NONLINEAR WAVE MODULATION IN A FLUID-FILLED THIN … · dan bendalir, modulasi gelombang tak linear dalam medium komposit dengan penghampiran gelombang panjang telah dilakukan dengan

NONLINEAR WAVE MODULATION IN A FLUID-FILLED

THIN ELASTIC STENOSED ARTERY

CHOY YAAN YEE

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

APRIL 2014

v

ABSTRACT

The aim of this research is to study the modulation of nonlinear wave in

a stenosed artery filled with blood. The artery is simulated as an incompressible,

stenosed, thin-walled elastic tube. By considering the blood as an incompressible

average inviscid fluid, average viscous fluid, full inviscid fluid and full viscous

fluid, four mathematical models of nonlinear wave modulation in a stenosed

artery are formed. The average inviscid and average viscous fluids correspond

to the approximate equations of inviscid and viscous fluids. The full inviscid

and full viscous fluids correspond to the exact equations of inviscid and viscous

fluids. By solving a set of nonlinear evolution equations of tube and fluid, the

modulation of nonlinear waves in such a composite medium in the long wave

approximation is carried out using the reductive perturbation method. The

evolution equation is obtained as the nonlinear Schrodinger equation with variable

coefficient (NLSVC) for the models of average inviscid and full inviscid fluids. The

governing equation for the models of average viscous and full viscous fluids are the

dissipative nonlinear Schrodinger equation with variable coefficient (DNLSVC).

By seeking the progressive wave solutions to the NLSVC and DNLSVC, it is

observed that the solution of NLSVC results in a dark-soliton whereas the solution

of DNLSVC produces a dark-soliton propagating with decaying amplitude as

the viscous effect increases. In order to illustrate the applicability of these four

models, quantitative analysis is performed for the variation of wave frequency,

wave speed and severity of stenosis of the wave propagation in a stenosed artery

and normal artery.

vi

ABSTRAK

Kajian ini bertujuan untuk mengkaji modulasi gelombang tak linear dalam

arteri berstenosis yang dipenuhi dengan darah. Arteri ini disimulasi sebagai tiub

elastik berdinding nipis yang tak mampat dan berstenosis. Dengan mengambil

kira darah sebagai bendalir tak mampat yang tak likat purata, bendalir likat

purata, bendalir tak likat penuh dan bendalir likat penuh, empat model matematik

untuk modulasi gelombang tak linear dalam arteri berstenosis terbentuk. Bendalir

tak likat purata dan bendalir likat purata merujuk kepada persamaan anggaran

untuk bendalir tak likat dan bendalir likat. Bendalir tak likat penuh dan bendalir

likat penuh merujuk kepada persamaan tepat untuk bendalir tak likat dan bendalir

likat. Dengan menyelesaikan satu set persamaan evolusi tak linear untuk tiub

dan bendalir, modulasi gelombang tak linear dalam medium komposit dengan

penghampiran gelombang panjang telah dilakukan dengan menggunakan kaedah

usikan penurunan. Persamaan evolusi diperoleh sebagai persamaan tak linear

Schrodinger dengan pekali pembolehubah (NLSVC) untuk model bendalir tak

likat purata dan bendalir tak likat penuh. Persamaan untuk model bendalir likat

purata dan bendalir likat penuh adalah persamaan tak linear disipatif Schrodinger

dengan pekali pembolehubah (DNLSVC). Dengan mencari penyelesaian gelombang

progresif bagi NLSVC dan DNLSVC, dapat diperhatikan bahawa penyelesaian

untuk NLSVC menghasilkan soliton gelap manakala penyelesaian untuk DNLSVC

menghasilkan soliton gelap yang merambat dengan amplitud yang berkurangan

apabila kesan kelikatan meningkat. Bagi menggambarkan aplikasi keempat-empat

model ini, analisis kuantitatif telah dilakukan bagi variasi frekuensi gelombang,

kelajuan gelombang serta kesan ketara stenosis untuk perambatan gelombang

dalam arteri berstenosis dan arteri normal.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xi

LIST OF FIGURES xii

LIST OF SYMBOLS xix

LIST OF APPENDICES xxiv

1 INTRODUCTION 1

1.1 Preview 1

1.2 Background of the Problem 6

1.3 Statement of the Problem 8

1.4 Objectives of the Study 9

1.5 Scope of the Study 9

1.6 Significance of the Study 10

1.7 Methodology 10

1.8 Outline of Thesis 11

viii

2 LITERATURE REVIEW 12

2.1 The Discovery of Solitary Waves and theKorteweg-de Vries (KdV) Equation 12

2.2 Review on the Nonlinear Schrodinger(NLS) Equation 19

2.2.1 Analytical Solution of the NLSEquation (Free System) 20

2.2.2 Properties of the Solution ofthe NLS Equation 21

2.3 Nonlinear Wave Modulation In An Elastic Tube 27

2.4 Conclusion 31

3 THE EQUATION OF A STENOSEDELASTIC TUBE 32

3.1 Introduction 32

3.2 Derivation of the Equation of a StenosedElastic Tube 33

3.3 Conclusion 43

4 NONLINEAR WAVES IN AN AVERAGEINVISCID FLUID CONTAINED IN ASTENOSED ELASTIC TUBE 44

4.1 Introduction 44

4.2 Equations of an Average Inviscid Fluid 46

4.3 Non-dimensionalization 47

4.4 Nonlinear Wave Modulation 48

4.5 Solution of the Field Equations 51

4.5.1 The Solution of O(ε) -order Equations 51

4.5.2 The Solution of O(ε2 ) -order Equations 53


4.6 Progressive Wave Solution 59

4.7 Analytical Results and Discussions 61

4.8 Conclusion 73

ix

5 NONLINEAR WAVES IN AN AVERAGEVISCOUS FLUID CONTAINED IN ASTENOSED ELASTIC TUBE 76

5.1 Introduction 76

5.2 Equations of an Average Viscous Fluid 78






5.8 Conclusion 108

6 NONLINEAR WAVES IN AN INVISCIDFLUID CONTAINED IN A STENOSEDELASTIC TUBE 111

6.1 Introduction 111

6.2 Equations of an Inviscid Fluid andBoundary Conditions 112









6.8 Conclusion 139

x

7 NONLINEAR WAVES IN A VISCOUSFLUID CONTAINED IN A STENOSEDELASTIC TUBE 142


7.2 Equations of a Viscous Fluid andBoundary Conditions 143









7.8 Conclusion 173

8 CONCLUSIONS AND RECOMMENDATIONS 177


8.2 Future Research 186

REFERENCES 188

Appendices A-E 196-220

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

5.1 Comparison of fluid’s viscosity, ν = 1 andν = 100 corresponding to the radial displacement,|U(t, z)| at certain space z and time t. 106

7.1 Comparison of fluid’s viscosity, ν = 1 andν = 100 corresponding to the radial displacement,|U(t, z)| at certain space z and time t. 171

8.1 Mathematical equations for the different types offluid flow in the stenosed elastic tube. 179

xii

LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Comparison between a healthy artery anda stenosed artery. 7

2.1 Schematic picture of the time evolution of thewater wave, driven by a piston moving downwardsor upwards (Dauxois and Peyrard, 2006). 12

2.2 A solitary wave (Debnath, 2007). 14

2.3 The solution of the periodic boundary-valueproblem for the KdV equation: (a) initial profileat t = 0, (b) profile at t = 1

πand (c) profile at

t = 3.6π

(Debnath, 2007). 15

2.4 Before interaction at t = −10. 16

2.5 Full interaction at t = 0. 17

2.6 After interaction at t = 10. 17

2.7 Self-modulation of plane wave. The dashed lineand solid line represented the envelope waveand travelling wave, respectively (Dauxois andPeyrard, 2006). 19

2.8 An envelope soliton (Debnath, 2005). 21

2.9 The envelope of the Ma-soliton for threedifferent time profiles (Grimshaw et al., 2010). 22

2.10 The bi-soliton for the Davey-Stewartsonequation with different time profiles(Ohta and Yang, 2013). 23

2.11 A dark soliton (Dauxois and Peyrard, 2006). 24

3.1 Situations that cause obstruction of a blood flow(Mazumdar, 1992). 33

3.2 The geometry of the stenosed tube in variousconfigurations. 35

xiii

3.3 Forces acting on the small tube element. 36

4.1 The modulus of the solution (4.61) versusspace τ for different travelling wave profile ξat δ = 0.30. 63

4.2 The modulus of the solution (4.61) versustravelling wave profile ξ for different space τat δ = 0.30. 64

4.3 The modulus of the solution (4.62) versusspace z for different time t at δ = 0.30. 64

4.4 The progressive wave solution (4.61) for the NLSequation with variable coefficient (4.49) versusspace τ for different travelling wave profile ξwhen (a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3in the presence of stenosis. 67

4.5 The progressive wave solution for the NLSequation with variable coefficient versus space τfor different travelling wave profile ξ when(a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3 in theabsence of stenosis. 68

4.6 The trajectory corresponding to the NLS equationwith variable coefficient (4.61) for slowvariables (ξ, τ). 69

4.7 The 3D-plot of the solution of the NLS equationwith variable coefficient (4.61) versus space τfor different travelling wave profile ξ atsharpness of the stenosis function δ = 0.30. 70

4.8 The trajectory corresponding to the NLS equationwith variable coefficient (4.62) for fastvariables (t, z). 71

4.9 The 3D-plot of the solution of the NLS equationwith variable coefficient (4.62) versus space zfor different time t at sharpness of thestenosis function δ = 0.30. 71

4.10 The wave speed, vp of the NLS equation withvariable coefficient (4.64). 72


5.2 The modulus of the solution (5.59) versustravelling wave profile ξ for different space τ

xiv

at δ = 0.30. 92

5.3 The comparison between (a) solution of the NLSequation with variable coefficient (4.61) and (b)solution of the dissipative NLS equation withvariable coefficient (5.59) at certaintravelling wave profile, ξ. 94

5.4 The comparison between (a) solution of the NLSequation with variable coefficient (4.61) and (b)solution of the dissipative NLS equation withvariable coefficient (5.59) at certain space, τ . 95

5.5 The modulus of the solution (5.61) versusspace z for different time t at δ = 0.30when ν = 1. 96


5.7 The comparison between (a) solution of the NLSequation with variable coefficient (4.62) and (b)solution of the dissipative NLS equation withvariable coefficient (5.61) at certain time, t. 97

5.8 The progressive wave solution (5.59) for thedissipative NLS equation with variablecoefficient (5.30) versus space τ for differenttravelling wave profile ξ when (a)ξ = 0, (b)ξ = 1,(c)ξ = 2 and (d)ξ = 3 in the presence of stenosis. 99

5.9 The progressive wave solution for the dissipativeNLS equation with variable coefficient versusspace τ for different travelling wave profile ξwhen (a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3in the absence of stenosis. 100

5.10 The trajectory corresponding to the dissipativeNLS equation with variable coefficient (5.59) forslow variables (ξ, τ). 102

5.11 The 3D-plot of the solution of the dissipative NLSequation with variable coefficient (5.59) versusspace τ for different travelling wave profile ξ atδ = 0.30. 102

5.12 The trajectory corresponding to the dissipative NLSequation with variable coefficient (5.61) forfast variables (t, z) when ν = 1. 103

5.13 The 3D-plot of the solution of the dissipative NLS

xv

equation with variable coefficient (5.61) versusspace z for different time t at δ = 0.30when ν = 1. 104


5.15 The 3D-plot of the solution of the dissipative NLSequation with variable coefficient (5.61) versusspace z for different time t at δ = 0.30when ν = 100. 105

5.16 The wave speed, vp of the dissipative NLSequation with variable coefficient (5.65)when ν = 1. 106




6.3 The modulus of the solution (4.62) versusspace z for different time t at δ = 0.30. 129

6.4 The comparison between (a) average inviscid fluidand (b) full inviscid fluid flowing through thestenosed tube at certain travelling wave profile, ξ. 130

6.5 The comparison between (a) average inviscid fluidand (b) full inviscid fluid flowing through thestenosed tube at certain time, t. 132

6.6 The progressive wave solution (4.61) for the NLSequation with variable coefficient (6.50) versusspace τ for different travelling wave profile ξwhen (a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3in the presence of stenosis. 133

6.7 The progressive wave solution for the NLSequation with variable coefficient versus space τfor different travelling wave profile ξ when(a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3in the absence of stenosis. 134

xvi

6.8 The trajectory corresponding to the NLS equationwith variable coefficient (6.50) for slowvariables (ξ, τ). 135

6.9 The 3D-plot of the solution of the NLS equationwith variable coefficient (6.50) versus space τfor different travelling wave profile ξ at δ = 0.30. 136

6.10 The trajectory corresponding to the NLS equationwith variable coefficient (6.50) for fastvariables (t, z). 137

6.11 The 3D-plot of the solution of the NLS equationwith variable coefficient (6.50) versus space zfor different time t at δ = 0.30. 137

6.12 The wave speed, vp of the NLS equation withvariable coefficient (6.50). 138



7.3 The comparison between (a) average viscous fluidand (b) full viscous fluid flowing through thestenosed tube at certain travelling wave profile, ξ. 159

7.4 The comparison between solution of (a) the NLSequation with variable coefficient (6.50) and(b) the dissipative NLS equation withvariable coefficient (7.39) at certaintravelling wave profile, ξ. 160

7.5 The comparison between solution of (a) the NLSequation with variable coefficient (6.50) and(b) the dissipative NLS equation withvariable coefficient (7.39) at certain space, τ . 161



7.8 The comparison between (a) average viscous fluidand (b) full viscous fluid flowing through the

xvii

stenosed tube at certain time, t. 163

7.9 The comparison between solution of (a) the NLSequation with variable coefficient (6.50) and(b) the dissipative NLS equation withvariable coefficient (7.39) at certain time, t. 164

7.10 The progressive wave solution (5.59) for thedissipative NLS equation with variablecoefficient (7.39) versus space τ for differenttravelling wave profile ξ when (a)ξ = 0, (b)ξ = 1,(c)ξ = 2 and (d)ξ = 3 in the presence of stenosis. 165

7.11 The progressive wave solution for the dissipativeNLS equation with variable coefficient versusspace τ for different travelling wave profile ξwhen (a)ξ = 0, (b)ξ = 1, (c)ξ = 2 and (d)ξ = 3in the absence of stenosis. 166

7.12 The trajectory corresponding to the dissipative NLSequation with variable coefficient (7.39) forslow variables (ξ, τ). 167

7.13 The 3D-plot of the solution of the dissipative NLSequation with variable coefficient (7.39) versusspace τ for different travelling wave profile ξ atδ = 0.30. 168






7.19 The wave speed, vp of the dissipative NLSequation with variable coefficient (7.39)

xviii

when ν = 100. 172

8.1 Comparison of wave modulation in differenttypes of fluid in the stenosed tube. 182

8.2 Variation of wave speed, vp for differentsharpness of the stenosis function, δ. 185

xix

LIST OF SYMBOLS

A - inner cross-sectional area

a - acceleration of the tube

c.c. - complex conjugate of the corresponding expressions

c0 - Moens-Korteweg wave speed

c∗kl - Finger deformation tensor

dkl - deformation rate tensor

dSZ - arclength along the meridional curve in the nature state

ds0z - arclength along the meridional curve after static deformation

dsz - arclength along the meridional curve after final deformation

dSΘ - arclength along the circumferential curve in the nature state

ds0θ - arclength along the circumferential curve after static

deformation

dsθ - arclength along the circumferential curve after finaldeformation

er - unit base vector in radial component

eθ - unit base vector in circumferential component

ez - unit base vector in axial component

F - total force acting to the tube element

FkK - deformation gradient

f ∗(z∗) - dimensional function characterizes the axisymmetric bumpon the surface of the arterial wall

f(z) - stenosis function after static deformation in spatial physicalcoordinates

H - thickness of elastic tube before deformation

h - thickness of elastic tube after deformation

h(τ) - stenosis function after static deformation in temporal

xx

stretched coordinates

I - identity operator

I1 - first invariant of Finger deformation tensor

I2 - second invariant of Finger deformation tensor

I3 - third invariant of Finger deformation tensor

i - imaginary number

ı - unit vector codirectional with the x axis

- unit vector codirectional with the y axis

k - wave number

m - mass of the tube

n - unit exterior normal vector along the deformed meridionalcurve

P - fluid reaction force acting per unit deformed area

P ∗0 (Z∗) - dimensional static inner pressure

P ∗ - dimensional fluid pressure in the inner surface of the tube

P ∗r - dimensional fluid reaction force in the radial direction

P ∗z - dimensional fluid reaction force in the axial direction

P - dimensional fluid pressure function

p - non-dimensional fluid pressure in the inner surface of the tube

p - non-dimensional fluid pressure function

pr - non-dimensional fluid reaction force in the radial direction

q - non-dimensional axial fluid velocity

R - position vector in the nature state

R0 - radius of the origin in the nature state

r∗ - dimensional radial coordinates in spatial configuration

rf - final inner radius of the tube after deformation

r - position vector after final deformation

r0 - position vector after static deformation

r0 - radius of the origin after static deformation

xxi

S - spatial smoothing operator

T1 - membrane force acting per unit length along themeridional curve

T2 - membrane force acting per unit length along thecircumferential curve

T1 - scalar membrane force along meridional curve

T2 - scalar membrane force along circumferential curve

T ∗ - axial tethering force

t - unit tangent vector along the deformed meridional curve

t∗ - dimensional time parameter

t - non-dimensional time parameter

tkl - Cauchy stress tensor

U - nonlinear evolution equation or equation considered

u∗ - dimensional radial displacement

u - non-dimensional radial displacement

V - volume of the tube

V ∗r - dimensional radial fluid velocity

V ∗z - dimensional axial fluid velocity

v - non-dimensional radial fluid velocity

vp - wave speed

w∗ - dimensional averaged axial fluid velocity

w - non-dimensional averaged axial fluid velocity

xk - motion

Z∗ - dimensional axial coordinates of a material point in thenature state

z∗ - dimensional axial coordinates after static deformation

z - non-dimensional axial coordinates after static deformation

Greek Symbol

α - material contanst

δ - characterize the sharpness of the stenosis

xxii

δkl - Kronecker delta

ε - small parameter

λ - group velocity

λz - constant axial stretch ratio along the tube axis

λθ - constant circumferential stretch ratio along the tube axis

λ01 - stretch ratio along the meridional curve after static

deformation

λ1 - stretch ratio along the meridional curve after finaldeformation

λ02 - stretch ratio along the circumferential curve after static

deformation

λ2 - stretch ratio along the circumferential curve after finaldeformation

µ1 - coefficient of the dispersive term

µ2 - coefficient of the nonlinear term

µ3 - coefficient of the variable coefficient term

µ4 - coefficient of the dissipative term

µ - shear modulus of the membrane material in the nature state

µv - dynamic viscosity

ν - non-dimensional kinematical viscosity

(= ε2ν in Chapters 5 and 7)

ν - non-dimensional kinematical viscosity

ω - angular frequency

Π - hydrostatic pressure

ρ - density

ρf - mass density of the fluid

ρ0 - mass density of the tube

Σ - strain energy density function of the membrane

τ - temporal variable in stretched coordinates

Θ - circumferential direction in material configuration

xxiii

θ - circumferential direction in spatial configuration

ξ - spatial variable in stretched coordinates

xxiv

LIST OF APPENDICES

APPENDIX TITLE PAGE

A Analytical Solution of the NonlinearSchrodinger Equation 196

B The Averaging Procedure 200

C Derivation of p1, p2 and p3 205

D Derivation of fn(kr) and Fi(2kr) 216

E Journals / Papers Published 220

CHAPTER 1

INTRODUCTION

1.1 Preview

The history of arterial wave mechanics is long and distinguished. The

history of arterial mechanics stretches back 400 years to William Harvey’s research

(1578-1657). Modern cardiovascular physiology and biophysics which began with

Harvey and his discovery of blood circulation was published in a book entitled

Exercitatio Anatomica De Motu Cordis et Sanguinis in Animalibus in 1628. All

the works of Harvey appeared before the microscope was invented.

Isaac Newton (1642-1727) was an English mathematician and physicist.

Newton’s monumental work Principia Mathematica contained the concept of fluid

viscosity. Newton introduced the concept of Newtonian viscosity in which the

shear stress is linearly proportional to strain rate. And this has now become cen-

tral to the consideration of blood flow through artery, pressure-flow relationships

and vascular resistance.

Reverend Stephan Hales (1677-1746) was the first to measure the arterial

pressure in an animal. He carried out an experiment for the response of arterial

pressure due to blood loss and the concept of peripheral resistance.

The development of the theoretical mathematical treatment of arterial

wave mechanics was further investigated in the eighteenth century. Leonhart Eu-

2

ler (1707-1783) made important contributions to the quantitative mechanics in

the cardiovascular system. Euler established one dimensional equations of con-

servation of mass (1.1) and momentum (1.2) for an inviscid fluid in a distensible

tube. These equations were expressed as (Parker, 2009)

(ds

dt

)+

(d · vs

dx

)= 0, (1.1)

2ρ

(dp

dz

)+ v

(dv

dz

)+

(dv

dt

)= 0, (1.2)

where s is the cross-sectional area of the tube, v is the average velocity, p is the

pressure, ρ is the density of blood, t is time and z is the axial distance.

The theory and application of arterial hemodynamics was further ad-

vanced in the nineteenth century. One of the most prominent researchers was

Thomas Young (1773-1829) who was the first to discover the wave speed for in-

compressible fluids contained in an artery in 1808. He studied the relationship

between the elastic properties of arteries and the velocity of propagation of the

arterial pulse. However, his research was obscured and the wave speed was not

formulated until it was rediscovered by German brothers, Ernst-Heinrich Weber

(1795-1878) and Wilhelm Eduard Weber (1804-1891). In 1850, they investigated

the speed of waves in elastic tubes by combining two first-order linear relations

for the elasticity of the tube equation of wave speed, c in the form

c =

√R

2kρ, (1.3)

where ρ is the density of the fluid, R is the radius of the tube and k is the elasticity

coefficient of the tube.

Joseph Fourier (1768-1830) was a French mathematician and physicist

who did not contribute directly to arterial haemodynamics. He asserted that

periodic functions can be expressed as the superposition of an infinite series of

sinusoidal functions and this observation has had such an impact on arterial

haemodynamics.

3

The development by Jean Louis Poiseuille (1799-1869) on his law of flow

in tubes is the next landmark in arterial mechanics. His major contribution was

introducing the relationship between the pressure gradient and dimensions of a

capillary tube. The Poiseuille’s law, Q written by Poiseuille was in the form of

Q =K

′′PD4

L, (1.4)

where K′′

is a function of temperature and the type of liquid flowing, P is the

driving pressure differential, D is the tube diameter and L is the tube length. In

fact, the more usual form of Poiseuille’s law is

Q =πPD4

128µL. (1.5)

The difference between equation (1.4) and (1.5) is that the Poiseuille’s constant,

K′′

is replaced by π/128µ where µ is the fluid viscosity.

In 1878, Adriaan Isebree Moens (1846-1891) investigated the wave propa-

gation in arteries by modifing the Weber equation (1.3). Later, Diederik Johannes

Korteweg (1848-1941) described the relationship between the arterial pulse wave

velocity and Young’s elastic modulus of the arterial wall based upon the Moens’

data, and thus is referred to as the Moens-Korteweg equation for the wave speed

c

c =

√Ehw

2riρ=

√Ehw

Dρ, (1.6)

where E is the Young’s modulus of the wall material, hw is the wall thickness, ρ

is the fluid density, ri and D are the inner radius and diameter of artery, c is the

wave speed.

Otto Frank (1865-1944) was one of the giants of quantitative physiology.

He worked primarily on the cardiovascular system and his work has had a lasting

effect on the practice of cardiology. He was the first to describe the influence

of ventricular dimension. Besides, Frank also contributed to arterial mechanics

4

where he sought a theoretical basis for arterial function with a ”lumped” disten-

sible section of Windkessel and ”lumped” peripheral resistance by ignoring the

wave propagation and wave reflection. Windkessel is a term which has stuck to

the description of the functional cushioning role of the arterial system (Nichols

and O’Rourke, 2005). In 1920, from Moens-Korteweg equation, Frank proposed

the wave speed, c in terms of elasticity

c =

√A

ρ · CA

, (1.7)

where A is the cross-sectional area of the vessel, CA = 4A4ρ

is the inverse of the

distensibility of the vessel and ρ is the density of the fluid. Equation (1.7) is often

called the Newton-Young equation.

Throughout the twentieth century, the study of blood flow in arteries, both

in normal and pathological conditions was notable for the great development

of the theoretical mathematical treatment of fluid dynamics. There has been

numerous investigations on arterial wave mechanics by modern workers and other

researchers. Most of the early works on blood flow in arteries concerned small

amplitude waves ignoring the nonlinear effects and focused on the dispersive

character of waves. For instance, Atabek and Lew (1966) investigated the wave

propagation in an initially stressed elastic tube filled with a viscous incompressible

fluid. The analysis is restricted to tubes with thin walls and to wavelengths

which are very large compared to the radius of the tube. They assumed that the

amplitude of the pressure disturbance is sufficiently small, therefore the nonlinear

terms in the inertia of the fluid are negligible compared to linear ones.

Rudinger (1970) studied the formation of shock wave for the model of

blood flow from the heart to aorta and made use of characteristic method to

solve the nonlinear equations for nonsteady blood flow. His research shows that

the wave pressure increases by increasing the wave velocity. However, the analysis

results for Rudinger’s model can only be applied to nontapered vessels.

5

Anliker et al. (1971) also employed the characteristic method to examine

the wave propagation in arteries. They considered blood as an incompressible

fluid and took the viscosity of fluid into account. The vessel is treated as a

tapered elastic tube. Such a combination of a solid and fluid is assumed to be a

model for blood flow in arteries. By using method of characteristic to transform

the governing equations of the fluid flow, they presented the theoretical results of

pressure profiles in semi-infinite tubes with different wave speed functions, effect

of friction in the laminar model, peripheral resistance and wave front velocity of

the pressure pulse.

In addition, Hashizume (1985) and Yomosa (1987) studied the pressure

wave propagation in a straight thin elastic tube filled with an incompressible

inviscid fluid in the long wave approximation. Utilizing the variable transforma-

tion, they showed that the governing equations can be reduced to the Korteweg-de

Vries (KdV) equation.

Recently in a series of works of Demiray (1996, 1998a, 1999, 2001a-2001c,

2004a-2004c, 2007a-2007c, 2008a-2008d, 2009a-2009d), Antar with Demiray (1999a,

2000) and Bakirtas with Demiray (2005) conducted since 1996 in which they

treated artery as an incompressible, prestressed, isotropic thin elastic, thick elas-

tic, thin viscoelastic, thick viscoelastic, or tapered elastic tube containing with

an incompressible inviscid, viscous or layered fluid as blood. By employing the

exact equations or approximate equations of fluid equations, the propagation of

weakly nonlinear waves in the long-wave approximation was studied. They ob-

tained various nonlinear evolution equations of Korteweg-de Vries, Burgers and

Korteweg-de Vries-Burgers type equations. In all these works, they considered

the arteries as circularly cylindrical long tubes with a constant cross-section.

Meanwhile, Tay and co-workers (2006, 2007, 2008) examined the propa-

gation of nonlinear waves in an incompressible, homogeneous, isotropic and thin-

6

walled elastic tube with a stenosis representing an artery filled with an incom-

pressible inviscid fluid, Newtonian fluid with constant viscosity and Newtonian

fluid with variable viscosity representing blood. Through the use of stretched co-

ordinate of the initial value problem and boundary value problem as well as the

reductive perturbation method, they showed that the governing equations can be

simplified to the forced KdV equation with variable coefficients, forced perturbed

KdV equation with variable coefficient and forced KdVB equation, respectively

for the three different studies.

Over the past 10 years, there has been a surge of interest in wave propa-

gation in the artery with stenosis through the blood. Findings give new insight

which leads on to more precision in the approach to understanding how stenosis

developed in the artery disrupts the blood flow pattern. However, for the work

on wave modulation, most of the studies concerned on the normal straight tube

without stenosis.

1.2 Background of the Problem

Cholesterol is a sterol occurring widely in animal tissues and also in some

plants and algae. It can exist as a free sterol or esterified with a long-chain

fatty acid. It serves principally as a constituent of blood plasma. High levels of

cholesterol in the blood will build up plaque on the inner walls of arteries. The

formation of plaque consists of lipid accumulation. Over time, the decomposition

of plaques will cause the arteries to become thicker and less flexible. This leads to

a narrowing of the arteries which is commonly referred as stenosis (Campbell and

Reece, 2005). Stenosis refers to an abnormal narrowing in a blood vessel or other

tubular organs. The stenosis will further cause atherosclerosis which is also known

as hardening of the arteries. Atherosclerosis restricts the blood flow in arteries

to tissues downstream and consequently, the arteries lose their elastic properties.

7

The appearance of atherosclerosis may lead to cerebral thrombosis, angina, heart

attack, cardiovascular disease and et cetera (Nichols and O’Rourke, 2005). Figure

1.1 illustrates the difference between a healthy artery and a stenosed artery.

Figure 1.1: Comparison between a healthy artery and a stenosed artery.

Atherosclerosis may alter the pattern of blood flow in arteries. Therefore,

the study of arterial wave mechanical properties together with the flow charac-

teristics of blood are indispensable in order to have a better understanding of the

important contributions of haemodynamics to the localization of atherosclerosis.

In recent years, many researchers have carried out studies on arterial wave

modulation with different perspectives related to blood flow. Most of the studies

were concerned with wave modulation in arteries without presence of stenosis (see

Section 2.2). The theoretical studies on blood flow in stenosed arteries are rather

limited since the mathematical modelling is relatively difficult to perform. In ad-

dition, most of the studies focused on wave propagation in the artery (see Section

1.1) rather than the wave modulation. The studies of nonlinear wave modulation

in a fluid-filled prestressed thin elastic tube with a symmetrical stenosis are rather

limited in the literature.

8

1.3 Statement of the Problem

The motivation of this research stems from the problem of modelling the

nonlinear wave in the stenosed artery to study the modulation of wave in the

stenosed artery filled with blood. In a realistic situation, the model of blood flow

in the artery with stenosis is not easy to generate due to the varying temperature,

concentration of fats in the blood, pulsating pressure as well as the multiplicity

of branches along the length of the artery. Besides, in reality, an artery is a

viscoelastic tube (Nichols and O’Rourke, 2005) whereas blood is essentially a

suspension of erythrocytes, leukocytes and platelets in plasma and shows anoma-

lous viscous properties (Nichols and O’Rourke, 2005). On the other hand, most

studies focused on wave modulation in the normal artery without stenosis. Nev-

ertheless, the study of wave modulation in a prestressed thin elastic tube with a

symmetrical stenosis filled with fluid has not been carried out yet.

Since the model is difficult to generate according to the realistic situa-

tion and the limitation of literature for the corresponding wave modulation in

the stenosed artery, the artery will be considered as an incompressible, isotropic,

prestressed elastic tube with a thin wall having a symmetrical stenosis in this

research. The blood which is flowing inside the artery is considered to be divided

into four types: (a) an incompressible average inviscid fluid, (b) an incompress-

ible average viscous fluid, (c) an incompressible full inviscid fluid, and (d) an

incompressible full viscous fluid. The nonlinear wave modulation in the stenosed

artery contained with blood can be observed by using the method of reductive

perturbation by Demiray (2002) and Bakirtas and Demiray (2004b). At the end

of this research, four different mathematical models for wave modulation in a

stenosed artery are developed.

9

1.4 Objectives of the Study

Objectives of this study are to:

(a) derive mathematical models of the nonlinear evolution equations in a thin

stenosed elastic tube filled with an average inviscid fluid, average viscous

fluid, full inviscid fluid and full viscous fluid by the stretched coordinates

of the boundary-value problem.

(b) obtain the progressive wave solutions for the nonlinear evolution equations

of average inviscid fluid, average viscous fluid, full inviscid fluid and full

viscous fluid with various graphical outputs.

1.5 Scope of the Study

The artery is made of thick viscoelastic material (Nichols and O’Rourke,

2005). For this research, the artery is treated as an incompressible, prestressed,

thin and long circularly cylindrical, elastic tube with a symmetrical stenosis as

shown in Figure 3.2 and the blood as an incompressible average inviscid fluid, an

incompressible average viscous fluid, a full inviscid fluid and a full viscous fluid.

If the density, ρ is constant which means density does not vary with pressure,

then the flow is said to be incompressible. Inviscid fluid is a fluid in which the

viscous effect is not present whereas viscous fluid is a fluid which has the property

of viscosity. Average inviscid fluid is the inviscid fluid where an approximate

equation of inviscid fluid without its boundary condition is used. For the full

inviscid fluid, an exact equation of inviscid fluid with its boundary condition is

applied. The average viscous fluid which is a viscous fluid without boundary

condition and the full viscous fluid which refers to a viscous fluid with boundary

condition, are introduced respectively.

10

1.6 Significance of the Study

The presence of stenosis in the artery leads to some health problems such

as atherosclerosis, stroke and some heart problems. In view of the increasing

awareness of the risks from the stenosed artery, the issues raised in the field of

research in stenosed artery include identification of the wave propagation in the

stenosed artery (Tay, 2007). The motivations of this research stem from the

problem of modelling the blood flow in the artery with the presence of stenosis

for testing the modulation of nonlinear wave in such a composite medium.

This research presents four mathematical models for nonlinear wave mod-

ulation through the stenosed artery. The results will be an interest in determining

whether the disorderly blood flow patterns can be used to detect localized arte-

rial disease in its early stages, particularly before it becomes clinically significant.

Besides, the mathematical models for blood flow in a stenosed artery provide

facilities for the testing of the wave modulation on a model scale.

1.7 Methodology

The research begins with the study of the knowledge of tensor analysis and

continuum mechanics in curvilinear coordinates in order to derive the equation

of a stenosed tube and fluids. Subsequently, the knowledge of long-wave ap-

proximation and perturbation method is used to solve the field quantities in the

governing equations (equations of the stenosed tube and fluids). By introducing

a set of stretched coordinates and expanding the field quantities into asymptotic

series of small parameter of nonlinearity and dispersion, a set of nonlinear differ-

ential equations at various orders will be obtained. These nonlinear differential

equations will be reduced to the nonlinear evolution equations by adopting the

reductive perturbation method. Later, the solution of the nonlinear evolution

11

equations is obtained as progressive wave solution which is an exact method.

Lastly, the graphical outputs of the progressive wave solutions will be carried out

by using MATLAB R2010a.

1.8 Outline of Thesis

This thesis focuses on the wave modulation in a thin stenosed elastic tube.

This section is about the main content of the thesis and it serves as an outline

for quick reference to the appropriate section. The thesis is divided into eight

chapters, including this introductory chapter. Chapter 2 presents the discov-

ery of solitary waves, the Korteweg-de Vries (KdV) equation and the nonlinear

Schrodinger (NLS) equation, particularly the derivation of the model equation.

Chapter 3 contains the derivation of the tube equation which considers the artery

as a thin-walled elastic tube with a symmetrical stenosis. In Chapters 4, 5, 6 and

7, a mathematical model is constructed to describe the wave modulation in a

prestressed thin-walled stenosed elastic tube filled with an average inviscid fluid,

average viscous fluid, full inviscid fluid and full viscous fluid, respectively. Chap-

ter 8 is about the concluding remarks and recommendations for future research.

CHAPTER 2

LITERATURE REVIEW

2.1 The Discovery of Solitary Waves and the Korteweg-de Vries (KdV)

Equation

More than 176 years ago, the phenomenon of the solitary wave was first

discovered by John Scott Russell (1808-1882) in the Edinburgh-Glasgow canal in

1834. When a barge abruptly stopped, he was struck by the sight of what he

called “the great solitary wave” that he followed for a few miles before losing

it in the meanders of the canal. The description that Russell gave shows the

enthusiasm of a scientist who then devoted about ten years of his life to investigate

this phenomenon and he reported his discovery to the British Association in his

“Report on Waves” (Drazin and Johnson, 1989).

Figure 2.1: Schematic picture of the time evolution of the water wave, driven by

a piston moving downwards or upwards (Dauxois and Peyrard, 2006).

13

Figure 2.1 shows the experiments carried out by John Scott Russell to

investigate “the great solitary wave”. By dropping the piston at one end of a

water cannel, the solitary waves are generated. Through this experiment, Russell

was able to deduce that:

(i) Depending on its amplitude, the initial perturbation can create one, two or

several solitary waves (Dauxois and Peyrard, 2006).

(ii) The speed, c, of the solitary wave is proportional to the amplitude, a, of the

wave. Therefore, the higher amplitude waves travel faster and the speed of

the wave is obtained as

c2 = g (h + a) , (2.1)

where g is the acceleration of gravity and h is the undisturbed depth of

water.

(iii) The waves are stable and can travel over large distance.

(iv) Negative amplitude of the wave does not create any solitary waves, that

means that would move as localised pits (Dauxois and Peyrard, 2006).

The observations of Russell led to controversy, which assumed that the

nonlinear effect played a main role. Observations of Russell were opposed by

Sir G.B. Airy (1801-1892), who concluded that the formula derived by John

Scott Russell has contradicted his own theory of shallow-water waves, which

predicted that a wave of finite amplitude cannot propagate without change of

form in 1845. Besides, G.G. Stokes showed that waves of finite amplitude and

permanent form can possibly exist in the deep water, with the condition that

they are periodic. This controversy ended when Joseph Valentine de Boussinesq

(1842-1929) and Lord Rayleigh (1842-1919) proposed a new theory of shallow-

water waves, which has the same observations as Russell. They showed that

14

by neglecting the dissipation, the increase in local wave velocity associated with

finite amplitude was balanced by the decrease associated with dispersion, leading

to a wave of permanent form (Remoissenet, 1994).

In 1895, Diederik Johannes Korteweg and his student, Gustav de Vries

investigated the water waves in a shallow canal and they derived a model equation,

given by

∂η

∂τ=

3

2

√g

h

[η∂η

∂ξ+

2

3ε∂η

∂ξ+

1

3σ

∂3η

∂ξ3

], σ =

1

3h3 − Th

ρg. (2.2)

Equation (2.2) described the undirectional propagation of weakly nonlinear shal-

low water waves where η is the height of the wave above the equilibrium level h,

T is the surface tension, g is the gravitational acceleration, ε is a small arbitrary

constant related to the uniform motion of the water and ρ is the density.

Figure 2.2: A solitary wave (Debnath, 2007).

They made a complete analysis on the solitary wave phenomenon and from

the equation (2.2), they found that the solitary wave solutions, with a shape that

is not changeable, vindicated the discovery made 51 years earlier of a solitary

channel wave by Russell in 1834. By introducing the following change of variables

into equation (2.2),

t =1

2

√g

hστ, x = − ξ√

σ, U =

1

2η +

ε

3, (2.3)

15

the dimensionless Korteweg-de Vries (KdV) equation

Ut + 6UUx + Uxxx = 0, (2.4)

was obtained. In general, the KdV equation describes the unidirectional propa-

gation of small but finite amplitude waves in a nonlinear dispersive medium.

The importance of the KdV equation was not greatly concerned until in

1965, when Martin Kruskal and Norman Zabusky reinvestigated the Fermi-Pasta-

Ulam’s problem by using the KdV equation,

Ut + UUx + δ2Uxxx = 0. (2.5)

Zabusky and Kruskal utilized centered difference, mass and energy conservation

scheme to solve the KdV equation (2.5) numerically with periodic boundary con-

ditions and initial condition

U (x, 0) = cos(πx), 0 ≤ x ≤ 2 (2.6)

as shown in Figure 2.3, in Profile (a).

Figure 2.3: The solution of the periodic boundary-value problem for the KdV

equation: (a) initial profile at t = 0, (b) profile at t = 1π

and (c) profile at t = 3.6π

(Debnath, 2007).

Initially, the wave steepens and almost produces a shock as in Figure 2.3,

profile (b). The dispersive term, δ2Uxxx becomes important and thus results in a

16

balance between the nonlinearity and dispersion. At later time, t = 3.6π

, the wave

forms a train of eight pulses travelling to the right, with the largest on the right.

The speed of the pulse is proportional to its amplitude. Since it is the periodic

boundary conditions, the pulses eventually reappear on the left boundary.

Physically, when two solitons of different amplitudes are placed apart on

a real line as in Figure 2.4, the taller wave travels faster and eventually catches

up with the shorter wave. Then, the taller one overlaps the shorter one as shown

in Figure 2.5. When this happens, they undergo nonlinear interaction according

to the KdV equation and emerge from the interaction completely unchanged in

shape and speed as shown in Figure 2.6. The taller wave travels at the right side of

the shorter wave. Both of these waves preserve their shape, amplitude and speed

with only a phase shift after the interaction. This is known as the recurrence

phenomenon. Zabusky and Kruskal coined these waves as solitons similar to

photon, proton, electron and so on to emphasize the particle-like character of

these waves.

0

0.5

1

1.5

2

2.5

-60 -40 -20 0 20 40 60

u

x

Figure 2.4: Before interaction at t = −10.

17

0

0.5

1

1.5

2

2.5

-60 -40 -20 0 20 40 60

u

x

Figure 2.5: Full interaction at t = 0.

0

0.5

1

1.5

2

2.5

-60 -40 -20 0 20 40 60

u

x

Figure 2.6: After interaction at t = 10.

Normally, the single soliton solution is referred as the solitary wave. When

there is more than one solitary wave in the solution, they are called solitons.

Based on the observation of Zabusky and Kruskal in the study of solitons, it is

concluded that solitons have the following features (Bhatnagar, 1979):

(i) These localized waves are bell-shaped and travel with permanent form and

constant speed.

(ii) Soliton’s speed is proportional to its amplitude. It means taller solitary

waves travel faster than the shorter waves.

18

(iii) The width of a soliton is inversely proportional to the square root of its

amplitude. In other words, the taller solitary waves are thinner than the

shorter solitons.

(iv) Three fundamental physical quantities which are mass, momentum and

energy of solitons are always conserved.

(v) Solitons can interact with each other without changing its shape and will

emerge from the interaction unchanged in waveform and amplitude, but

phase shift occurs.

(vi) The taller soliton will overtake the shorter ones and continue on its way

intact and undistorted.

Another canonical equation, the nonlinear Schrodinger (NLS) equation is

formulated during the 1960s due to its importance to the physical problems in

the nonlinear optics (Agrawal, 2001). Nonlinear optics deal with the study of

how high intensity light interacts with and propagates through matter. There is

a growing interest in studying the propagation of optical soliton pulses in fibers.

This is because of their potential applications in fiber optic based communication

systems, soliton laser and switching devices (Debnath, 2007). Besides, the NLS

equation can be applied to describe the evolution of modulation of water waves

with weak nonlinearity (Zakharov, 1968). It also appears in many branches of

physics and applied mathematics, including nonlinear quantum field theory, hy-

drodynamic (Infeld, 1984) and plasma waves, the propagation of a beat pulse

in solid, fluid mechanics, theory of turbulence and the propagation of solitary

waves in piezoelectric semiconductors (Borhanifar and Abazari, 2010). The de-

tails about the NLS equation will be discussed in the following section.

19

2.2 Review on the Nonlinear Schrodinger (NLS) Equation

The nonlinear Schrodinger (NLS) equation is given by

iUt + βUxx ± α|U |2U = 0, (2.7)

where β and α are real constants in the nonlinear partial differential equations.

The NLS equation is the simplest representative equation describing the self-

modulation of plane waves in dispersive media. This equation exhibits a balance

between the nonlinearity, α|U |2U and the dispersion, βUxx. Under certain condi-

tions such a balance leads to the occurrence of stable structures for the amplitude

of the modulated waves, such as envelope soliton waves. The NLS equation ad-

mits the envelope soliton solution because these solitons consist of a travelling

wave (carrier wave) modulated by the envelope wave as shown in Figure 2.7.

Figure 2.7: Self-modulation of plane wave. The dashed line and solid line repre-

sent the envelope wave and travelling wave, respectively (Dauxois and Peyrard,

2006).

In the following subsection, the review for the analytical solution of the

NLS equation is presented. Besides, the properties of the solution of the NLS

equation are also discussed.

20

2.2.1 Analytical Solution of the NLS Equation (Free System)

Zakharov and Shabat (1972) solved the NLS equation analytically using

the inverse scattering method. They showed that this equation can be solved

exactly by reducing it to the inverse scattering problem for a certain linear dif-

ferential operator. The abbreviation of ZS will be used to denote Zakharov and

Shabat.

Drazin and Johnson (1989) used the ZS scheme and the following matrix

operator

∆(1)0 = I

(iα∗

∂

∂t− ∂2

∂x2

), ∆

(2)0 =

l 0

0 m

∂

∂x, (2.8)

to solve the NLS equation. The l, m and α∗ are real constants, and I is the 2× 2

unit matrix. By applying direct integration method, they obtained the solution

of the NLS equation as below:

u(x, t) = ±aexp

[i

{c

2(x− ct) + nt

}]sech{a (x− ct) /

√2}, (2.9)

where n = 14(2a2 + c2).

Ong (2002) solved the NLS equation analytically by taking the NLS equa-

tion in the form of

i∂U

∂t+ β

∂2U

∂x2+ α|U |2U = 0, i =

√−1. (2.10)

He proposed that the travelling wave solution to equation (2.10) was given in the

following form:

U (x, t) = exp [i(kx− ωt)] V (ξ), ξ = x− ct. (2.11)

By adopting equation (2.11) and using direct integration method, he obtained

travelling wave solution of the NLS equation as follows:

U(x, t) =

√2θ

αexp [i(kx− ωt)] sech

(√θ

β(x− ct)

). (2.12)

21

Detailed discussion for the analytical solution of the NLS equation obtained by

Ong can be found in Appendix A.

Applying a series of coordinate transformations,

x = x− Ut, t = t, U = Uexp

{−iU

2β

(x− 1

2Ut

)}, (2.13)

Debnath (2005) showed that the exact solution of the NLS equation (2.10) was

given by

U(x, t) =

(−2c

α

) 12

sech

[√−c

βx

]exp

[i

(U

2β

)− i

2

{(U

2β

)2

+2c

}t

], (2.14)

where c is an independent constant.

2.2.2 Properties of the Solution of the NLS Equation

The properties of the solution depends on the sign of β and α. If both of

them are in the same sign, which means αβ > 0, then the equation (2.10) is called

a self-focusing NLS equation and the solution of (2.10) tends to zero as |ξ| → ∞.

The solitary wave solution for self-focusing NLS equation is represented in (2.12)

where the solution decreases very rapidly as |ξ| → ∞. The solution (2.12) is often

called a bright soliton (or envelope soliton) as illustrated in Figure 2.8, where the

travelling wave (carrier wave) represented by the exponential function and the

amplitude of the sech profile propagates with different velocities.

Figure 2.8: An envelope soliton (Debnath, 2005).

22

Besides bright soliton, self-focusing NLS equation has another two types

of soliton solutions which are Ma-soliton and bi-soliton as shown in Figures 2.9

and 2.10, respectively. Ma-soliton is the soliton solution obtained by Ma in 1979

where this solution tends to a uniform solution. This solution is mostly used to

describe the modulation of a wide spread wave field (Peregrine, 1983). Zakharov

and Shabat discovered the bi-soliton where this soliton solution corresponds to two

isolated solitons which cannot be separated and are considered “bound” solitons.

The combination acts like a soliton.

Figure 2.9: The envelope of the Ma-soliton for three different time profiles

(Grimshaw et al., 2010).

23

Figure 2.10: The bi-soliton for the Davey-Stewartson equation with different time

profiles (Ohta and Yang, 2013).

When the signs of β and α are different, which means αβ < 0, the equation

(2.10) is called a defocusing NLS equation. The solution for defocusing NLS

equation is named dark soliton as shown in Figure 2.11, where this solution tends

to a uniform solution. It is called “dark” because its modulus is always less

than that of the uniform solution in which it propagates (Peregrine, 1983). The

bright soliton leads to more attention and interest compared to the dark soliton

because the solution of the defocusing NLS equation is similar to the solution of

the Korteweg-de Vries (KdV) equation (Peregrine, 1983).

24

Figure 2.11: A dark soliton (Dauxois and Peyrard, 2006).

In 1972, Zakharov V.E. and Shabat A.B. demonstrated an ingenious method

to solve the NLS equation analytically. They adopted the inverse scattering

method, following Lax (1968). The inverse scattering approach may be applied

in analogy with the method for solving the KdV solution (Whitham, 1999). By

considering the NLS equation in the form

iUt + Uxx + 2|U |2U = 0, (2.15)

and by introducing the spectral problem

v1x = −iςv1 + qv2, v2x = rv1 + iςv2, (2.16)

they changed equation (2.16) into the form of matrix notation

v1

v2

x

=

−iς q

r iς

v1

v2

, (2.17)

where bounded functions q(x) and r(x) are potentials, and ς is the eigenvalue.

By solving equations (2.15) and (2.17) through inverse scattering method, some

remarkable results can be concluded as follows (Debnath, 2005):

(i) An initial envelope wave pulse of arbitrary shape splits into a number of

solitons of shorter scales and an oscillatory tail. The soliton is a progressive

wave without changes in shape.

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NONLINEAR WAVE MODULATION IN A FLUID-FILLED THIN … · dan bendalir, modulasi gelombang tak linear dalam medium komposit dengan penghampiran gelombang panjang telah dilakukan dengan