the generalized kortewed de vries (kdv)...

14

THE GENERALIZED

KORTEWED DE VRIES (KdV) EQUATION

2.1 INTRODUCTION

The Korteweg-de Vries (KdV) equation is a nonlinear partial differential

equation of third order as

0 xxxxt uuuu , (2.1.1)

where and are positive parameters and the subscripts denote partial

differentiation. The KdV equation first appeared in an article written by Dutch

mathematicians Korteweg and de Vries [62]. They formulated the KdV equation to

describe long wave propagation on shallow water. The KdV equation is one of the

simplest nonlinear model equations for solitary waves. In the KdV equation (2.1.1),

),( txuu measures the elevation at time t and position x , i.e. the height of the water

above the equilibrium level. The second and the third terms in the equation are the

nonlinearity and the dispersion. The term xuu describes the sharpening of the wave

and xxxu the dispersion i.e. waves with different wave lengths propagate different

velocities. Remarkably the balances between these two effects allow solutions in

terms of propagating waves with unchanged form. Indeed, the KdV equation is a

special case of the generalized Korteweg-de Vries (gKdV) equation of the form

0 xxxx

p

t uuuu , (2.1.2)

where p is a positive integer. When 2p , the gKdV equation (2.1.2) takes the form

02 xxxxt uuuu (2.1.3)

and it is known as the Modified Kortweg-de Vries equation (mKdV). In the study of

the KdV equation, Zabusky and Kruskal [99] indicated that the wave solutions

persisted after interactions and the wave solutions were called ‘solitons’. The soliton

concept is very important in the study of nonlinear wave phenomena. Physically,

15

when two solitons of different amplitudes (and hence, of different speeds) are placed

far apart on the real line, the taller (faster) wave is on the left of the shorter (slower)

wave, the taller one eventually catches up to the shorter one and then overtakes it.

When this happens they undergo a nonlinear interaction according to the KdV

equation and emerge from the interaction completely preserved in form and speed

with only a phase shift. Thus these two remarkable features: (i) steady progressive

pulse like solitons and (ii) the preservation of their shapes and speeds confirmed the

particle like property of the waves.

Many exact solutions of the KdV equation with appropriate initial condition

by using scattering theory were given by Gardner et al. [39]. They discover the first

integrable nonlinear partial differential equation. Gardner et al. [40] and Hirota

[48,49] constructed analytic solutions of the KdV equation that provide the

description of the interaction among N solitons for any positive integer N . Certain

other methods for finding the exact solutions of the nonlinear PDEs include the

Painleve analysis [62], the Lie group theoretic methods [72], the direct algebraic

method [50] and tangent hyperbolic method [68].

Kruskal and Zabusky, and then Miura [65] considered the equation of the

existence of conservation laws of the KdV equation namely equations of the form

0

x

J

t

N, (2.1.4)

where ,.....),,( xxx uuuNN is the density and ,.....),,( xxx uuuJJ is the associated

flux. If 0J as x (or if it is periodic in space) then the total number Ndx

is conserved, because

0,.....),,(

dxuuuNdt

dxxx .

16

It was conjecture that the KdV equation had an infinite number of

conservation laws, and this was latter proved by Kruskal and Miura and

simultaneously by Gardner[40].

For, 3p it was found that the gKdV equation (2.1.2) had only three

conservation laws [7] and it is also found non-integrability [21]. Miura conjecture that

Eq. (2.1.3) has an infinite number of conservation laws and this might imply that the

KdV equation (2.1.1) and the mKdV Eq. (2.1.3) are related. He proceed to obtain the

transformation between them, namely if v satisfies (2.1.3)

02 xxxxt vvvvQv ,

and xvvu 2 , (2.1.5)

then ),( txu satisfies

0 xxxxt uuuuPu .

Hence, ),( txu satisfies the KdV equation (2.1.1). The transformation formula

(2.1.5) is known as Miura Transformation.

2.2 REVIEW OF THE NUMERICAL METHODS FOR SOLVING

gKdV EQUATION

Zabusky and Kruskal [99] were the pioneers in studying the KdV equation

numerically, by using the leapfrog method as an explicit finite difference scheme.

Tappert [92] has proposed a split-step Fourier method to solve the KdV equation

numerically. One of the most important numerical methods used to solve the KdV

equation was presented and exposed by Gereig and Morris [45]; this was the

hopscotch method. This hopscotch is an implicit finite difference method.

Fornberg and Whitham [37] gave an extensive study for the gKdV equation,

by using a pseudo-spectral method or what is called the finite Fourier transform.

17

Sanz-Serna and Christe [84] has applied a spectral method, namely the

Galerkin method. Taha and Ablowitz [93] in a series of papers have studied and

applied the famous and well known finite element method to solve the KdV equation.

They also gave an extensive comparison between the different numerical methods,

especially the finite difference and finite element methods. Although this is a wide

comparison, no one is able to say which is the best numerical method, for the

numerical solution of the KdV equation.

Chan and Kerkhoven [14] have discussed alternative time discretization of the

KdV equation. They showed that, with the leapfrog method for the advection term

and Crank-Nicolson method for the linear term, the stability limit is independent of

x (the space step) for any finite time interval.

Helal [53] has applied the tau method for obtaining a semi-analytical solution

for the KdV equation. He used the Chebyshev orthogonal polynomials as a basis for

this method. A numerical comparison showed that this method is more efficient than

the hopscotch method.

Abdur Rashid [78] has applied the Fourier Pseudospectral method for solving

the KdV equation accurately. He constructed the discrete representation of the

solution through interpolate trigonometric polynomial of the solution at collocation

points. Recently Rathish Kumar and Mani Mehra [79] have solved the KdV equation

by using Wavelet Galerkin method. They verified the asymptotic stability of the

proposed scheme.

Gardner et al. [41] has obtained the solutions of the Modified Korteweg-de

Vries (mKdV) equation by Galerkin method with quadratic B-spline finite elements.

The motion, interaction and generation of solitary waves are studied using the

18

method. T. Geyikli and D. Kaya [44] had used finite element technique for the

numerical solution of the mKdV equation.

Dogan Kaya and El-Sayed [59] had used Adomain Decomposition Method

(ADM) for the generalized KdV equation. They also proved the convergence of ADM

applied to the generalized KdV equation. The remarkable accuracy had shown by

comparing the numerical solution with the known analytic solution.

2.3 EXACT SOLUTION TO THE gKdV EQUATION

The simplest mathematical wave is a function of the form )(),( ctxftxu

which for example, is a solution to the simplest partial differential equation

0 xt cuu where c denotes the speed of the wave. For the well known wave

equation 02 xxtt ucu the famous d’Alembert solution leads to two wave fronts

represented by terms )( ctxf and )( ctxf .

Hence we start here with a trial solution

)()(),( fctxftxu . (2.3.1)

Substituting the trial solution (2.3.1) into the gKdV equation (2.1.2) we are led to the

ordinary differential equation

03

3

d

fd

d

dff

d

dfc p , (2.3.2)

which can written as

01 3

31

d

fdf

d

d

pd

dfc p .

Integrating, with respect to , it follows that

Ad

fdf

pcf p

2

21

1

,

where A is the constant of integration. In order to obtain a first order equation of f a

multiplication of d

df2 is done, i.e. :

19

d

dfA

d

fd

d

df

d

dff

pd

dfcf p 22

1

22

2

21

d

dfA

d

df

d

df

d

d

ppf

d

dc p 2

)2)(1(

22

22

Integrating both sides with respect to leads to

BAfd

dff

ppcf p

2

)2)(1(

22

22

, (2.3.3)

where B is another constant of integration.

Now it required that in case x we should have 0,0,02

2

d

fd

d

dff .

From these requirements it follows that 0 BA . With 0 BA , equation (2.3.3)

can be written as

pkfcf

d

df

22

, where )2)(1(

2

ppk

.

By separation of variables we may write

d

kfcf

df

p 21

.

Using the transformation, 2sechckf p , so that dhcdfkpf p 21 sec2 , we get

ddcp

2.

0

2x

cp

, where 0x is another constant of integration.

0

2

2sec

2

)2)(1(xctx

cph

ppcf p

p xctxcp

hppc

ctxftxu

0

2

2sec

2

)2)(1()(),(

(2.3.4)

20

This is the exact solution of the gKdV equation (2.1.2). Taking 1p and 2p in

equation (2.3.4) it will give the exact solutions of the KdV equation (2.1.1) and the

mKdV equation (2.1.3) respectively.

2.4 THE PETROV-GALERKIN METHOD

For convenience, the gKdV equation (2.1.2) is rewritten in the form

01

1

xxxx

p

t uup

u

. (2.4.1)

Periodic boundary conditions on the region bxa are assumed in the form

0),(),( tbutau . The space interval bxa is discretized by uniform )1( N

grid points jhax j , where Nj ,,2,1,0 and the grid spacing is given by

Nabh /)( . Let )(tU j denote the approximation to the exact solution ),( txu j .

Using Petrov-Galerkin method, we assume the approximate solution of Eq. (2.4.1) as

N

j

jjh xtUtxu0

)()(),( . (2.4.2)

The product approximation technique [16] is used for the nonlinear term in the

following manner

N

j

j

p

j

p

h xtUtxu0

11 )()(),( , (2.4.3)

where Njxj ,,2,1,0),( are the usual piecewise linear “hat” function given by

otherwise

xxxhjhx

xxxhjhx

x jj

jj

j

,0

],[,/)(1

],[,/)(1

)( 1

1

The unknown functions NjtU j ,...,2,1,0,)( are determined from the variational

formulation

0,)(,)(1

,)( 1

jxxxhjx

p

hjth uup

u

, (2.4.4)

where j , Nj ,,2,1,0 are test functions, which are taken to be cubic B-splines

given by

21

otherwise

xxxxx

xxxxxxxhxxhh

xxxxxxxhxxhh

xxxxx

hx

jjj

jjjjj

jjjjj

jjj

j

,0

,)(

,)(3)(3)(3

,)(3)(3)(3

,)(

1)(

21

3

2

1

3

1

2

11

23

1

3

1

2

11

23

12

3

2

3

and , denote the usual inner product b

a

dxxgxfgf )()(, .

Integrating by parts and using the fact that 0)()()()( baba xx , Eq.

(2.4.4) leads to the formulation

0)(,)(,)(1

,)( 1

xxjxhjx

p

hjth uup

u

. (2.4.5)

Performing the integrations on (2.4.5) will give the following system of ordinary

differential equations (ODEs)

0)1(1260

13

CBAhhp

, (2.4.6)

where 1112 266626 jjjjj UUUUUA ,

1

1

1

1

1

1

1

2 1010

p

j

p

j

p

j

p

j UUUUB

and 2112 22 jjjj UUUUC , .,....,3,2,1,0 Nj

Now to solve the ODEs, we assume n

jU to be a fully discrete approximation to the

exact solution ),,( nj txu where tntn and t is the time step size. Using the

Crank-Nicholson approach 21 nn UUU and the forward difference scheme for

the time derivative, tUUU nn

1 , Eqn. (2.4.6) is reduced to the system of

nonlinear equations

.)1(1230

1

)1(1230

1

3

1

3

11

nnn

nnn

hhp

t

hhp

t

CBA

CBA

(2.4.7)

The nonlinear system (2.4.7) can be solved by Newton’s method and the required

solution to the gKdV equation can be found.

22

2.5 STABILITY ANALYSIS

To apply the Von Neumann stability for the system (2.4.7), we must first

linearize this system. Assuming u in the nonlinear term x

puu of the gKdV equation

(2.1.2) as locally constant, u~ , the linearized scheme is

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

aUbUcUdUeU

eUdUcUbUaU

2112

1

2

1

1

11

1

1

2

(2.5.1)

where 312

~

30

1

h

t

h

uta

p

,

3

2

12

~10

30

26

h

t

h

utb

p

,

30

66c ,

3

2

12

~10

30

26

h

t

h

utd

p

,

312

~

30

1

h

t

h

ute

p

.

Substituting the Fourier mode

),exp( ijU nn

j hk and 1i , (2.5.2)

where k is the mode number and h the space step size into the linearized system

(2.5.1), we obtained the amplification factor, g as

iYX

iYXg

n

n

1

, (2.5.3)

where cos2cos dbeacX

and sin2sin dbeaY .

Thus, for all values of ,, th and we have

122

22

YX

YXggg .

Hence, the proposed method is unconditionally stable in the linear sense.

2.6 NUMERICAL TESTS

It has been shown that the gKdV equation has an analytic solution of the form

p xctxcp

hppc

txu

0

2

2sec

2

)2)(1(),(

,

23

where c and 0x are constants. It has mention that the gKdV equation has three

conservation laws. These are given below:

b

a

udxI1 , b

a

dxuI 2

2 and

b

a

x dxuuI 23

3

3

.

In this section, we present some numerical experiments to find the solution of single

solitary wave, in addition to the two soliton interactions at different time levels. Also

we present the birth of solitons from an initial pulse.

2.6.1 SINGLE SOLITARY WAVE

First we take 1p , so that the gKdV equation is the KdV equation and the

analytic solution takes the form:

0

2

2

1sec

3),( xctx

ch

ctxu

, (2.6.1)

where c and 0x are constant. The 2L and L error norms are used to compare the

numerical solutions with the exact solution and the quantities 1I , 2I and 3I are shown

to measure the conservation for the scheme. We choose )3/1(c , 2.0h , 1.0t ,

1 , 1 and 200 x over the domain ]80,0[ . Thus the solitary wave has

amplitude unity and the simulations are done up to 80t . Values of the three

invariants as well as 2L and L error norms from our method has been computed and

reported in Table 2.1. The chances of the invariants 1I , 2I and 3I from their initial

values are less than 0.00012, 0.00001 and 0.00005 respectively. Error deviations are

changed in the range 000062644.00000756628.0 error . The motion of a single

solitary wave using the proposed scheme corresponding to the above set of parameters

has been computed and plotted in Fig. 2.1.

24

Table 2.1: Invariants and error norms for single solitary wave 1p , 3/1c ,

2.0h , 1.0t , 1 , 200 x , 800 x .

t

1I 2I 3I 3

2 10L 310L

0

20

40

60

80

6.92814

6.92824

6.92825

6.92820

6.92823

4.6188

4.6188

4.6188

4.6188

4.6188

2.77128

2.77129

2.77129

2.77129

2.77126

0

0.203188

0.255819

0.327420

0.404813

0

0.342689

0.472876

0.581553

0.756628

Fig. 2.1: Single solitary wave with 1p , 3/1c , 200 x , 800 x at level

time 80,40,0t


time 80,40,0t .

25


time 80,40,0t .


2.0h , 1.0t , 1 , 200 x , 800 x .

t 1I 2I 3I 3

2 10L 310L

0

10

20

30

40

50

60

70

80

6.92814

6.92823

6.92824

6.92825

6.92825

6.92821

6.92820

6.92820

6.92823

4.6188

4.6188

4.6188

4.6188

4.6188

4.6188

4.6188

4.6188

4.6188

2.77128

2.77129

2.77129

2.77127

2.77129

2.77128

2.77129

2.77127

2.77126

0

0.173189

0.203188

0.229209

0.255819

0.304573

0.32742

0.38525

0.404813

0

0.238921

0.342689

0.311464

0.472876

0.541176

0.581553

0.766973

0.756628

Secondly, we take 2p , so that the gKdV equation becomes the modified

KdV equation and the equation has the analytic solution as

0sec

6),( xctx

ch

ctxu

. (2.6.2)

In this case, we consider the following parameters for the simulation of the problem:

)2/1(c , 2.0h , 1.0t , 3 , 1 and 200 x over the domain ]80,0[ . Thus

the solitary wave has amplitude unity and the simulations are done up to 80t .

26

Values of the three invariants as well as 2L and L error norms from our method has

been computed and reported in Table 2.2. The chances of the invariants 1I , 2I and 3I

from their initial values are less than 0.00005, 0.00002 and 0.00003 respectively.

Error deviations are changed in the range of 00122617.000122794.0 error .

The motion of a single solitary wave using the proposed scheme corresponding to the

above set of parameters has been computed and plotted in Fig. 2.2.

Finally, we take 3p and consider the following set of parameters for the

simulation of the problem. We take, )5/1(c , 2.0h , 1.0t , 2 , 1 and

200 x for the simulation of the problem over the domain ]80,0[ . Thus the solitary

wave has amplitude unity and the simulations are done up to 80t . Values of the

three invariants as well as 2L and L error norms from our method has been

computed and reported in Table 2.3. The chances of the invariants 1I , 2I and 3I from

their initial values are less than 0.00014, 0.00003 and 0.00009 respectively. Error

deviations are changed in the range of 000116961.000016448.0 error . The

motion of a single solitary wave using the proposed scheme corresponding to the

above set of parameters has been computed and plotted in Fig. 2.3.


2.0h , 1.0t , 3 , 1 , 200 x , 800 x .

t 1I 2I 3I 3

2 10L 310L

0

10

20

30

40

50

60

70

80

4.44288

4.44288

4.44289

4.44292

4.44292

4.44291

4.44289

4.44286

4.44286

2.8284

2.8281

2.8281

2.8281

2.8281

2.8281

2.8281

2.8281

2.8281

1.74998

1.74997

1.74997

1.74997

1.74997

1.74997

1.74997

1.74996

1.74997

0

0.710642

1.35740

2.01775

2.68235

3.35304

4.00643

4.67266

5.33633

0

0.165455

0.311979

0.465156

0.618102

0.774527

0.920563

1.07373

1.22794

27


2.0h , 1.0t , 3 , 1 , 200 x , 800 x .

t 1I 2I 3I 3

2 10L 310L

0

10

20

30

40

50

60

70

80

6.27083

6.27083

6.27096

6.27101

6.27100

6.27088

6.27098

6.27084

6.27030

3.85663

3.85663

3.85663

3.85663

3.85663

3.85663

3.85664

3.85663

3.85663

2.48559

2.48557

2.48551

2.48556

2.48563

2.48558

2.48555

2.48552

2.48551

0

0.958627

0.973995

0.957792

0.957425

0.980087

0.957492

0.965510

0.971789

0

0.169812

0.149183

0.144506

0.151097

0.141056

0.140572

0.140047

0.116961

2.6.2 INTERACTION OF TWO gKdV SOLITARY WAVES

Here we study the interaction of two well separated solitary waves having

different amplitudes and travelling in the same direction. The initial condition is given

by

2

1

2

2sec

2

)2)(1()0,(

i

pi

ii xxcp

hppc

xu

, (2.6.3)

where ic and 2,1, ixi are arbitrary constants. In this case also we take

32,1 andp . For all values of p , the following parameters:

1,1.0,2.0 th have been chosen with range ]160,0[ .

28

Fig. 2.4 (a) Fig. 2.4 (b)

Fig. 2.4(c) Fig. 2.4(d)

Fig. 2.4(e)

Fig. 2.4 (a)-(d): Interaction of two gKdV solitary waves at different time levels and

Fig. 2.4 (e) three dimensional plot for .1p

29

First we take 1p , so that the gKdV equation (2.1.2) becomes the KdV

equation and the initial condition (2.6.3) takes the form

i

i

i

i xxc

hc

xu 2

1sec

3)0,( 2

2

1

(2.6.4)

Here, we chose the following parameters for our simulation: 1 , ,3/11 c

6/12 c , 201 x and 602 x . Then the amplitudes of the two solitary waves are in

the ratio 1:2 . The simulation is done up to the time t 320, The three invariant of

motion are tabulated in Table 2.4 and Fig. 2.4(a)-2.4(d) show the interaction of the

two solitons at different time and Fig. 2.4(e) shows its three dimensional plot.

Table 2.4: Invariants for two solitary wave interaction for 1p .

t 1I 2I 3I

0

40

80

120

160

200

240

280

320

11.8271

11.8272

11.8272

11.8272

11.8272

11.8271

11.8272

11.8272

11.8272

6.25179

6.25180

6.25180

6.25180

6.25181

6.25187

6.25182

6.25180

6.25180

3.26119

3.26116

3.26116

3.26117

3.26123

3.26123

3.26122

3.26119

3.26117

Secondly, we consider the case for 2p . In this case the initial condition

(2.22) takes the form:

i

i

i

i xxc

hc

xu

sec6

)0,(2

1

. (2.6.5)

Here, we chose the following parameters for our simulation: 1,3 , ,2/11 c

8/12 c , 201 x and 602 x . The amplitudes of the two solitary waves are in the

ratio 2:1. The simulation is done up to the time t 200, The three invariant of motion

30

are tabulated in Table 2.5 and Fig. 2.5(a)-2.5(d) show the interaction of the two

solitons at different time and Fig. 2.5(e) shows its three dimensional plot.

.

Fig. 2.5(a) Fig. 2.5(b)

Fig. 2.5(c) Fig. 2.5(d)

Fig. 2.5(e)


Fig. 2.5 (e): three dimensional plots for .2p

31

Fig. 2.6(a) Fig. 2.6(b)

Fig. 2.6(c) Fig. 2.6 (d)

Fig. 2.6 (e)


Fig. 2.6 (e) three dimensional plots for .3p

32

Finally, we take 3p . In this case the initial condition (2.6.3) takes the form:

2

1

32

2

3sec

10)0,(

i

iii xx

ch

cxu

. (2.6.6)

Here, we chose the following parameters for our simulation: 1,2 , ,5/21 c

10/12 c , 201 x and 602 x . The amplitudes of the two solitary waves are in the

ratio 2:1. The simulation is done up to time 320t , The three invariant of motion are

tabulated in Table 2.6 and Fig. 2.6(a)-2.6(d) show the interaction of the two solitons at

different time and Fig. 2.6(e) shows its three dimensional plot.


t 1I 2I 3I

0

40

80

120

160

200

8.88576

8.88581

8.88574

8.88571

8.88567

8.88568

4.24263

4.24263

4.24276

4.24267

4.24263

4.24263

2.24646

2.24611

2.24832

2.24875

2.24544

2.24646


t 1I 2I 3I

0

40

80

120

160

200

240

280

12.6253

12.6252

12.6253

12.6253

12.6253

12.6273

12.6200

12.6219

7.76476

7.76476

7.76476

7.76512

7.76488

7.76475

7.76480

7.76466

4.99077

4.99013

4.99077

4.99022

4.99203

4.99665

4.99257

4.99264

33

2.6.3 SPLITTING OF SOLITONS FROM A SINGLE INITIAL PULSE

Here we study the splitting of solitons from a single initial pulse. In this case

also we consider three cases viz. 1p , 2p and 3p .

First we take 1p . The initial condition in this case takes the form:

108

1sec

3

2)( 2 x

hxf (2.6.7)

The parameter chosen are: 1 , 410 , 01.0 th and the movement

of the initial pulse is consider over the interval ]3,0[ . This simulation is performed up

to the time step 3t . The three invariants of motion 1I , 2I and 3I are tabulated in

Table 2.7. Fig. 2.7(a) and 2.7(b) show the splitting of the soliton at time 0t and at

time 3t respectively. Fig. 2.7(c) shows the three dimensional plot of this case.

Fig. 2.7(a) Fig. 2.7(b)

Fig. 2.7(c)

Fig. 2.7 (a) and (b): Splitting of gKdV solitary waves from a single initial pulse and

Fig. 2.7 (c) its three dimensional plot for .1p

34

Secondly we take 2p and consider the same initial condition as given in

equation (2.6.7) for the case 1p . The parameters chosen are: 1 , 410 ,

01.0 th and the movement of the initial pulse is consider over the interval ]3,0[ .

This simulation is performed up to the time step 8t . The three invariants of motion

1I , 2I and 3I are tabulated in Table 2.8. Fig. 2.8(a)-(c) showed the splitting of the

soliton from the initial pulse at different time 4,0t and 8 respectively. Also, Fig.

2.8(d) shows the three dimensional plot of this case.

Fig. 2.8(a) Fig. 2.8(b)

Fig. 2.8(c) Fig. 2.8(d)

Fig. 2.8 (a)-(c): Splitting of gKdV solitary waves from a single initial pulse and Fig.

2.8 (d) its three dimensional plot for .2p

35

Table 2.7: Invariance for the splitting of gKdV solitary waves for .1p

t 1I 2I 3I

0

1

2

3

4

0.138564

0.138565

0.138564

0.138564

0.138563

0.0615837

0.0614930

0.0649040

0.0614902

0.0614901

0.0314758

0.0314142

0.0313945

0.0314100

0.0313999


t 1I 2I 3I

0

2

4

6

8

0.217647

0.217649

0.217638

0.217720

0.217641

0.092376

0.092318

0.092312

0.092312

0.092311

0.047513

0.048489

0.049179

0.049464

0.049597


t 1I 2I 3I

0

2

4

6

8

0.217647

0.217655

0.217651

0.217705

0.217609

0.092376

0.092355

0.092325

0.092331

0.092341

0.0475125

0.0493923

0.0505958

0.0507069

0.0507019

Lastly, we consider the case .3p In this case also we consider the same

initial condition as above. The parameters chosen are: 1 , 410 , 01.0 th

and the movement of the initial pulse is consider over the interval ]3,0[ . This

simulation is performed up to the time step 8t . The three invariants of motion 1I ,

2I and 3I are tabulated in Table 2.9. Fig. 2.9(a)-(c) showed the splitting of the soliton

from the initial pulse at different time 4,0t and 8 respectively. Also, Fig. 2.9(d)

36

shows the three dimensional plot of this case. It is observed from the Tables 2.7-2.9

that the third invariance,

b

a

x dxuuI 23

3

3

is little fluctuated in all the cases.

Fig. 2.9(a) Fig. 2.9(b)

Fig. 2.9(c) Fig. 2.9(d)

Fig. 2.9 (a)-(c): Splitting of gKdV solitary waves from a single initial pulse and

Fig. 2.9 (d): its three dimensional plot for .3p

2.7 CONCLUSION

The Petrov-Galerkin method using the linear hat function and cubic B-spline-

function as trial and test functions has been successfully implemented to study the

solitary waves of the gKdV equation. It has been shown that the scheme is accurate

37

and efficient. It has also been shown that the scheme so developed is unconditionally

stable. We have tested our scheme using single solitary wave for which the exact

solution is known and then extended this scheme to the study of the interaction of two

solitary waves and also breaking of the solitary waves from an initial pulse. It is also

observed that the solution of the gKdV equation does not give true soliton solution

when 3p . We have also shown the appearance of small tail after the collision of

two solitons in the case of 3p . Thus we require further research work in this area

through both analytical and numerical solution of this equation type.

the generalized kortewed de vries (kdv)...

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