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 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

v 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.

vi 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 eightsoliton 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 xtplane. 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 twosoliton 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

- Interaction of two solitons xv

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 19 th 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 REFERENCES Backlund, A. and Weston, D. (2010). Analytical and Numerical Study of Soliton Collisions. Degree Project in Engineering Physics, First Level. Royal Institute of Technology. Brauer, K. (2000). The Korteweg-de Vries Equation:History, exact solutions, and graphical representation. University of Osnabrück. Germany. Drazin, P.G. and Johnson, R.S. (1989). Solitons: an introduction. Cambridge University Press. Dimakis, A. and Muller-Hoissen, F. (2013). KdV Solitons Interactions: A Tropical View. Physics and Mathematics of Nonlinear Phenomena 2013. Cornell University Library. Druitt, F. (2005). Hirota s Direct Method and Sato s Formalism in Soliton Theory. Honour Thesis. University of Melbourne. Australia. Eilbeck, C. (1995). John Scott Russell and the solitary wave. Heriot-Watt University. Goldstein, P.P.(2007). Hints on the Hirota Bilinear Method. Acta Physica Polonica A. Andrzej Soltan Institute. Poland. Hietarinta, J. (2008). Hirota s bilinear method and intergrability. Department of Physics, University of Turku, FIN-20014 Turku, Finland. Marchant, T.R. and Smyth, N.F.(1996). Soliton interaction for extended Korteweg-de Vries equation. IMA Journal of Applied Mathematics,157-176. Oxford University Press.

129 Mat Zin, S., Ariffin, W.N.M, and Hashim Ali, S.A. (2012). Simulation of Korteweg de Vries Equation. International Journal of Modern Physics: Conference Series, 574-580. World Scientific Publishing Company. Miin, Y.H. (2012). Korteweg-de Vries Equation in the Modelling of Tsunami Wave. Undergraduate Thesis. Universiti Teknologi Malaysia, Skudai. Ong, C.T.,(1993). Various Aspects of Soliton Interactions. Master Degree Thesis. Universiti Teknologi Malaysia, Skudai. Riseborough, P.S. (2010). The Korteweg-de Vries equation: Its place in the development of nonlinear physics. Philosiphical Magazine. Taylor and Francis. Siti Zarifah Binti Sarif (2012). Multi Solitons Solutions of Korteweg de Vries (KdV) Equation: Six Solitons. Master Thesis. Universiti Teknologi Malaysia. Skudai. Peterson, P.(2001). Construction and decomposition of multi-soliton solutions of KdV type equations. Research Report Mech. Institute of Cybernetics. Talinn University of Technology. Wadati, M.(2001). Introduction to solitons. Indian Academy of Sciences, Vol. 57, pp.841-847. Pramana-Journal of Physics. Wazwaz, A.M. (2008). The Hirota s direct method for multiple-soliton solutions for three model equations of shallow water waves. Applied Mathematics and Computation 201, 489-503. Science Direct. Yang, J.R., and Mao, J.J.(2008). Soliton solution and interaction property for a coupled modified Korteweg-de Vries (mkdv) system. Chinese Physics B. IOP Publishing Ltd.

130 Zhao, H.Q.(2012). Soliton propagation and collision in a variable-coefficient coupled Korteweg-de Vries Equation. The Europian Physical Journal B. Springer-Verlag.