HIROTA BILINEAR COMPUTATION OF MULTI SOLITON SOLUTIONS

KORTEWEG de VRIES EQUATION

ANNIZA BINTI HAMDAN

UNIVERSITI TEKNOLOGI MALAYSIA

HIROTA BILINEAR COMPUTATION OF MULTI SOLITON SOLUTIONS

KORTEWEG de VRIES EQUATION

ANNIZA BINTI HAMDAN

A thesis submitted in partial fulfilment of the requirements for the award of Master of

Science (Mathematics).

Faculty of Science

Universiti Teknologi Malaysia

2014

iv

To my beloved family

A very understanding mother, Jamaliah Hassan

A supportive husband, Mohammad Misrul Ain

and my child, Muhammad Ian Shaheem

To all my dearest friends

and

To my thoughtful supervisor

Assoc. Prof. Dr. Ong Chee Tiong

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ACKNOWLEDGEMENT

First of all, I would like to express my deepest appreciation to my supervisor,

Assoc. Prof. Dr. Ong Chee Tiong for his wonderful guidance and advice. His sharing on

ideas, knowledge and experiences had given me another view of this world which

completes one step achievement in my life.

I am grateful to have a very understanding and supportive mother, Jamaliah

Hassan for always by my side from the beginning of this journey. A special thanks to

my beloved husband for his moral supports and considerations all the way through

completing this research.

Furthermore, I would like to thanks my dear friends, on their sharing ideas in

preparing this thesis writing.

Last but not least, I wish to thank all those who had involved directly or indirectly in

bringing this text successful. Thank you for every single opinions and thoughts that had

been shared with me.

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ABSTRACT

Soliton is the solution of nonlinear partial differential equation that exists due to

the balance between nonlinearity and dispersive effects. The existence of these two

effects in Korteweg de Vries (KdV) equation enables us to obtain solitons solutions. The

purpose of this research is to obtain the multi soliton solutions of KdV equation by using

Hirota bilinear method. This method can produce the explicit expression for soliton

solutions of KdV equation. From these solutions, a general pattern of F function in

Hirota bilinear method is revealed. The amplitude of interacting soliton will determine

the phase shift pattern. Various interactive graphical outputs produced by MAPLE

computer programming can illustrate the solutions of these multi soliton up to eight-

soliton solutions of KdV equation.

vii

ABSTRAK

Soliton adalah penyelesaian bagi persamaan pembezaan separa tak linear yang

wujud akibat keseimbangan di antara kesan tak linear dan penyelerakan. Kewujudan dua

kesan ini dalam persamaan Korteweg de Vries (KdV) membolehkan kita untuk

mendapat penyelesaian soliton. Tujuan kajian ini adalah untuk mendapatkan

penyelesaian multi soliton bagi persamaan KdV dengan menggunakan kaedah Hirota

bilinear. Kaedah ini boleh menghasilkan ungkapan eksplisit bagi penyelesaian soliton

dalam persamaan KdV. Daripada penyelesaian ini, corak umum bagi fungsi F

didedahkan. Amplitud bagi soliton yang berinteraksi akan menentukan corak anjakan

fasa. Pelbagai paparan grafik yang interaktif dihasilkan melalui pengatucaraan komputer

MAPLE dapat memberi ilustrasi penyelesaian multi soliton sehingga lapan soliton

dalam persamaan KdV.

viii

TABLE OF CONTENT

CHAPTER TITLE PAGE

DECLARATION

DEDICATION

ACKNOWLEDGMENT

ABSTRACT vi

ABSTRAK vii

TABLE OF CONTENTS viii

LIST OF FIGURES x

LIST OF TABLES xiii

LIST OF ABBREVIATIONS/SYMBOLS xiv

1 INTRODUCTION

1.0 Introduction

1.1 Preface

1.2 Background of Study

1.3 Problem Statement

1.4 Objectives of Study

1.5 Scope of Study

1.6 Significance of Study

1

1

2

3

4

4

5

2 LITERATURE REVIEWS

2.0 Introduction

2.1 Soliton theory and discovery of Korteweg de Vries

(KdV) equation

6

6

ix

2.2 Methods applied in solving Korteweg de Vries (KdV)

equation in solitons

2.3 Hirota bilinear method

2.4 Solitons Interaction and Phase Shift

9

14

18

3 SOLUTIONS OF KdV EQUATION

3.0 Introduction

3.1 One-Soliton Solution of KdV Equation

3.2 Two-Soliton Solution of KdV Equation

4.2.1 Phase Shift of Two-Soliton Interaction

3.3 Multi Solitons Solutions

3.3.1 Three-Soliton Solution of KdV Equation

3.3.2 Four-Soliton Solution of KdV Equation

3.3.3 Five-Soliton Solution of KdV Equation

3.3.4 Six-Soliton Solution of KdV Equation

3.3.5 Seven-Soliton Solution of KdV Equation

3.3.6 Eight-Soliton Solution of KdV Equation

21

21

26

37

41

43

47

54

63

77

96

4 SUMMARY AND CONCLUSION

4.0 Introduction

4.1 Summary

4.2 Conclusion

4.3 Suggestions and Recommendations

125

125

126

127

REFERENCES

128

x

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 The wave profile of two colliding Korteweg de Vries

(KdV) solitons

10

2.2 Solution of KdV equation for one soliton, 11

2.3 Solution of KdV equation for two soliton,

11

2.4 Solution of KdV equation for three soliton,

and

12

2.5 A soliton collision via KdV equation 19

2.6 Asymptotic structure of M-soliton KdV solution in xt-

plane.

20

3.1 MAPLE coding to solve one-soliton solution of KdV

equation

24

3.2 Location of one-soliton at time, and

.

25

3.3 The 3D plot of one-soliton solution. 25

xi

3.4 MAPLE coding to solve two-solitons solution of KdV

equation.

35

3.5 Location of two-solitons solution at time,

and .

36

3.6 The 3D plot of two-solitons solution. 37

3.7 Phase shift pattern (represent by bold line) of two-

soliton solutions with negative and positive phase shift

respectively

38

3.8 Phase shift pattern of two-soliton interaction 38

3.9 Relationships of F function with phase shift pattern of

two-soliton solutions.

38

3.10 Contour plot of different values of first soliton

amplitude

40

3.11 Number of coefficient in F function according to N-

soliton developing Pascal triangle.

42

3.12 MAPLE coding to solve three-solitons solution of KdV

equation.

45

3.13 Location of three-solitons solution at time,

and .

46

3.14 The 3D plot of three-solitons solution. 47

3.15 MAPLE coding to solve four-solitons solution of KdV

equation.

51

3.16 Location of four-soliton solution at different time,

xii

and . 53

3.17 The 3D plot of four-solitons solution. 53

3.18 MAPLE coding to solve five-solitons solution of KdV

equation.

61

3.19 Location of five-soliton solution at different time,

and .

62

3.20 The 3D plot for five-solitons solution. 63

3.21 MAPLE coding to solve six-solitons solution of KdV

equation.

74

3.22 Location of six-soliton at different time,

and ..

75

3.23 The 3D plot for six-solitons solution. 76

3.24 MAPLE coding to solve seven-solitons solution of KdV

equation.

94

3.25 Location of seven-soliton at different time,

and .

95

3.26 The 3D plot for seven-solitons solution. 96

3.27 MAPLE coding to solve eight-solitons solution of KdV

equation.

122

3.28 Location of eight-soliton at different time,

and .

123

3.29 The 3D plot for eight-solitons solution. 124

xiii

LIST OF TABLES

TABLE NO. TITLE PAGE

3.0 The relationship of solitons amplitude towards its phase

shift in two-soliton solutions.

39

3.1 Combination of coefficients in multi solitons solutions. 43

3.2 The list of conditions test for six-soliton solution. 65

xiv

LIST OF ABBREVIATIONS/SYMBOLS

DNA - Deoxyribo Nucleic Acid.

KdV - Korteweg de Vries.

PDE - Partial Differential Equation.

2D - Two dimensional.

3D - Three dimensional.

- The elongation of the wave function with

variable and .

- Space domain.

- Time domain.

- The height of the peak.

- Speed of the wave.

- Phase shift of soliton.

- The depth of water.

- The gravitational acceleration.

- Amplitude of soliton.

- Existence of soliton wave

xv

- Interaction of two solitons

1

CHAPTER 1

INTRODUCTION

1.0 Introduction

This chapter introduces the topic chosen for the study. It presents the background

of study followed by the problem statement. The objectives of the study are listed and

the scope of the study is also discussed. The chapter ends with the significance of the

study.

1.1 Preface

The existence of solitary waves was first discovered by Scottish engineer, John

Scott Russell in year 1834 (Miin, 2012). It is called as solitary waves as it often occurs

as this single entity and is localized. He was observed this solitary wave phenomenon on

the Edinburgh Glasgow canal and called it ‘great wave translation’. Throughout this

phenomenon, he had performed some laboratory experiments and found that taller

waves travel faster and are narrower (Drazin and Johnson, 1989).

A breakthrough came later in the 19th

century, when Diederik Korteweg and

Gustav de Vries in year 1895 starting a hydrodynamic description, derived from a

nonlinear partial differential equation which had solutions in which the nonlinearity

2

is counterbalanced by a dispersive term which stabilizes the shape of solitary wave,

thereby vindicating John Scott Russell’s observations. The partial differential equation

arrived by Korteweg and de Vries’s reasoning had the same form as one previously

studied by Joseph Boussinesq in year 1871 (Riseborough, 2010).

In addition, Riseborough (2010) also stated that Zabusky and Kuskal had found

an amazing discovery in year 1965 by showing that, in continuum limit, the experiment

conducted by Fermi-Pasta-Ulam in year 1955 yielded the Korteweg de-Vries (KdV)

equation. They had found that although the equation was nonlinear, the solitary waves

described by Korteweg de Vries (KdV) equation appeared as if they did not interact with

each other. That is, after two solitons collide, they emerge with their shape and velocities

unchanged. The only signature of the collision was phase shift. Due to this particle-like

attributes of the wave pulse excitations of the Korteweg de Vries (KdV) equation,

Zabusky and Kuskal first introduced the term “soliton” to describe them.

As solitons can be interacting with each other, thus the study on these interacting

solitons had been conducted extensively by researchers since 1970s. According to Zhao

(2012), the sign and absolute value of velocity determine the propagation direction and

speed of soliton. He found that the propagation direction and behavior of these

interacting solitons followed the asymptotic forms.

1.2 Background of study

As mentioned above, the concept of the soliton was initiated by Zabusky and

Kuskal in year 1965. As the result, theory of the inverse scattering transform was

developed. Apart from that, many numerical studies were used to indicate the soliton

behavior towards nonlinear wave solutions. According to Drazin and Johnson (1989),

here are three main numerical methods which are finite difference method, finite

element method and spectral method.

3

The Korteweg de Vries equation can model the dynamics of solitary waves. The

KdV equation can be expressed as

This is a nonlinear partial differential equation where is the amplitude of the

wave, is the spatial term and , is the time evolution. The term is a dispersive

term and the term is a nonlinear term (Druitt, 2005).

Therefore, this research will investigate the analytical solution of KdV equation

by using Hirota bilinear method for multi soliton solutions. Hirota bilinear method was

introduced by Japanese researcher, Hirota in year 1971. This new direct method was

developed to construct multi soliton solutions to integrable nonlinear evolution equation.

The Hirota bilinear method turned out to be the fastest and easiest way to obtain the

results of KdV equation solution for multi soliton (Hietarinta, 1997).

Nowadays, most of the researchers applied Hirota bilinear method to further

revise on the multi soliton solutions. Yang and Mao (2008) had conducted a study on

soliton solution and interaction property by using the Hirota bilinear method. They had

found that in two solitons interactions, both solitons display the states before collision,

exhibit the impacted and superposed states in the interactive area and after collision the

solitons represent the identical states before collision. They also argued that from the

changes of amplitude and velocity, it shows that solitons interactions do not exchange

their physical quantities but undergo a phase shift.

1.3 Problem Statement

The Korteweg de Vries equation is a nonlinear partial differential equation that

can be solved numerically and analytically. However, to obtain the analytical solutions

of KdV equation is not easy. Several methods had been used earlier to obtain the

solutions of KdV equation. In this research, we will observe the soliton ladder of

4

solutions. We have to produce the permutation parameters of solitons interactions in

order to obtain all the eight soliton solutions of KdV equation. As these solutions are

difficult to calculate manually, thus we need a computer programming tools to derive the

function and produce the various graphical outputs for up to eight-soliton solutions of

KdV equation.

1.4 Objective of Study

The objectives of this research are:

1) to solve KdV equation analytically by using Hirota bilinear method.

2) to obtain the phase shift pattern in multi solitons solution of KdV equation.

3) to obtain the general pattern of F function for multi solitons solution of KdV

equation.

1.5 Scope of Study

This study focuses on the multi soliton solutions of KdV equation by Hirota

bilinear method that transforms the nonlinear partial differential equation into bilinear

equation via the transformation of dependent variable. Meanwhile, the explicit

expression of multi soliton solutions for KdV equation is obtained. The solutions will be

investigated up to eight-solitons and computer programming; MAPLE will be used to

obtain the graphical outputs of multi soliton solutions for KdV equation. The solution is

discussed until eight-soliton as KdV equation mostly applied for shallow water waves

which consist of ripple. Eight-soliton solutions already give a long calculation but it is

sufficient to be applied for multi soliton solutions. Besides that, the phase shift pattern of

two-soliton interaction is discussed in order to observe the relationship of amplitude

towards soliton phase shift. The general pattern of F function is explored as well as to

5

help us in determining the accurate function applied in solving multi soliton solutions of

KdV equation.

1.6 Significance of Study

In detail, this study will discuss the multi soliton solutions of KdV equation up to

eight-solitons. The Hirota bilinear method will be used to obtain these solutions of KdV

equation which developed from the concept of soliton. Thus, it can be applied in many

areas such as shallow water and deep water waves, fibre optics, DNA and protein, and

biological model.

The characteristics of KdV equation itself which are nonlinear and dispersive

could gives important thoughts in solving a critical phenomenon such as tsunami. The

balancing of the nonlinearity and dispersion effects in tsunami phenomenon is important

to understand the occurrence of wave’s dispersion near beaches as it travels in a long

distance. The travelling of tsunami waves which behaves like soliton can be modelled as

KdV equation.

Through this research, we will able to obtain the solution of multi soliton up to

eight-solitons which shows the soliton ladders of KdV equation by using Hirota bilinear

method. The phase shift pattern and general pattern of F function will gives another

thoughts which convenience others to solve multi soliton solutions.

128

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Physics, University of Turku, FIN-20014 Turku, Finland.

Marchant, T.R. and Smyth, N.F.(1996). Soliton interaction for extended Korteweg-de

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Mat Zin, S., Ariffin, W.N.M, and Hashim Ali, S.A. (2012). Simulation of Korteweg de

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Equation: Six Solitons. Master Thesis. Universiti Teknologi Malaysia. Skudai.

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