both the nonlinear
Schrodinger (NLS) and modified Korteweg-de Vries (mKdV) equations, as it is
complex generalization of the mKdV equation and it is a part of the NLS hierarchy of
the integrable equation. Also, its soliton solution has a very similar form to the NLS
soliton. The Hirota equation (1) has a two-parameter soliton family, with amplitude
and velocity. The exact solution of Hitora equation (1) is
u
( x, t) β sec h[ κ( x − s − vt)
] exp( iϕ)
= , (2)
2γ
β = κ , φ = a ( x −bt − s ) , (3)
α
v a b a
2 2 2 2
= γ ( κ − 3 ) , = γ (3 κ − ) , (4)
β is the amplitude of the wave, κ is related to the width of the wave envelope and
v is the velocity. The parameter a is the wave number of the phase and b is related to
the frequency of the phase. Also the solution is at x
has the conserved quantities
= s at t = 0 .The Hirota equation
∞
∞
2
⎛ α 4 2 ⎞
1
= ∫ = constant,
2
= ∫ ⎜ −
x ⎟ = constant
2
−∞
−∞ ⎝
⎠
(5)
I u dx I u u dx
The Hirota equation (1) has been solved analytically by sine-cosine and tanh methods
by Wazwaz [33] and showed that this equation admits sech-shaped soliton solutions
whose amplitudes and velocities are free parameters, and tanh solution (kink type).
Hirota method can be used [9]. To avoid complex computation[13,14,15], we assume
where u ( x t ),
u ( x,
t )
coupled system
( ) ( ) ( )
2
u x , t = u x , t + iu x , t , i = − 1
(6)
1 2
1
,
2
are real functions. This will reduce Hirota equation to the
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3
∂u1 ∂u1 ∂ u1
+ 3 αf ( u1 , u
2 ) + γ = 0,
3
∂t ∂x ∂x
3
∂u 2
∂u 2
∂ u
2
+ 3 α f ( u1 , u
2 ) + γ = 0,
3
∂t ∂x ∂x
(7)
where
2 2
( u , u ) = u .
f +
1 2 1
u2
The numerical solution of nonlinear wave equations has been the subject of many
studies in recent years. Although many numerical schemes have been proposed for
some well- known integrable equations, such as the Korteweg-de Vries (KdV)
equation [5,25,29,30] and the nonlinear Schrodinger equation [28]. Numerical
solution of coupled differential is a fertile area, as an example, the coupled nonlinear
Schrodinger equation admits soliton solution and it has many applications in
communication and optical fibers, this system has been solved numerically by Ismail
[13,14,15,17], the coupled Korteweg-de Vries equation [7,8,18,34]. The nonintegrable
2
variant of Hirota in which the nonlinear term in (1) is replaced by( u u ) , is solved
numerically by [19,23].
x
This thesis consists of three chapters. In Chapter 1, we present in details, this
equation and the exact solution, also we study its conserved quantities. The solution
of the block penta-diagonal system and block septa-diagonal system are derived. We
described the fixed point method and Newton's method for solving the nonlinear
system. In Chapter 2, two numerical schemes are derived for solving Eq. (1), linear
and nonlinear implicit schemes, both are of second order in both direction and they
are unconditionally stable. The implicit nonlinear scheme produced a block nonlinear
penta diagonal system solved by Newton's and fixed point methods. Also we use a
linearized implicit technique for solving the linear penta diagonal scheme. numerical
results for single soliton and the interaction of two solitons are given. In Chapter 3,
two numerical schemes are derived for solving Eq. (1), linear and nonlinear implicit
schemes, both are of fourth order in space and second order in time and they are
unconditionally stable. The implicit nonlinear scheme produced block nonlinear septa
diagonal system solved by Newton's and fixed point methods. Also we use a
linearized implicit technique for solving the linear septa diagonal scheme. numerical
results for single soliton and the interaction of two solitons are given.
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TABLE OF CONTENTS
ABSTRACT ……………………………………………………………...………...
ii
ABSTRACT (Arabic) ………………………………………………………...…. iii
DEDICATION ……………….. ……………………………………………………... iv
ACKNOWLEDGEMENTS ……………………………………………………….... v
LIST OF TABLES …………………………………………………………………… viii
LIST OF FIGURES………………………………………………………………….. ix
I THE HIROTA EQUATION ………………………………………………… 1
1.1 Introduction ……………………………………………………………………... 1
1.2 The Hirota Equation ………………………………….……………………….… 1
1.3 Conserved Quantities …………………………………………………………… 2
1.4 Solution of Block Penta-diagonal System………………………………………. 4
1.5 Solution of Block Septa-diagonal System ……………………………………… 7
1.6 Nonlinear System ……………………………………………………………….. 12
1.6.1 Newton’s Method ………………………………………………………….. 13
1.6.2 Fixed Point Method ………………………………………………………… 14
II NUMERICAL SOLUTION OF HIROTA EQUATION (SECOND
ORDER) ………………………………………………………....………………….. 17
2.1 Introduction ………………………………………………………....................... 17
2.2 Numerical Method …………………………………………………………….…. 17
2.3 Iterative technique……………………………………………………………….... 20
2.3.1 Newton.s Method …………………………………………………………... 20
2.3.2 Fixed point method ………………………………………………………… 23
2.3.3 Accuracy of the scheme ……………………………………………………. 26
2.3.4 Stability of the Scheme .……………………………………………………. 28
2.4 linearly implicit method .…………………………………………………………. 30
2.4.1 Accuracy of the scheme ……………………………………………………. 34
2.4.2 Stability of the Scheme …………………………………………………….. 35
2.5 Numerical Results ………………………………………………………………... 37
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2.5.1 Single soliton ………………………………………………………………. 37
2.5.2 Collision of Two solitons …………………………………………………... 45
III NUMERICAL SOLUTION OF HIROTA EQUATION (FOURTH
ORDER) ……………………………………………………………………………... 55
3.1 Introduction ………………………………………………………………………. 55
3.2 Numerical Method ………………………………………………………………...55
3.3 Iterative technique …………………………………………………………………58
3.3.1 Newton’s Method …………………………………………………………... 58
3.3.2 Fixed point method ………………………………………………………… 61
3.3.3 Accuracy of the scheme ……………………………………………………. 65
3.3.4 Stability of the Scheme …………………………………………………….. 67
3.4 linearly implicit method ………………………………………………………….. 70
3.4.1 Accuracy of the scheme ……………………………………………………. 75
3.4.2 Stability of the Scheme …………………………………………………….. 77
3.5 Numerical Results ………………………………………………………………... 79
3.5.1 Single soliton ………………………………………………………………. 79
3.5.2 Collision of Two solitons …………………………………………………... 86
Conclusion………………………………………………………………….……….. 96
References…………………………………………………………………………… 97
Arabic Summary
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