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New Non-traveling Solitary Wave Solutions for a Second-order

Korteweg-de Vries Equation

Woo-Pyo Hong and Young-Dae Junga

Department of Physics, Catholic University of Taegu-Hyosung,

Hayang, Kyongsan, Kyungbuk 712-702, South Koreaa Department of Physics, Hanyang University, Ansan, Kyunggi-Do 425-791, South Korea

Reprint requests to Prof. W.-P. H.;E-mail: wphong@cuth.cataegu.ac.kr

Z. Naturforsch.54 a,375378 (1999); received March 12, 1999

Modeling the propagation of two different wave modes simultaneously, the second-order KdVequation is of current interest. Applying a tanh-typed method with symbolic computation, we havefound certain new analytic soliton-typed solutions which go beyond the the previously obtainedtraveling wave solutions.

Key words:Nonlinear Evolution Equations; Second-order KdV Equation; Solitonic Solutions;Symbolic Computation.

We investigate the second-order KdV equation pro-posed by Korsunsky [1], which is assumed to gov-ern propagation in the same direction of two wavemodes with the same dispersion relation, but withdifferent phase velocities, nonlinearity and dispersionparameters:

u

x x

+ ( c 1+ c 2)ux t

+ c 1 c 2 ux x

+h

( 1+ 2)

t

+ ( 1 c 2+ 2 c 1)

x

i

u u

x

+

h

(

1+

2)

t + (

1c

2+

2c

1)

x

i

u u

x x x x

= 0

(1)

where u (x t ) is a field function, ci

are the phase ve-locities,

i

the parameters of nonlinearity, and i

thedispersion parameters for the first (

i

= 1) and second(

i

= 2) mode. This equation exhibits two importantfeatures: (i) if one of the modes is absent, the otherobeys the ordinary KdV equation, and (ii) on applica-tion of the perturbation techniqu,e this equation leadsto the uncoupled KdV equations for each mode ona corresponding temporal and spatial scale [1]. Wecan show that in the absence of the other wave theevolution of each mode is described by its own KdV

equation

u

t

+c

i

u

x

+

i

u u

x

+

i

u

x x x

= 0 (2)

09320784 / 99 / 06000375 $ 06.00 c

Verlag der Zeitschrift fur Naturforschung, Tubingen

www.znaturforsch.com

with the traveling solitary wave solutions

u (x t ) = A sech2 f [x ; (ci

+1

3 1 A )t ]= L

i

g

where 12L 2i

= A i

=

i

:

(3)

To simplify the analysis, we transform (1), using thetransformations

= ( 1+ 2);

1=

2(x ; c 0 t ) T = ( 1+ 2);

1=

2t

c 0 = 1

2

(c 1+ c 2) U ( T ) = ( 1+ 2)u (x t )

(4)

into

U

T T

; s

2U

(5)

+

T

+ s

U U

+

T

+ s

U U

= 0

where

s = 1

2(c 1 ; c 2) =

2 ; 1

2+ 1 =

2 ; 1

2+ 1(6)

with s > 0, j j 1, and j j 1.Two families of traveling wave solutions have been

found in [1] for (5):

mailto:wphong@cuth.cataegu.ac.krmailto:wphong@cuth.cataegu.ac.kr

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376 W.-P. Hong and Y.-D. Jung Solutions for a Second-order Korteweg-de Vries Equation

U

I(x t

) =U 0+ 3 U 0 sech

2h

s

U 0( 1)

4(

1)

( 1+ 2)

; 1= 2(x ; c 0 t ) ; t

i

(7)

whereU 0 is a constant wave amplitude and = s corresponds to the two modes represented in (1) and

U

II(x t

) =A

sech2h

s

A ( s ; )

12( s ;

)

( 1+ 2)

;

1=

2(x ; c 0 t ) ; t

i

(8)

where the two roots of

2 =s

2 + 1=

3A

( s ;

) correspond to the two solitary wave modes, andA

is the waveamplitude.

In this work we apply the tanh-typed method [2 - 4] with symbolic computation to (5) to find somenon-traveling solitary wave solutions. As an Ansatz we assume that the physical field

U

( T

) has the form

U

( T

) =

N

X

n =0

A

n

(T

)

tanhn [G (T ) + H (T )] (9)

where N is the integer determined via the balance of the highest-order contributions from both the linear and

nonlinear terms of (5) as N = 2, while A N (t ), G (T ), and H (T ) are the non-trivial differentiable functions tobe determined.

With the symbolic computation package Maple we substitute the Ansatz (9), together with the aboveconditions, into (5) and collect the coefficients of like powers of tanh:

(tanh6) : 10 A 2(T )G (T )[A 2 (T ) s G (T ) + A 2(T )G (T )T + A 2(T )H (T )T + 12 s G (T )3 (10)

+ 12G

(T

)2G

(T

)T

+ 12G

(T

)2H

(T

)T

]

(tanh5) :;

2A 2(T )

2G

(T

)T

;

24A 2(T )

T T

G

(T

)3 + 12A 1(T )G (T )

T

A 2(T )G (T ) (11)

+ 24A 1(T )G (T )3

G (T )T

+ 24 s A 1(T )G (T )4

; 72A 2(T )G (T )2

G (T )T

+ 24 A 1(T )G (T )3

H (T )T

+ 12 s A

1(T

)G

(T

)

2A

2(T

) + 12A

1(T

)H

(T

)T

A

2(T

)G

(T

);

4A

2(T

)T T

A

2(T

)G

(T

)

(tanh4) :;

6A 1(T )

T

(G

(T

))3 + 6A 2(T )H (T )

T

2;

6s

2A 2(T )(G (T ))

2 + 3A 1(T )

2G

(T

)T

G

(T

)

(12)

;

3A 1(T )

T

A 2(T )G (T ) + 6 A 0(T )A 2(T )G (T )H (T )T

;

240A 2(T )(G (T ))

3H

(T

)T

; 240 A 2(T )G (T )3

G (T )T

; 18 A 1(T )G (T )2

G (T )T

+ 6 A 2(T )G (T )T

2

2

+ 6A 0(T )A 2(T )G (T )G (T )T ; 240 s A 2(T )G (T )

4;

16A 2(T )

2H

(T

)T

G

(T

)

+ 12A 2(T )G (T )

T

H

(T

)T

;

3A 2(T )

T T

A 1(T )G (T ) ; 16 s (A 2(T ))2(

G

(T

))2

+ 3A 1(T )

2H

(T

)T

G

(T

) + 3 s

(A 1(T ))

2G

(T

)2;

16 (A 2(T ))

2G

(T

)T

G

(T

)

; 3 A 1(T )A 2(T )G (T )T + 6 s A 0(T )A 2(T )(G (T ))2

(tanh3) :;

18A 1(T )H (T )T A 2(T )G (T ) ; 40A 1(T )G (T )

3H

(T

)T

;

40A 1(T )G (T )

3G

(T

)T

(13)

;

18A 1(T )G (T )

T

A 2(T )G (T ) ; 4 A 2(T )T

H

(T

)T

+ 120A 2(T )G (T )

2G

(T

)T

; 2 A 2(T )G (T )T T

; A 1(T )2

G (T )T

; 2 A 2(T )H (T )T T

+ 2 A 1(T )H (T )T

2

+ 2A 1(T )G (T )T

2

2;

2A 0(T )A 2(T )G (T )T ; 4 A 2(T )T G (T )T ; 2 A 1(T )T A 1(T )G (T )

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W.-P. Hong and Y.-D. Jung Solutions for a Second-order Korteweg-de Vries Equation 377

+ 4 A 2(T )T

A 2(T )G (T ) ; 2 A 0(T )A 2(T )T

G (T ) ; 2 A 0(T )T

A 2(T )G (T ) + 40 A 2(T )T

G (T )3

;

40 s A 1(T )G (T )

4;

18 s A 1(T )G (T )

2A 2(T ) ; 2 s

2A 1(T )G (T )

2

+ 2A 0(T )A 1(T )G (T )G (T )

T

+ 2 s A 0(T )A 1(T )G (T )

2 + 2A 0(T )A 1(T )G (T )H (T )

T

+ 4A

1(T

)G

(T

)T

H

(T

)T

+ 2A

2(T

)2

G

(T

)T

(tanh2) : A 2(T )T T ; 4A 1(T )2

H (T )T

G (T ) + 6A 2(T )2

H (T )T

G (T ) + 136 A 2(T )G (T )3

H (T )T

(14)

+ 3A 1(T )A 2(T )G (T )

T

;

8A 2(T )G (T )

T

2

2 + 24A 1(T )G (T )

2G

(T

)T

;

4A 1(T )

2G

(T

)T

G

(T

)

+ 3A 1(T )

T

A 2(T )G (T ) ; 16 A 2(T )G (T )T

H

(T

)T

;

8A 0(T )A 2(T )G (T )H (T )

T

; 8A 0(T )A 2(T )G (T )G (T )T

+ 136 A 2(T )G (T )3

G (T )T

+ 136 s A 2(T )G (T )4

;

4 s A 1(T )

2G

(T

)2;

8 s A 0(T )A 2(T )G (T )

2 + 6A 2(T )

2G

(T

)T

G

(T

)

+ 8s

2A 2(T )G (T )

2

; A 1(T )G (T )T T

; A 0(T )A 1(T )G (T )T

+ 6 s A 2(T )

2G

(T

)2 + 8A 1(T )

T

G

(T

)3

;

8A 2(T )H (T )

T

2; A 0(T )

T

A 1(T )G (T ) + 3 A 2(T )T

A 1(T )G (T ) ; 2 A 1(T )T

G

(T

)T

; A 0(T )A 1(T )T G (T ) ; 2 A 1(T )T H (T )T ; A 1(T )H (T )T T

(tanh1) :A 1(T )T T + 6 A 1(T )H (T )T A 2(T )G (T ) + 16 A 1(T )G (T )

3H

(T

)T

+ 16A 1(T )G (T )

3G

(T

)T

(15)

+ 6A 1(T )G (T )

T

A 2(T )G (T ) + 4 A 2(T )T

H

(T

)T

;

48A 2(T )G (T )

2G

(T

)T

+ 2A 2(T )G (T )

T T

+A 1(T )

2G

(T

)T

+ 2A 2(T )H (T )

T T

;

2A 1(T )H (T )

T

2;

2A 1(T )G (T )

T

2

2

+ 2 A 0(T )A 2(T )G (T )T + 4 A 2(T )T G (T )T + 2 A 1(T )T A 1(T )G (T ) + 2 A 0(T )A 2(T )T G (T )

+ 2A 0(T )

T

A 2(T )G (T ) ; 16 A 2(T )T

G

(T

)3 + 16 s A 1(T )G (T )

4 + 6 s A 1(T )G (T )

2A 2(T )

+ 2s

2A 1(T )G (T )

2;

2A 0(T )A 1(T )G (T )G (T )

T

;

2 s A 0(T )A 1(T )G (T )

2

; 2A 0(T )A 1(T )G (T )H (T )T ; 4 A 1(T )G (T )T H (T )T

(tanh0) :A 0(T )T T ; 2 A 1(T )T G (T )

3 + 2A 1(T )T H (T )T + 2 A 2(T )H (T )T

2 +A 1(T )H (T )T T (16)

+A 0(T )A 1(T )G (T )

T

+ 2A 1(T )

T

G

(T

)T

+ 2A 2(T )G (T )

T

2

2;

2s

2A 2(T )G (T )

2

+A 1(T )

2H

(T

)T

G

(T

) +A 0(T )

T

A 1(T )G (T ) + A 0(T )A 1(T )T

G

(T

) +A 1(T )G (T )

T T

; 16 A 2(T )G (T )3

H (T )T

; 6 A 1(T )G (T )2

G (T )T

+ 4 A 2(T )G (T )T H (T )T

;

16 s A 2(T )G (T )

4 + s A 1(T )

2G

(T

)2 + 2 s A 0(T )A 2(T )G (T )

2 +A 1(T )

2G

(T

)T

G

(T

)

+ 2A 0(T )A 2(T )G (T )G (T )

T

+ 2A 0(T )A 2(T )G (T )H (T )

T

;

16A 2(T )G (T )

3G

(T

)T

where the subscriptT

denotes time derivative.Our goal is to find the conditions forA

N

(T

) G

(T

),and

H

(T

) which simultaneously let the above termsbecome zero. After dealing with some complicatedsymbolic calculations using Maple, we obtained anew family of non-traveling solitary-wave solutions

as

U

new( T

) =A 0(T ) + A 1(T ) tanh

1[G

(T

)

+H

(T

)]

+ A 2(T ) tanh2[G (T ) + H (T )] (17)

where

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378 W.-P. Hong and Y.-D. Jung Solutions for a Second-order Korteweg-de Vries Equation

Fig. 1. Beyond traveling solitary-wave solutionU

new(x t

)with the parameters

2 = 10 1 = 001 2 = 10, 1 = 001, c 1 = 1 c 2 = 001 C 1 = 01 C 2 = 001,and

C 3 = 001, satisfying the solitary wave property thatU

new(x t

) tends to zeroj x j

as approaches infinity.

G

(T

) =G

= nonzero constant

(18)

H

(T

) =C 1 T

2 +C 2 T + C 3 (19)

where Ci

are arbitary constants

A 2(T ) = ; 12G 2

(20)

A 1(T ) = 0 (21)

A 0(T ) R

(T

)

S (T )

(22)

R

(T

) ;

24G

(2C 1 T + C 2)

4 + (192G

4;

48G

2s

)

(2 C 1 T + C 2)3 + 576 G 5 s (2 C 1 T + C 2)

2

+ (576s

2G

6 + 48s

3G

4)(2C 1 T + C 2)

+ 24s

4G

5 + 192s

3G

7

S

(T

)

3G

3s

(2C 1 T + C 2)

2 + 3s

2G

4(2C 1 T + C 2)

G

2(2C 1 T + C 2)

3 +s

3G

3

and the following auxiliary conditions are requiredfor

U

new( T

) to be a solution of (5):

[1] S. V. Korsunsky, Phys. Lett. A 185, 174 (1994).[2] B. Tian, K. Zhao and Y. T. Gao, Int. J. Engng. Sci.

(Lett.)35, 1081 (1997).

[3] Y. T. Gao and B. Tian, Acta Mechanica 128, 137(1998).

[4] E. Parkes and B. Duffy, Computer Phys. Comm.98,

288 (1996).

Fig.2. A typical sech2 -typedsolitary wave solutionU

II (x t

)with

A

= 1 2 = 10 1 = 001 2 = 10 1 = 001 c 1 = 1,

and c 2 = 001.

=

= 1 or

=

=;

1

(23)

which imply that

2

1 and

2

1 for

=

= 1 or 1 2 and 1 2 for = = ; 1.

Physically, these conditions indicate that two solitarywave modes propagate in the medium, where onemodes nonlinearity and dispersion parameters aremuch bigger than the other ones.

Finally we present two figures with some selectedparameters. We set

2 = 10 1 = 0:01 2 = 10 1 =0

:

01 c 1 = 1 c 2 = 0: 01 C 1 = 0: 1 C 2 = 0: 01, and

C 3 = 0:01 for the new solitary-wave solutions (17)and plot U new(x t ) in Figure 1. The new solutionssatisfy solitary wave property that U new(x t ) tendsto zero as

j x j

approaches infinity. For comparison in

Fig. 2 we plot the traveling wave solution ofU

I I

(x t

)(8) with

A

= 1 2 = 10 1 = 0: 01 2 = 10 1 =

0:

01 c 1 = 1, and c 2 = 0: 01.

To sum up, the tanh method and symbolic computa-tions lead to the new analytic solitary-wave solutions(17), different from the previously obtained results [1]for the second order KdV equation.

Acknowledgements

This research was supported by the Korean Re-search Foundation through the Basic Science Re-search Institute Program (1998-015-D00128).