J Spectral Method for Solving the Equal Width Equation Based on Chebyshev Polinomials

8/3/2019 J Spectral Method for Solving the Equal Width Equation Based on Chebyshev Polinomials

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Nonlinear Dyn (2008) 51:59–70

DOI 10.1007/s11071-006-9191-0

O R I G I N A L A R T I C L E

Spectral method for solving the equal width equation based

on Chebyshev polynomialsA. H. A. Ali

Received: 10 November 2006 / Accepted: 8 December 2006 / Published online: 24 January 2007C Springer Science+Business Media B.V. 2007

Abstract A spectral solution of the equal width (EW)

equation based on the collocation method using Cheby-

shev polynomials as a basis for the approximate so-

lution has been studied. Test problems, including the

migration of a single solitary wave with different am-

plitudes are used to validate this algorithm which is

found to be accurate and efficient. The three invariants

of the motion are evaluated to determine the conser-

vation properties of the algorithm. The interaction of

two solitary waves is seen to cause the creation of a

source for solitary waves. Usually these are of smallmagnitude, but when the amplitudes of the two inter-

acting waves are opposite, the source produces trains

of solitary waves whose amplitudes are of the same or-

der as those of the initial waves. The three invariants of

the motion of the interaction of the three positive soli-

tary waves are computed to determine the conservation

properties of the system. The temporal evaluation of a

Maxwellian initial pulse is then studied. Comparisons

are made with the most recent results both for the error

norms and the invariant values.

Keywords EW equation . Collocation method .

Chebyshev polynomials . Spectral method

A. H. A. Ali

Mathematics Department, Faculty of Science, Menoufia

University, Shebein El-Koom, Egypt

e-mail: ahaali [email protected]

1 Introduction

The regularized long-wave (RLW) equation is an alter-

native description of nonlinear dispersive waves to the

more usual Korteweg–de Vries (KdV) equation [1]. It

has solitary wave solutions of a rather general type [1,

2]. A less well-known alternative, proposed by Morri-

son et al. [3], is the equal width equation (EWE) which

also has solitary wave solutions, but of a less general

type.

Solitary waves are wave packets or pulses whichpropagate in nonlinear dispersive media. The dynam-

ical balance between the nonlinear and the dispersive

effects of these waves retain a stable waveform. A soli-

ton is a very special type of solitary wave which also

keeps its waveform after collision with other solitons.

In practical physics and quantum mechanics, it is stan-

dard practice to use the term soliton to designate both

solutions to wave equations integrable via the inverse

scattering transform, such as KdV, and also to designate

localized solutions of nonintegrable equations, such as

RLW and EWE.Few analytic solutions for solving EWE are known

under certain conditions. Approximate solutions for

solving EWE using Galerkin’s method with both cu-

bic B-spline finite elements [4, 5], a Petrov-Galerkin

method using quadratic B-spline finite elements [6],

Zaki [7, 8] has solved EW equation by a least-

square technique using linear space-time finite ele-

ments and Petrov–Galerkin finite element scheme with

shape functions taken as quadratic B-spline functions,

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60 Nonlinear Dyn (2008) 51:59–70

respectively. Recently, Raslan [9] has solved the EW

equation using collocation method with quartic B-

spline finite elements and solved the resulting system

of first-order ordinary differential equations using the

fourth-order Runge–Kutta method. Soliman [10] has

solved the corresponding generalized regularized long-

wave equation by He’s variational iteration method.Also, Soliman and Hussein [11] have solved the corre-

sponding RLW equation using the collocation method

with septic spline. In this paper, we set up the spec-

tral method based on collocation method with basis

Chebyshev polynomials. The resulting system will be

a system of ordinary differential equations which can

be solved using the Runge–Kutta algorithm.

2 Governing equation and numerical method

The EW equation for the long waves propagating in the

positive X -direction can take the form:

ut + εuu X − νu X X t = 0, a ≤ X ≤ b, t > 0. (1)

with the boundary conditions

u(a, t ) = u(b, t ) = 0, t > 0 (2)

and the initial condition

u( X , 0) = f ( X ), (3)

where ε and ν are positive parameters, subscripts X

and t denote differentiation, and f ( X ) is a localized

disturbance inside the considered interval.

Using a linear transformation to transfer the inter-

val [a, b] into the interval [−1, 1], Equations (1)–(3)

become:

α2ut + 2εαuu x − 4νu xxt = 0, −1 ≤ x ≤ 1, t > 0,

(4)

where x is the new variable and α = b − a, the bound-

ary conditions

u(−1, t ) = u(1, t ) = 0, t > 0 (5)

and the initial condition

u( x, 0) = f ( x), (6)

The approximate solution u N ( x, t ) to the

exact solution u( x, t ) can be written in the

form:

u N ( x, t ) =

N

n=0

δn (t )T n( x), (7)

where δn are the time-dependent quantities to be deter-

mined, T n ( x) are the Chebyshev functions and is de-

fined as:

T n( x) = cos(n cos−1( x)). (8)

From Equation (7) and the property of the

Chebyshev functions, we can get the following

[12]:

ut =

N n=0

δn(t )T n ( x), (9)

u x =

N −1n=0

δn(t )T n( x) =

N −1n=0

δ(1)n (t )T n( x), (10)

u xxt =

N −2n=0

δn(t )T n ( x) =

N −2n=0

δ(2)n (t )T n ( x). (11)

Using Equations (9)–(11) and substituting in Equation

(4) we obtain

α2 N

n=0

δn (t )n T n ( x)

+ 2ε α

N n=0

δn(t ) T n( x)

N −1k =0

δ(1)k (t ) T k ( x)

− 4ν

N −2n=0

δ(2)n (t ) T n ( x) = 0, (12)

where δn denotes the derivative of δn

with respect to t and δ(1), δ(2) are defined

by

δ(1)n =

2

cn

n+2 j−1≤ N j=1

(n + 2 j − 1)δn+2 j−1,

δ(2)n =

2

cn

n+2 j≤ N j=1

j (n + j )(n + 2 j )δn+2 j . (13)

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Nonlinear Dyn (2008) 51:59–70 61

The nonlinear term T n ( x)T k ( x) can be expressed as

a linear combination which is defined as:

T n ( x)T k ( x) =1

2

T n+k ( x) + T |n−k |( x)

(14)

We substitute Equation (14) into Equation (12) and

hence use the inner product with the weight func-

tion ψ j ( x). If we choose the weight function ψ j ( x) =

δ( x − xi ), where δ( x − xi ) is the Dirac Delta function,

then this equation can be deduced using the property

of the Dirac Delta function to obtain [12]

α2 N

n=0

δn(t )T n( xi )

+ ε α N −1n=0

⎡⎢⎣ N , N −1l,k =0l+k =n

δl (t )δ(1)k (t )T l+k ( xi )

+

N −1l,k =0|l−k |=n

δl (t )δ(1)k T |l−k |( xi )

⎤⎥⎦

− 4ν

N −2n=0

δ(2)n (t ) T n( xi ) = 0, i = 1, . . . , N − 1

(15)

where the collocation points of the Chebyshev func-

tions are calculated from Equation (8) at x = xi .

Equation (15) can be written in the recurrence rela-

tion as:

α2δn+εα

⎡⎢⎣ N , N −1

l,k =0l+k =n

δl δ(1)k +

N , N −1l,k =0|l−k |=n

δl δ(1)k

⎤⎥⎦−4ν δ

(2)n =0.

(16)

Hence, we can write Equation (16) in the matrix

form as a system of ordinary differential equations

(α2 I − 4ν A1)δ∼=−(εα A2(δ∼

)) δ∼

, (17)

where A1 and A2 are the coefficient matrices for the

second derivative and the nonlinear term, respectively.

Equation (17) can be written in the simple form:

A4 δ∼

(t ) = B1 δ∼

(t ), (18)

where

A4 = α2 I − 4ν A1, B1 = −εα A2(δ∼

)

are two matrices of order ( N + 1)× ( N + 1). Multi-

plying both sides of Equation (18) by the matrix S, we

get [13]

S A4 δ∼

(t ) = S B1 δ∼

(t ), (19)

where

S = (sin) = (T n ( xi )), i = 1, . . . , N − 1,

n = 0, 1, . . . , N (20)

the matrix S of order ( N − 1) × ( N + 1). The system

given in the Equation (19) consists of N − 1 equations

in N + 1 unknowns and to obtain a unique solution for

this system we need two further equations. For this, we

add the two boundary conditions:

u(−1, t ) =

N n=0

an T n (−1) = a0 − a1 + a2 − a3

+ a4 + · · · + a N = 0

u(1, t ) =

N n=0

an T n (1) = a0 + a1 + a2 + a3

+ a4 + · · · + a N = 0

to Equation (20) to obtain a new system:

A δ∼

(t ) = F ∼

(t , δ∼

(t )), (21)

where δ∼

(t ) and F ∼

are N + 1 vectors with N + 1 com-

ponents and A is a matrix of order ( N + 1)× ( N + 1).

This system of first order of ordinary differential equa-

tions can be solved numerically using the Runge–Kutta

algorithm to get the numerical solution δ∼

(t ).

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62 Nonlinear Dyn (2008) 51:59–70

3 The initial state

From the initial condition u( x, 0) on the function

u( x, t ), we must determine the initial vector∼δ0 so

that the time evolution of ∼δ, using Equation (21),

can be started.

We rewrite Equation (7) for the initial condition as

u N ( xi , 0) =

N n=0

δ0n T n ( xi ), i = 0, 1, . . . , N . (22)

Equation (22) gives a system of N + 1 equations

which can be written in the matrix form as:

Sδ∼

0 = b∼

, (23)

where S is a matrix of order ( N + 1) × ( N + 1) and

is defined by Equation (20) and the vector is b∼=

( f ( x0), f ( x1), . . . , f ( x N ))T.

4 Test problems

A computer program using FORTRAN language with

algorithm to calculate the nonlinear term was written

for the purpose of obtaining soliton solutions and mod-

eling the undular bore to the EW equation. Accuracy

of the method is measured by the L2- and L∞- error

norms

L22 = ||uexact − u N ||22 =

h

N i=0

uexacti − u N

i

2 ,

L∞ = ||uexact − u N ||∞ = maxi

uexacti − u N

i

, (24)

and the conservation properties of the proposed algo-

rithm are examined by calculating the invariants which

was shown by Olver [15] and corresponds to mass, mo-

mentum, and energy, respectively,

I 1 =

b

a

u d x, I 2 =

b

a

(u2 + νu2 x ) d x,

I 3 =

b

a

u3 d x . (25)

The analytic values of the invariants can be found from:

I 1 = 6c/k , I 2 = 12c2/k + 48k νc2/5,

I 3 = 144c3/5k . (26)

4.1 Single solitary wave motion

We now validate our algorithm by studying the mo-

tion of solitary waves. The solitary wave solution of the

EWE Equation (1) is [3]:

u( x, t ) = 3c sec h2[k ( x − vt − x0)], (27)

where v = ε c is thewave velocity, andk 2 = 1/4ν. This

equation represents a single soliton of magnitude 3c

and width k , initially centered at x0. Here k dependsonly on ν and not c as does the corresponding constant

for RLW equation; thus, for a given equation (fixed ν)

all solitary waves have the same width, hence the name

EW equation. Waves exist with all possible velocities

c, −∞ ≤ c ≤ ∞, unlike the RLW equation for which

there is the forbidden region 0 ≤ c ≤ 1.

First, we study the motion of a single solitary wave

Equation (27) of amplitude 3 through a region 0 ≤ x ≤

80 with x0 = 15 and ε = ν = 1. To make a comparison

with earlier simulation results, Equation (27) is taken as

the initial condition with t = 0, x = 0.4, t = 0.1.The simulation is run up to time t = 20. The analytic

values of the invariant quantities are: I 1 = 12, I 2 =

28.8, I 3 = 57.6.

In the simulation of a solitary wave of amplitude

3, the present approach algorithm leads, at t = 20,

to L2- error norm as given in Table 1 with value

L2 = 0.0055× 10−3 which is very small, while the

quantities I 1, I 2, and I 3 are shown in Table 2. The con-

stants of motion vary little from the analytic value: I 1does not change from the analytic, I 2 varies by less

than 0.00006% and I 3 varies by less than 0.0002%. Ina corresponding simulation using a collocation method

with quartic spline elements [9], the L2-error norm at

t = 20 is less than 0.6463× 10−3 and the quantity I 1does not change from the analytic, I 2 varies by less than

0.043%, and I 3 varies by less than 0.005%.

Second, we model the motion of a single solitary

wave with the three different amplitudes 0.3, 0.09, and

0.03 and compare with results given in [6, 7, 9] at times

t = 40, 80; see Tables 3–7, using the region 0 ≤ x ≤

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Nonlinear Dyn (2008) 51:59–70 63

Table 1 The L2 × 103 error norm for a single solitary wave of amplitude 3, 0 ≤ x ≤ 80

t 0.2 2 4 6 8 10 12 14 16 18 20

Present method 0.0001 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0036 0.0043 0.0049 0.0055

N = 200, t = 0.1

Method [9] 0.0111 0.0940 0.1567 0.2009 0.2441 0.2938 0.3502 0.4135 0.4835 0.5608 0.6462

N = 400, t = 0.2

Table 2 Invariant values for a single solitary wave: amplitude = 3, 0 ≤ x ≤ 80

t Method I 1 I 2 I 3 Method I 1 I 2 I 3

0.2 Present 11.999998 28.800000 57.599999 [9] 12.000000 28.788570 57.599950

2 N = 200 12.000001 28.799999 57.599992 N = 400 12.000000 28.788500 57.599730

4 t = 0.1 12.000000 28.799999 57.599984 t = 0.2 12.000000 28.788420 57.599480

6 12.000000 28.799998 57.599975 12.000000 28.788330 57.599210

8 12.000000 28.799996 57.599967 12.000000 28.788240 57.598960

10 12.000000 28.799994 57.599958 12.000000 28.788150 57.598700

12 12.000000 28.799992 57.599951 12.000000 28.788070 57.598440

14 12.000000 28.799989 57.599943 12.000000 28.787990 57.59817016 12.000000 28.799987 57.599934 12.000000 28.787900 57.597900

18 12.000000 28.799985 57.599926 12.000000 28.787810 57.597650

20 12.000000 28.799983 57.599918 12.000000 28.787720 57.597380

Table 3 Invariants and error for single solitary wave amplitude = 0.3, 0 ≤ x ≤ 30

Method T I 1 I 2 I 3 L2 × 103

Present x = t = 0.25 5 1.199935 0.288000 0.057600 0.0067

10 1.199961 0.288000 0.057600 0.0041

20 1.199988 0.288000 0.057600 0.0016

30 1.200002 0.288000 0.057600 0.0015

40 1.200019 0.288000 0.057600 0.003880 1.201120 0.288001 0.057600 0.2052

Collocation [9] x = 0.03, t = 0.2 40 1.199992 0.292159 0.057599 0.0795

Least-square [7] x = 0.03, t = 0.03 40 1.1967 0.2860 0.0570 3.475

Least-square [7] x = 0.03, t = 0.03 80 1.1964 0.2858 0.0569 7.444

Petrov–Galerkin [6] x = 0.03, t = 0.05 80 1.1910 0.2855 0.0558 3.849

Table 4 Error norms for a

single solitary wave:

amplitude= 0.09,

0 ≤ x ≤ 30

t Method L2 × 103 L∞ × 103 Method L2 × 103 L∞ × 103

0 Present 0.0000 0.0000 [7] 0.0000 0.0200

10 x = 0.25 0.0024 0.0120 x = 0.1 0.0200 0.0200

20 t = 0.25 0.0018 0.0089 t = 0.05 0.0400 0.0300

40 0.0010 0.0049 0.0900 0.070080 0.0004 0.0015 0.2200 0.1600

Table 5 Invariant values

for a single solitary wave:

amplitude= 0.09,

0 ≤ x ≤ 30


0 Present 0.3600 0.0259 0.00156 [7] 0.3600 0.0259 0.00156

10 x = 0.25 0.3600 0.0259 0.00156 x = 0.1 0.3599 0.0259 0.00156

20 t = 0.25 0.3600 0.0259 0.00156 t = 0.05 0.3599 0.0259 0.00155

40 0.3600 0.0259 0.00156 0.3597 0.0259 0.00155

80 0.3600 0.0259 0.00156 0.3593 0.0259 0.00155

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64 Nonlinear Dyn (2008) 51:59–70

Table 6 Error norms for a

single solitary wave:

amplitude= 0.03,

0 ≤ x ≤ 30

t Method L2 × 103 L∞ × 103 Method L2 × 103 L∞ × 103

0 Present 0.0000 0.0000 [7] 0.0017 0.0055

40 x = 0.25 0.0007 0.0036 x = 0.1 0.0084 0.0063

80 t = 0.25 0.0005 0.0024 t = 0.1 0.0177 0.0127

Table 7 Invariant values for a single solitary wave: amplitude = 0.03, 0 ≤ x ≤ 30


0 Present 0.1200 0.00288 0.000058 [7] 0.1200 0.00288 0.000058

40 x = 0.25 0.1200 0.00288 0.000058 x = 0.1 0.1200 0.00288 0.000058

80 t = 0.25 0.1200 0.00288 0.000058 t = 0.1 0.1200 0.00288 0.000058

30, x0 = 10, ε = ν = 1, x = t = 0.25. To examine

the behavior of the present numerical algorithm for the

solitary wave with the three different amplitudes 0.3,0.09, and 0.03, we use the L2- error norm to measure

the accuracy and the quantities I 1, I 2, and I 3 to measure

conservation. We discuss the three cases:

(i) The case with amplitude 0.3, the analytic values

of the invariants are: I 1 = 1.2, I 2 = 0.288, and

I 3 = 0.0576. By the time t = 40, see Table 3 where

the L2-error norm is less than 0.0038 × 10−3 and

the constants of motion vary little from the ana-

lytic value: I 1 varies by less than 0.002%, I 2 and

I 3 are conserved during the experiment. In a cor-responding simulation using a collocation method

with quartic spline elements [9], the L2-error norm

is less than 0.0796× 10−3 and the quantity I 1changes by less than 0.0007%, I 2 by less than 1.5%,

and I 3 varies by less than 0.002%, and using a

least-squares method with linear spline elements

[7] the L2-error norm is less than 3.476 × 10−3

and the quantity I 1 changes by less than 0.28%,

I 2 by less than 0.7%, and I 3 varies by less than

1.1%. At time t = 80, the L2-error norm is less than

0.2053 × 10−3, which is smaller than the previousresults [6, 7] 3.849 × 10−3 and 7.444× 10−3, re-

spectively. The invariant value I 1 changes by less

than 0.094%, I 2 by less than 0.0004%, and I 3 does

not change and in the corresponding simulation us-

ing Petrov–Galerkin method with cubic spline fi-

nite elements [6] the invariant values I 1, I 2, and I 3change by less than 0.75, 0.87, and 3.13%, respec-

tively, and using a least-squares method with linear

spline elements [7] the quantity I 1 changes by less

than 0.3%, I 2 by less than 0.74%, and I 3 varies by

less than 1.22%.

(ii) The case with amplitude 0.09 and time t = 80,the analytic values of the invariants are: I 1 = 0.36,

I 2 = 0.02592, and I 3 = 0.001555. The computed

values of the L2-error norm and the quantities I 1,

I 2, and I 3 are given in Tables 4 and 5, respec-

tively, with the corresponding previous results. This

simulation of a solitary wave of amplitude 0.09

leads, with the present algorithm, to the L2-and

L∞-error norms of 0.0004 × 10−3 0.0015 × 10−3,

respectively, while I 1, I 2, and I 3 are constants

during the experiment. In corresponding simula-

tion using least-squares method with linear splinefinite elements [7], the L2-and L∞ error norms are

0.22× 10−3 0.15 × 10−3 respectively, while the

invariant values change by less than 0.65% during

the experiment.

(iii) The case with small amplitude 0.03 and time

t = 80, the analytic values of the invariants are:

I 1 = 0.12, I 2 = 0.00288, and I 3 = 0.000058. The

computed values of the L2-error norm and the

quantities I 1, I 2, and I 3 are given in Tables 6

and 7, respectively, with the corresponding previ-

ous results. This simulation of a solitary wave of amplitude 0.03 leads, with the present algorithm,

to the L2-and L∞-error norms of 0.0005× 10−3

0.0024× 10−3, respectively, while I 1, I 2,and I 3 are

constants during the experiment. In corresponding

simulation using least-squares method with linear

spline finite elements [7], the L2- and L∞-error

norms are 0.0177 × 10−3 0.0127 × 10−3, respec-

tively, while the invariant values are constants dur-

ing the experiment.

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Nonlinear Dyn (2008) 51:59–70 65

Table 8 Invariant values for the interaction of positive and negative solitary waves, 0 ≤ x ≤ 80


1 Present −2.399998 97.919679 −58.521339 [9] −2.399999 97.909970 −58.521330

4 N = 320 −2.399999 97.919605 −58.521151 N = 800 −2.399996 97.908200 −58.521110

6 t = 0.1 −2.400001 97.919581 −58.521044 t = 0.1 −2.399999 97.694180 −58.521440

8 −2.399941 97.925487 −58.522231 −2.399998 97.012280 −58.526150

10 −2.399900 97.915904 −58.522098 −2.399998 97.149240 −58.526940

4.2 The interaction of solitary waves

Consider the initial condition of two solitary waves:

u( x, 0) = u1 + u2 (28)

where

ui = 3ci sec h2(0.5( x − xi − ci )), i = 1, 2 (29)

and solving the EW equation over the region a ≤ x ≤ b

taking ε = ν = 1 and appropriate boundary conditions.

4.2.1 The interaction of positive and negative solitary

waves

Gardner [5] and Raslan [9] have studied the interac-

tion of a positive and negative solitary waves for the

EW equation, and observed the collision to produceadditional pairs of daughter solitary waves emanat-

ing from the point of initial contact, an observation

confirmed by [13–15]. We have repeated those exper-

iments using the appropriate initial condition Equa-

tions (27)–(28) and solved the EW equation over the

region 0 ≤ x ≤ 80 taking, c1 = 1.2 x1 = 23, c2 =

−1.4, x2 = 38, x = 0.25, and t = 0.1. The an-

alytical valued of the invariant quantities are I 1 =

12(c1 + c2) = −2.4, I 2 = 28.8(c21 + c2

2) = 97.92, and

I 3 = 57.6(c31 + c3

2) = −58.5216. In Fig. 1 weshow the

behavior of interaction of positive and negative soli-

tary waves at time t = 10. The values of I 1, I 2, and I 3throughout the simulation are shown in Table 8 com-

paring with Raslan [9] and all are satisfactorily con-

served; I 1 changes by less than 0.0042%, I 2 by lessthan 0.0042%, and I 3 varies by less than 0.0009%. In

corresponding simulation using a collocation method

with quartic spline finite elements [9], the invariant

quantity I 1 changes by less than 0.00009%, I 2 by less

than 0.79%, and I 3 by less than 0.0092% during the

experiment.

4.2.2 The interaction of two positive solitary waves

Raslan [9] has studied the interaction of two positivesolitary waves for the EW equation with initial con-

dition and observed the two waves have apparently

passed through one another and emerged unchanged

by the encounter. We have chosen to study a similar

situation using the initial condition given by Equations

(27)–(28). The EW equation was solved over the region

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

- 5

- 4

- 3

- 2

- 1

0

1

2

3

u

X

Fig. 1 The interaction of

two opposite solitary waves

at t = 10

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66 Nonlinear Dyn (2008) 51:59–70

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

a: t=0.5

u

X

0 10 20 30 40 50 60 70 80

0

1

2

3

4

5

b: t=15

u

X

0 20 40 60 80

0

1

2

3

4

5

c: t=20

u

X

0 20 40 60 80

0

1

2

3

4

5

d: t=25

u

X

Fig. 2 The motion of two positive solitary waves with amplitudes ratio two to one (2:1) before the interaction and after the interaction

0 ≤ x ≤ 80with c1 = 1.5, x1 = 10, c2 = 0.75, x2 =

25, x = 0.4, and t = 0.1. The analytical valued of

the invariant quantities are I 1 = 27.0, I 2 = 81.0, and

I 3 = 218.7. The configuration at time t = 25, which is

sometime after the interaction is complete, is shown in

Fig. 2a–d. The waves have apparently passed through

one another and emerged unchanged by encounter. The

simulations are run to time t = 25, and the invariants

I 1, I 2, and I 3 are recorded in Table 9 comparing with

Raslan [9] and all are satisfactorily conserved; I 1 does

not changes, I 2 by less than 0.0004, and I 3 varies by less

than 0.0002%. In corresponding simulation using a col-

location method with quartic spline finite elements [9],

the invariant quantity I 1 changes by less than 0.47%, I 2by less than 0.28%, and I 3 by less than 0.0008% during

the experiment.

Now, we study the interaction of two solitary waves

as above but with amplitudes ratio three to one (3:1), in

this case it changes only the value of c1 = 2.25. In Fig.

3a–d, the interaction of these two solitary waves are

plotted at different times. The analytical valued of the

invariants can be found as I 1 = 36.0, I 2 = 162.0, and

I 3 = 680.4. The configuration at time t = 15, which is

some time after the interaction is complete, is shown in

Fig. 3a–d. The waves have apparently passed through

one another and emerged unchanged by encounter. The

simulations arerun to time t = 15, and the invariants I 1,

I 2, and I 3 are given in Table 10 comparing with Raslan

[9] and all are satisfactorily conserved; I 1 does not

changes, I 2 by less than 0.007%, and I 3 varies by less

than 0.0093%. In corresponding simulation using a col-

location method with quartic spline finite elements [9],

the invariant quantity I 1 changes by less than 0.003%,

I 2 by less than 0.012%, and I 3 by less than 0.0089%

during the experiment.

Now, we study the interaction of two solitary waves

as earlier but with amplitude ratio four to one (4:1),

in this case the change is only in the value of c1 = 2

and c2 = 0.5. In Fig. 4a–d, the interaction of these two

solitary waves are plotted at different times. The ana-

lytical valued of the invariant quantities are I 1 = 30,

I 2 = 122.4, and I 3 = 468. The configuration at time

t = 15, which is sometime after the interaction is com-

plete, is shown in Fig. 4a–d. The waves have apparently

Springer



Nonlinear Dyn (2008) 51:59–70 67

Table 9 Invariant values for the interaction of two positive solitary waves with amplitudes ratio 2:1, 0 ≤ x ≤ 80


0.5 Present 26.999961 81.000402 218.702754 [9] 26.999730 81.010280 218.701200

5 N = 200 27.000001 81.000244 218.702045 N = 800 27.000080 81.000370 218.700600

10 t = 0.1 27.000000 81.000075 218.701267 t = 0.1 27.000050 80.992990 218.699700

15 27.000000 80.999990 218.700697 27.001290 80.996670 218.699000

20 27.000000 80.999877 218.700413 27.026770 82.408130 218.698900

25 27.000000 80.999703 218.699658 27.124800 81.220630 218.698300



0.5 Present 36.000011 162.000888 680.408994 [9] 36.000060 162.015400 680.399500

5 N = 200 36.000001 161.996975 680.382228 N = 800 36.000030 161.979500 680.372400

10 t = 0.1 36.000000 161.994645 680.366003 t = 0.1 36.000060 161.980400 680.354600

15 36.000000 161.990234 680.336773 36.000990 161.982000 680.339500

0 1 0 2 0 3 0 4 0 5 0 60 7 0 8 0

0

1

2

3

4

5

6

7

8

a : t= 1 5

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7

8

b : t = 5

u

X

0 1 0 2 0 3 0 4 0 5 0 60 7 0 8 0

0

1

2

3

4

5

6

7

8

c : t= 1 0

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7

8

d : t= 1 5

u

X

Fig. 3 The motion of two positive solitary waves with amplitudes ratio three to one (3:1) before the interaction and after the interaction.

passed through one another and emerged unchanged

by encounter. The simulations are run to time t = 15,

and the invariants I 1, I 2, and I 3 are given in Table 11

comparing with Raslan [9] and all are satisfactorily

conserved; I 1 does not changes from the analytic value,

I 2 by less than 0.003%, and I 3 varies by less than

0.0045%. In corresponding simulation using a collo-

cation method with quartic spline finite elements [9],

the invariant quantities I 1 changes by less than 0.029%,

I 2 by less than 0.012%, and I 3 by less than 0.32% dur-

ing the experiment.

At the end of the study of the interaction of two

solitary waves, we study the interaction of three pos-

itive solitary waves with amplitudes ratio 9:3:1, in

this case we put c1 = 4.5, c2 = 1.5, c3 = 0.5, x =

0.4, t = 0.1, x1 = 10, x2 = 25, x3 = 35, and the

Springer



68 Nonlinear Dyn (2008) 51:59–70

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7a : t = 5

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7

b : t = 5

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7

c : t = 5

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

0

1

2

3

4

5

6

7

d : t= 1 5

u

X

Fig. 4 The motion of two positive solitary waves with amplitudes ratio four to one (4:1) before the interaction and after the interaction

interval 0 ≤ x ≤ 100. The analytical value of the

invariant quantities are I 1 = 12(c1 + c2 + c3) = 78.0,

I 2 = 28.8(c21 + c2

2 + c23) = 655.2, and I 3 = 57.6(c3

1 +

c32 + c3

3) = 5450.4. The simulations are run to time

t = 15, which is some time after the interaction is

complete, as shown in Fig. 5a–d. The waves have ap-

parently passed through one another and emerged un-

changed by encounter. Thevalues taken by the invariant

quantities I 1, I 2, and I 3 over the period of simulation

are given in Table 12 comparing with Raslan [9] andall are satisfactorily conserved; I 1 varies by less than

0.00003%, I 2 by less than 0.43%, and I 3 by less than

0.71%. In corresponding simulation using a colloca-

tion method with quartic spline finite elements [9], the

invariant quantities I 1 changes by less than 0.006%, I 2by less than 0.37%, and I 3 by less than 0.72% during

the experiment.

4.3 Maxwellian initial condition

Consider the Maxwellian initial condition

u( x, 0) = exp(−( x − 7)2). (30)

We have solved the EW equation with initial con-

dition (30) and various values of the parameter ν.

We discuss the numerical solution in the cases: ν =

0.2, 0.04, 0.001.



0.5 Present 30.000000 122.400616 468.004881 [9] 29.999930 122.41500 468.000200

5 N = 200 30.000001 122.399078 467.995550 N = 800 30.000040 122.386300 467.991000

10 t = 0.1 30.000000 122.398102 467.989175 t = 0.1 30.000070 122.389500 467.98370015 30.000000 122.396379 467.979127 30.008420 122.789500 467.974600

Table 12 Invariant values for the interaction of three positive solitary waves with amplitudes ratio 9:3:1, 0 ≤ x ≤ 100


0.5 Present 78.000015 655.168150 5449.679333 [9] 78.001320 655.118800 5449.065000

5 N = 250 77.999999 654.543602 5440.868793 N = 1000 78.000240 654.349400 5438.952000

10 t = 0.1 77.999998 653.483686 5426.678714 t = 0.1 77.997310 653.418900 5426.081000

15 77.999984 652.411538 5412.231849 77.995390 652.810400 5411.639000

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Nonlinear Dyn (2008) 51:59–70 69

Table 13 Invariant values

for Maxwellian initial

condition

ν t I 1 (Present) I 2 (Present) I 3 (Present) I 1 [9] I 2 [9] I 3 [9]

0.2 0.5 1.772454 1.503977 1.023327 1.772454 1.503339 1.023327

1.0 1.773063 1.508032 1.023328 1.772457 1.503303 1.023327

2.0 1.773604 1.520078 1.023329 1.772531 1.503190 1.023328

3.0 1.774116 1.539771 1.023329 1.774443 1.505232 1.023329

4.0 1.774653 1.566542 1.023330 1.796324 1.583276 1.023345

0.04 0.5 1.772454 1.303447 1.023327 1.772454 1.303289 1.023328

1.0 1.773063 1.308312 1.023328 1.772453 1.303178 1.023331

2.0 1.773603 1.322768 1.023329 1.772454 1.302571 1.023355

3.0 1.774114 1.346399 1.023329 1.772447 1.301551 1.023400

4.0 1.774648 1.378525 1.023330 1.765447 1.301041 1.023424

0.001 0.5 1.772453 1.254567 1.023327 1.772453 1.254567 1.023326

1.0 1.772454 1.259631 1.023328 1.772454 1.254566 1.023327

2.0 1.772453 1.274673 1.023329 1.772453 1.254304 1.023330

3.0 1.772434 1.299265 1.023329 1.772434 1.253925 1.023327

4.0 1.760800 1.332696 1.023330 1.760800 1.255103 1.023311

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

0

2

4

6

8

1 0

1 2

1 4

u

X

a : t = 0 . 5

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

0

2

4

6

8

1 0

1 2

1 4

b : t= 5

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

0

2

4

6

8

1 0

1 2

1 4

c : t= 1 0

u

X

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0

0

2

4

6

8

1 0

1 2

1 4

d : t= 1 5

u

X

Fig. 5 The motion of three positive solitary waves with amplitudes ratio 9:3:1 before the interaction and after the interaction

For ν = 0.2, the simulations are run to time t = 4,

while the invariant quantities I 1, I 2, and I 3 are given

in Table 13 comparing with Raslan [9]. The three in-

variant quantities change from the initial by a fac-

tor of 0.0022, 0.063, and 0.000003, respectively. In


with quartic spline finite elements [9], the three in-

variant quantities change from the initial by a factor

of 0.024, 0.08, and 0.000018, respectively, during the

experiment.





tor of 0.0022, 0.075, and 0.000003, respectively. In


with quartic spline finite elements [9], the three in-

variant quantities change from the initial by a factor

of 0.007, 0.0023, and 0.0001, respectively, during the

experiment.

Springer



70 Nonlinear Dyn (2008) 51:59–70





tor of 0.012, 0.078, and 0.000003 respectively. In


with quartic spline finite elements [9], the three invari-ant quantities change from the initial by a factor of

0.012, 0.00065, and 0.00002, respectively, during the

experiment.

5 Discussion

It has been shown that the numerical solution for solv-

ing the EW equation using the spectral method based on

Chebyshev polynomials within the collocation methodis more accurate compared to the recent results dur-

ing all run to the simulations. The error norms com-

puted by the present algorithm with different ampli-

tudes compared to the previous results were found to

be smaller. The three invariants of motion are satis-

factorily constant in all the computer simulations de-

scribed here, so that the algorithm can fairly be de-

scribed as conservative. So, we deduce that this algo-

rithm is accurate and more efficientthan theprevious al-

gorithms and we believe that this approach will also be

useful for solving similar nonlinear partial differentialequations.

It is worthwhile noticing that all our computa-

tions have been conducted on a 32-bit machine, which

means that the accuracy could have been much bet-

ter if we used a mainframe with 256 or 512-bit

processors.

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3. Morrison, P.J., Meiss, J.D., Carey, J.R.: Scattering of RLWsolitary waves. Physica D 11, 324–336 (1981)

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(1990)

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