Solitary Wave Solutions of High Order Scalar Fields and Coupled Scalar Fields

Solitary Wave Solutions of High Order Scalar Fields and Coupled Scalar Fields

Jian-song Yang and Sen-yue Lou^a

Physics Department of Hangzhou Normal College, Hangzhou 310000, P. R. China

aInstitute of Mathematical Physics, Ningbo University, Ningbo 315211, P. R. China Z. Naturforsch. 54 a, 195–203 (1999); received December 13, 1998

An arbitrary Klein-Gordon field with a quite general constrained condition (which contains an arbitrary function) can be used as an auxilialy field such that some special types of solutions of high order scalar fields can be obtain by solving an ordinary differential equation (ODE). For a special type of constraint, the general solution of the ODE can be obtained by twice integrating.

The solitary wave solutions of the⁵model are treated in an alternative simple way. The obtained solutions of the⁵model can be changed to those of the⁸field and coupled scalar fields.

1. Introduction

The Lagrangian density of a generalized nonlinear Klein-Gordon (NKG) field

⁽

x

¹

;x

²

;:::x

D

;t

^{) in}

(

D

+ 1)-dimensions has the form

L[

;

^∂

^{] =∂}

^∂

^;

V

⁽

⁾

;

⁽¹⁾

and the corresponding equation of wave motion reads

2

^XD

i⁼¹

xⁱx^i;

tt⁼

F

⁽

⁾

;

F

⁽

^{) d}

V

d

:

⁽²⁾

For not very large

, the potential

V

V

⁽

^{) in (1)}

can be replaced by a polynomial function of

^:

V

^=XN

n⁼⁰

1

nv

ⁿ

ⁿ

;

v

n⁼ ₍

n

^;11)!dn

v

d

ⁿ

⁼⁰

;

⁽³⁾

and then the related wave motion equation becomes

2

^=XN

n⁼¹

v

n

^n;1

:

⁽⁴⁾

Some well known NKG models are just spe- cial cases of (3) (or (4)), say, the

^{4 model cor-}

responding to

N

^{= 4}

; v

^{1 =}

v

³ = 0 [1], the

⁶

model to

N

^{= 6}

; v

^{1 =}

v

^{3 =}

v

⁵ = 0 [2], and the Reprint requests to Dr. Sen-yue Lou;

E-mail: sylou@public.nbptt.zj.cn.

0932–0784 / 99 / 0300–0195 $ 06.00^c Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingenwww.znaturforsch.com

³⁺

⁴ (or Friedberg-Lee (FL)) model [3] is related to

N

^{= 4}

; v

¹= 0. In [4], the authors discuss the inter- esting properties of the self-exited soliton motion by using a

¹⁰⁽

N

^{= 10}

; v

²i^{+1 = 0}

;

⁽

i

^{= 0}

:::

^{4)) model.}

In [5] the generalized

^{2mmodel (}

N

^{= 2}

m; v

²i⁺¹⁼ 0

;

⁽

i

^{= 0}

;

¹

;:::;m

^;1)) is used to study the effective kink-kink interaction mediated by phonon exchange.

However, to our knowledge there are no known exact solutions of (4) for

N

^{= 5 and}

N >

6. In Sects. 2 and 3 of this paper we study the exact solitary wave solutions of (4) for

N

= 5. In Sect. 2, using an arbi- trary scalar field with a quite general constraint as a basic equation system, we solve the (

D

+ 1)-dimen- sional

⁵ model by a second order ordinary differ- ential equation (ODE). For a special constraint, the general solution of the ODE can be expressed sim- ply by an integration. The solitary wave solutions of the

⁵ model are discussd in an alternative way in Section 3. In Sect. 4, we give special solutions of the

^{8 model (}

v

^{1 =}

v

^{3 =}

v

^{5 =}

v

⁷ = 0 in (4)) by means of the

⁵ model. In Sect. 5, we discuss some special solutions of a coupled scalar field model by means of the

⁵model. The last section is a short summary.

2. Base Equation Approach for the

^5Model

For the

⁵model, the motion eq. (4) becomes

2

⁼

v

²

⁺

v

³

²⁺

v

⁴

³⁺

v

⁵

⁴

;

⁽⁵⁾

where we have set

v

¹ = 0 without loss of generality because we can make transformation

^!

⁺

c

^by

selecting the constant

c

appropriately. In order to get some interesting special solutions of (5), we introduce a basic equation system [4, 5]

2 =

A

^{( )}

;

⁽⁶⁾

∂ ^∂

D

X

i⁼¹⁽ x^i)2;( t^{)2 =}

B

^{( )}

;

⁽⁷⁾

where

A

A

^{( ) and}

B

B

⁽ ) are arbitrary func- tions of . Now we suppose that

is only a function of . In other words, the space-time dependence of

⁼

⁽ ) results from the auxiliary field , which is given by (6) and (7) with two arbitrary functions

A

and

B

. Using the basic eqs. (6) and (7) and the above ansatz, (5) becomes an ODE:

A

⁺

B

⁼

v

²

⁺

v

³

²⁺

v

⁴

³⁺

v

⁵

⁴

:

⁽⁸⁾

Furthermore, after introducing

C

^{( ) = expn2Z}

B

⁽¹⁰⁾

h

A

^{( 0);1}₂^d

B

^{( 0)}

d ⁰

i

d ⁰

o

;

(9) the ODE is changed to

^{= (}

v

²

⁺

v

³

²⁺

v

⁴

³⁺

v

⁵

⁴⁾

C

^{( (}

⁾⁾

;

⁽¹⁰⁾

where

^and are related by

^=Z

d

⁰

p

B

^{( 0)}

C

^{( 0)}

:

⁽¹¹⁾

Generally, to solve the ODE (10) is still quite difficult.

However, if we select

A

^{= 1}₂^d

B

d

;

⁽¹²⁾

then

C

⁽ ) = 1, and (10) is reduced to

⁼

v

²

⁺

v

³

²⁺

v

⁴

³⁺

v

⁵

⁴

;

^=Z

d

⁰

p

B

^{( 0)}

(13) with the general solution

^=Z

d

⁰

q

C

¹⁺

v

²

⁰²⁺²3

v

³

⁰³⁺¹2

v

⁴

⁰⁴⁺²5

v

⁵

⁰⁵

+

C

²

;

⁽¹⁴⁾

where

C

^1and

C

²are two arbitrary integral constants.

Now the remaining key problems are how to solve the basic eqs. (6) and (7) and to finish the integra- tion (14). For the first problem, the concrete solutions of (6) are dependent on the selection of the arbitrary function

B

. If we selected the function

B

as the poten- tial of the known NKG fields, say, sine-Gordon (sG),

^{4 or}

⁶ models, various types of special solutions have been given^[6]. The simplest nontrivial selection of (6) and (7) reads

2 =

²

;

⁽¹⁵⁾

∂ ^∂ ₌

^{2 2}

:

⁽¹⁶⁾

In this simple case we have

^{= ln}

:

⁽¹⁷⁾

The simplest solution of (15) and (16) has the form

=

h^XN ⁼¹

c

^exp

¹

i¹

;

⁽¹⁸⁾

where

;

^1and

c ;

^{= 1}

;

²

;:::;N

are arbitrary con- stants and

^=XD

i⁼¹

P

_i

x

i^;

! t;

⁽¹⁹⁾

while

P

_i^and

!

satisfy the conditions D

X

i⁼¹

P

_i

P

_i^0;

! !

^{0 = 1}

; ;

^{0 = 1}

;

²

;:::;N:

⁽²⁰⁾

3. Solitary Wave Solutions of the

^5Model

Generally, the integration of (14) can not be ex- pressed by simple functions. For the solitary wave solutions with some special parameters (

v

i) and inte- gration constant (

C

¹) we can take an alternative way proposed, in [7] to express (14) by some known func- tions.

Equation (14) is equivalent to

^{2 =}

C

¹⁺

v

²

²⁺²₃

v

³

³⁺¹₂

v

⁴

⁴⁺²₅

v

⁵

⁵

:

⁽²¹⁾

To find the solitary wave solutions of (21), we take a

truncated series expansion of

around an extended singular manifold

^{= 0, while}

is determined by [7]

^=XK

k⁼⁰

a

k

^k

;

⁽²²⁾

where

a

k

; k

^{= 0}

;

¹

;:::K

are arbitrary functions of

^.

If the integer

K

is fixed, the possible expansions of

read

⁼

2 3(XK^;1)

p⁼⁰

b

p

^{p (23)}

with

b

p

; p

^{= 0}

;

¹

;:::

²3(

K

^;1) being functions of

^.

For simplicity, we take

K

^{= 4}

:

⁽²⁴⁾

Substituting (23) with (22) and (24) into (21) and putting to zero the coefficients of

^k

;k

^{= 0}

;:::

^{, we}

have eleven complicated equations of the functions

a

⁰

;a

¹

;a

²

;a

³

;a

⁴

;b

⁰

;b

¹

;b

²and the constant

C

^{1. If the}

parameters

v

²

;v

³

;v

^4and

v

⁵satisfy the conditions

W

¹¹²³²

v

²5

v

²4

v

²3^;5400

v

⁵

v

4⁴

v

^{3+ 9504}

v

5²

v

²

v

4³

; 35712

v

³5

v

²

v

⁴

v

^{3+ 31104}

v

⁴5

v

²2 (25) + 675

v

^64;2048

v

³⁵

v

^{33 6= 0}

and

(640

v

⁵

v

³3^;150

v

²4

v

²3+ 675

v

³4

v

^2;3240

v

³

v

⁵

v

⁴

v

²

+ 5832

v

⁵²

v

^{22) (;128}

v

⁵²

v

^33;45

v

⁴⁴

v

^{3+ 222}

v

³²

v

⁵

v

²⁴

+ 27

v

⁵

v

4³

v

^2;504

v

³

v

5²

v

⁴

v

^{2+ 216}

v

5³

v

²2) = 0

;

⁽²⁶⁾

we have an unique solution for

a

i

;b

i ^and

C

^{1 being}

constants:

a

⁴⁼ ₁₀^1p10

v

⁵

b

²

b

²

; a

^{3= 1}₅

b

^1p10

v

⁵

b

²

;

⁽²⁷⁾

a

²⁼ ₈^p₁₀¹

v

⁵

b

²⁽⁷

v

⁵

b

²1+ 20

v

⁵

b

⁰

b

^{2+ 5}

v

⁴

b

²⁾

;

⁽²⁸⁾

a

¹⁼

b

¹

8

q

10

v

⁵

b

³2

(^;

v

⁵

b

²1+ 20

v

⁵

b

⁰

b

^{2+ 5}

v

⁴

b

²⁾

;

⁽²⁹⁾

a

^{0= 1}

384

q

10

v

³5

b

⁵2

9

v

5²

b

⁴1^;120

v

5²

b

⁰

b

²1

b

²

; 30

v

⁴

b

²1

b

²

v

^{5+ 720}

v

5²

b

²0

b

²2 (30) + 360

v

⁴

b

⁰

b

²2

v

^5;75

v

4²

b

²2+ 320

v

³

b

²2

v

⁵

; C

^1=;2₃

v

³

b

³0^;

2

5

v

⁵

b

⁵0^;

v

²

b

²0+

b

²1

a

²0^;

1

2

v

⁴

b

⁴0

;

⁽³¹⁾

b

^{0 =;}₄

b

²

v

¹⁵

W

51840

v

5²

b

²

v

²

v

4^4;9504

v

5³

v

²

v

³4

b

²1

+ 35712

v

5⁴

v

²

v

⁴

b

²1

v

^{3+ 98304}

v

⁴5

b

²

v

²

v

3²

; 31104

v

⁵5

v

²2

b

²1^;233856

v

³5

b

²

v

²

v

²4

v

³

+ 196992

v

⁴5

b

²

v

²2

v

^{4+ 2048}

v

³3

v

⁴5

b

²1 (32) + 5400

v

5²

v

4⁴

v

³

b

²1^;11232

v

5³

v

4²

v

²3

b

²1

; 51200

v

³⁵

b

²

v

⁴

v

^33;30600

v

⁵

b

²

v

⁵⁴

v

³

+ 81120

v

5²

b

²

v

³4

v

²3+ 3375

b

²

v

4^7;675

v

4⁶

b

²1

v

⁵

while

b

^{1 and}

b

²remain free.

Now the remaining problem is to solve the ODE (22) with (27) - (30) and (32). According to the dif- ferent relations among the model parameters

v

^2,

v

^3,

v

^4,

v

⁵and the constants

b

^1and

b

², there may be nine types of possible

^solutions:

1.) If the quartic equation

4

X

k⁼⁰

a

k

^{k = 0}

;

⁽³³⁾

a

i being given by (27) - (30), possesses four non- degenerate real roots, say^f

c

¹

;c

²

;c

³

;c

^4g, the general solution of (22) reads

4

X

k⁼⁰

ln(

^;

c

k⁾

Qj⁶⁼k⁽

c

j^;

c

k^);

a

⁴⁽

^;

^{0) = 0}

;

⁽³⁴⁾

where

a

iare related to

c

i^by

a

⁰

=a

⁴⁼

c

¹

c

²

c

³

c

⁴

;

⁽³⁵⁾

a

¹

=a

^4=;

c

¹

c

²

c

^4;

c

¹

c

²

c

^3;

c

²

c

³

c

^4;

c

¹

c

³

c

⁴

;

⁽³⁶⁾

a

²

=a

⁴⁼

c

¹

c

²⁺

c

²

c

⁴⁺

c

²

c

³⁺

c

¹

c

⁴⁺

c

¹

c

³⁺

c

³

c

⁴

;

⁽³⁷⁾

a

³

=a

^4=;

c

^1;

c

^2;

c

^3;

c

⁴

:

⁽³⁸⁾

ln(

^;

c

²⁾

(

c

^2;

c

¹⁾²⁽

c

^3;

c

^2);(

c

^2;2

c

¹⁺

c

^{3) ln(}

^;

c

¹⁾

(^;

c

¹⁺

c

^2)2(;

c

¹⁺

c

^{3)2 + ln(}

^;

c

³⁾

(

c

^2;

c

³⁾⁽

c

^3;

c

¹⁾²⁺⁽

c

^2;

c

¹⁾⁽

c

^3;1

c

¹⁾⁽

^;

c

¹⁾⁺

a

⁴⁽

^;

⁰⁾

= 0

:

⁽³⁹⁾

3.) If three of the four roots of (33) are equal,

c

^{3 =}

c

^{4 =}

c

¹, the general solution of (22) becomes ln(

^;

c

¹⁾

(^;

c

¹⁺

c

^{2)3 +(}

c

^2;

c

¹⁾¹²⁽

c

^1;

^{) ;2(}

c

^2;

c

¹¹⁾⁽

^;

c

^{1)2 ;ln(}

^;

c

²⁾

(

c

^2;

c

^{1)3 +}

a

⁴⁽

^;

^{0) = 0}

:

⁽⁴⁰⁾

4.) If all four real roots of (33) are degenerate, the function

^becomes

^=;₍₃

a

⁴⁽

^;1

^{0))1=3 +}

c

¹

:

⁽⁴¹⁾

5.) If there are two sets of degenerate roots of (33) say,

c

^{4 =}

c

¹

; c

³⁼

c

², the solution of (22) has the form

c

^1;2

c

^2ln

^;

c

¹

^;

c

^{2 +}

^;1

c

^{1 +}

^;1

c

^{2 +}

a

⁴⁽

c

^2;

c

¹⁾²⁽

^;

^{0) = 0}

:

⁽⁴²⁾

6.) If two of the roots are conjugate complex and the other two are nondegenerate real, we have

d

¹

c

^{3+ 2}

c

³

c

^{4+ 2}

d

²⁺

d

²1+

d

¹

c

⁴

(

d

¹

c

³⁺

c

²3^;

d

²⁾⁽

d

^2;

d

¹

c

^4;

c

²4)

q

;4

d

^2;

d

²1

arctan

2

⁺

d

¹

q

;4

d

^2;

d

²1

;

ln(

^;

c

⁴⁾

(

c

^3;

c

⁴⁾⁽

d

^2;

d

¹

c

^4;

c

²4) + ln(

^;

c

³⁾

(

c

^3;

c

⁴⁾⁽

d

^2;

d

¹

c

^3;

c

²3)^;

(

c

³⁺

d

¹⁺

c

^{4) ln(}

²⁺

d

¹

^;

d

²⁾

(

d

^2;

d

¹

c

^4;

c

²4)(^;

d

¹

c

^3;

c

²3+

d

^{2) +}

a

⁴⁽

^;

^{0) = 0}

:

⁽⁴³⁾

where

d

²1+ 4

d

²

<

⁰

;

⁽⁴⁴⁾

and the

a

iare linked with^f

d

¹

;d

^2gandf

c

³

;c

^4gby

a

⁰

=a

^4=;

d

²

c

³

c

⁴

;

⁽⁴⁵⁾

a

¹

=a

⁴⁼

d

²

c

⁴⁺

d

²

c

³⁺

d

¹

c

³

c

⁴

;

⁽⁴⁶⁾

a

²

=a

⁴⁼

d

^2;

d

¹⁽

c

⁴⁺

c

^{3) +}

c

³

c

⁴

;

⁽⁴⁷⁾

a

³

=a

⁴⁼

d

^1;

c

^3;

c

⁴

:

⁽⁴⁸⁾

7.) If two of the roots are conjugate complex and the other two are degenerate real (

c

⁴⁼

c

³in (45) - (48)), the related solution of (22) reads

;

2

d

¹

c

³⁺

d

²1+ 2

c

²3+ 2

d

²

(

d

^2;

d

¹

c

^3;

c

²3)²

q

;4

d

^2;

d

²1

arctan 2

⁺

d

¹

q

;4

d

^2;

d

²1

+(2

c

³⁺

d

^{1) ln(}

^;

c

³⁾

(

d

^2;

d

¹

c

^3;

c

²3)²

;

(2

c

³⁺

d

^{1) ln(}

²⁺

d

¹

^;

d

²⁾

2(

d

^2;

d

¹

c

^3;

c

²3)²

;

1

(

d

^2;

d

¹

c

^3;

c

²3)(

^;

c

³⁾⁺

a

⁴⁽

^;

^{0) = 0}

:

⁽⁴⁹⁾

2.) If (33) possesses four real roots and two roots are degenerate, say,

c

⁴⁼

c

¹, then the solution of (22) has the form

8.) If (33) possesses two sets of nondegenerate conjugate complex roots, the general solution of (22) becomes (2

d

⁴⁺

d

¹

d

^3;

d

²1^;2

d

²⁾

q

;4

d

^2;

d

²1

arctan 2

⁺

d

¹

q

;4

d

^2;

d

²1

+1

2(

d

^3;

d

^{1) ln(}

²⁺

d

¹

^;

d

²⁾

+ 2

d

^2;2

d

^4;

d

²3+

d

¹

d

³

q

;4

d

^4;

d

²3

arctan 2

⁺

d

³

q

;4

d

^4;

d

²3

;

1

2(

d

^3;

d

^{1) ln(}

²⁺

d

³

^;

d

^{4) (50)}

+ (

d

^24;2

d

²

d

^4;

d

²³

d

²⁺

d

¹

d

³

d

⁴⁺

d

^22;

d

²¹

d

⁴⁺

d

¹

d

²

d

³⁾

a

⁴⁽

^;

^{0) = 0}

;

where

d

²1+ 4

d

²

<

⁰

; d

²3+ 4

d

⁴

<

⁰

;

⁽⁵¹⁾

and

a

i^and

d

iare related by

a

⁰

=a

⁴⁼

d

²

d

⁴

; a

¹

=a

^4=;

d

²

d

^3;

d

¹

d

⁴

;

⁽⁵²⁾

a

²

=a

^4=;

d

^2;

d

⁴⁺

d

³

d

¹

; a

³

=a

^{4 =}

d

¹⁺

d

³

:

⁽⁵³⁾

9.) Finally, if

d

¹⁼

d

^3and

d

²⁼

d

⁴in (52) and (53), the

function should satisfy the relation 2

⁺

d

¹

(

²⁺

d

¹

^;

d

^2)+q;4

d

^42;

d

²1

arctan 2

⁺

d

¹

q

;4

d

^2;

d

²1

+

a

^4(;4

d

^2;

d

²1)(

^;

^{0) = 0}

:

⁽⁵⁴⁾

If the condition (25) is not satisfied, we have five further possible cases:

(i) If the parameters

v

²

;v

^3and

C

¹are taken as

C

^{1= 0}

; v

^{3 = 15}

v

²4

64

v

⁵

; v

^{2 = 0}

;

⁽⁵⁵⁾

then the

function is given by 1

p

10(2

b

²

⁺

b

^1);

p

v

⁵

5

p

;

b

²

v

^{4 arctanh}

p

2

v

⁵⁽²

b

²

⁺

b

¹⁾

p

;5

b

²

v

⁴

+

v

⁴

32

p

v

⁵

b

²⁽

^;

^{0) = 0}

;

⁽⁵⁶⁾

and the corresponding solution of the

⁵model with (55) reads

⁼

b

²1

4

b

^{2 +}

b

¹

⁺

b

²

^{2 (57)}

with three arbitrary constants

b

¹

; b

^2and

^0.

(ii) If the parameters

v

²

;v

^3and

C

¹are given by

v

^{2 =;9}

v

4³

32

v

5²

; v

^{3 =;3}

v

4²

16

v

⁵

; C

^{1 = 27}

v

4⁵

640

v

5⁴

;

⁽⁵⁸⁾

the related

obeys the form

;

1 2

b

²

⁺

b

^{1 ;}

p

v

⁵

p

5

b

²

v

^4arctanh

;

p

v

⁵⁽²

b

²

⁺

b

¹⁾

p

5

b

²

v

⁴

+

p

10

v

⁴

16

p

v

⁵

b

²⁽

^;

^{0) = 0}

;

⁽⁵⁹⁾

and

has the form

^{= ;3}

v

⁴

b

²⁺

v

⁵

b

²1

4

v

⁵

b

^{2 +}

b

¹

⁺

b

²

²

:

⁽⁶⁰⁾

(iii) The constraints on the parameters for the third special solution read

C

^{1= 0}

; v

^{2 = 25}

v

4³

864

v

5²

; v

^{3 = 5}

v

4²

16

v

⁵

;

⁽⁶¹⁾

while the corresponding

^and

are given by

; p

10

100(2

b

²

⁺

b

^{1); 150}

v

⁵

500

p

b

²

v

^4arctanh

; p

15

v

⁵⁽²

b

²

⁺

b

¹⁾

p

b

²

v

⁴

=^;

v

⁴

480

p

v

⁵

b

²⁽

^;

^{0) (62)}

and

^{= 3}

v

⁵

b

²1^;5

v

⁴

b

²

12

v

⁵

b

^{2 +}

b

¹

⁺

b

²

²

;

⁽⁶³⁾

respectively.

(iv) For the fourth special solution we have constraints on the parameters

v

²

; v

³ and the first integration constant

C

^1:

v

^{3 =3(}

p

5 + 1)

v

4²

32

v

⁵

; v

^{2 =;}

v

³4(29

p

5^;60) 16

v

²5(7

p

5^;27)

; C

^{1= 0}

:

⁽⁶⁴⁾

The related

function is given by

s

1

p

5+ 3 arctanh

q

v

^{5(5 + 3p5)(2}

b

²

⁺

b

¹⁾

p

10

v

⁴

b

²

; p

2 arctanh

5^p

v

⁵⁽²

b

²

⁺

b

¹⁾

q

p

5

v

⁴

b

²

;

v

4³⁼²(

p

5^;1)(

^;

⁰⁾

5¹=⁴

v

^{5 = 0}

;

⁽⁶⁵⁾

and the solution of the

equation is

⁼

v

⁵

b

²1^;

p

5

v

⁴

b

²

4

v

⁵

b

^{2 +}

b

¹

⁺

b

²

²

:

⁽⁶⁶⁾

(v) The final special solution has the form

⁼

v

⁵

b

²1^;2

v

⁴

b

^2+p5

v

⁴

b

²

4

v

⁵

b

^{2 +}

b

¹

⁺

b

²

²

;

⁽⁶⁷⁾

where the function

is related to

implicitly by

s

5 +

p

5 2 arctanh

q

v

^{5(5 + 3p5)(2}

b

²

⁺

b

¹⁾

p

;

b

²

v

⁴

;5¹=⁴_arctanh

v

⁵⁽²

b

²

⁺

b

¹⁾

q

; p

5

b

²

v

⁴

+

p

10(

p

5^;1)

v

^4p;

v

⁴⁽

^;

⁰⁾

32

v

^{5 = 0}

;

⁽⁶⁸⁾

while the constrained conditions for the model parameters

v

²

; v

³and the integral constant

C

^1are

v

^{3= 3(}

p

5 + 1)

v

4²

32

v

⁵

; v

^{2 =;}

v

³4(29

p

5^;60) 16

v

5²(7

p

5^;27)

; C

¹⁼

v

4⁵(4525655302009

p

5^;10119671799191) 1280

v

5⁴(204037668464

p

5^;456248189781)

:

⁽⁶⁹⁾

The parameters

b

¹

;b

^{2, and}

⁰in the above special cases are all arbitrary constants.

4. Solutions of the

^8Model

If we take

N

^{= 8}

; v

¹⁼

v

^{3 =}

v

^{5 =}

v

⁷ = 0 in (4), we get the

^8model

2

⁼

V

²

⁺

V

⁴

³⁺

V

⁶

⁵⁺

V

⁸

⁷

;

⁽⁷⁰⁾

where we have written

v

i

;i

^{= 2}

;

⁴

;

⁶

;

^{8 and}

^as

V

i and

for convenience later. By means of the basic equation approach proposed in Sect. 2, some types of special solutions of the

⁸model (70) can be obtained by solving a similar ODE:

A

⁺

B

⁼

V

²

⁺

V

⁴

³⁺

V

⁶

⁵⁺

V

⁸

⁷

;

⁽⁷¹⁾

where the auxilialy field and the arbitrary functions

A

⁼

A

^{( ) and}

B

⁼

B

⁽ ) are related by (6) and (7).

Especially, if

A

^and

B

satisfy (12), we have

^{2 =}

V

²

²⁺¹₂

V

⁴

⁴⁺¹₃

V

⁶

⁶⁺¹₄

V

⁸

^{8 (72)}

;

c

12

24

V

^{2+ 18}

V

⁴

c

^{+ 16}

V

⁶

c

^{2+ 15}

V

⁸

c

³

;

where

is given in (13) and

c

is an arbitrary integration constant.

It is interesting and straightforword to see that the solutions of (72) can be obtained simply by using the transformation

^=p

c

⁺

;

⁽⁷³⁾

where

is a solution of the

⁵model (21) with

v

^{5 =5}₂

V

⁸

;

⁽⁷⁴⁾

v

^{4 =8}₃

V

^{6+ 10}

cV

⁸

;

⁽⁷⁵⁾

v

^{3 = 3}

V

^{4+ 16}

cV

^{6+ 15}

c

²

V

⁸

;

⁽⁷⁶⁾

v

^{2 = 4}

V

^{2+ 6}

cV

^{4+ 8}

c

²

V

^{6+ 10}

c

³

V

⁸

;

⁽⁷⁷⁾

C

^{1= 4}

c

²⁽

V

²⁺

cV

⁴⁺

c

²

V

⁶⁺

c

³

V

⁸⁾

:

⁽⁷⁸⁾

So, all the special solutions obtained in Sects. 2 and 3 can be transformed to those of the

^8model.

5. Some Special Solutions of Coupled Scalar Fields Actually, the special solutions of the

^{5model ob-}

tained in sections 2 and 3 may be used to get exact

solutions of other physically significant models, say the coupled nonlinear scalar field model

2

f

⁼

A

¹

f

⁺

A

²

f

³⁺

A

³

fg

²

;

⁽⁷⁹⁾

2

g

⁼

B

¹

g

⁺

B

²

g

³⁺

B

³

gf

^{2 (80)}

which appears in some physical fields such as parti- cle physics and field theory [8] and condense matter physics [9].

After some complicated but direct calculations, we can change the solutions of the

⁵model obtained in Sections 2 and 3 to those of the coupled scalar fields

f

^and

g

for some special parameters

A

i^and

B

i^{. Here} are five possible examples.

Case 1

If the parametes

B

^{1 and}

B

³ are related to other parameters by

B

^1=;

A

¹

B

²

11

B

^2;6

A

³

; B

^{3= 5}₉

A

²

;

⁽⁸¹⁾

a special type of the coupled scalar fields reads

g

⁼

;

⁽⁸²⁾

f

^{2= 9}

50

A

^2p

A

¹

3(6

A

^3;11

B

²⁾⁽

A

^3;5

B

²⁾

(15

B

^2;8

A

³⁾¹⁼²

³⁺₂₅⁹

A

²⁽¹⁵

B

^2;8

A

³⁾

²

; 9 25

A

²

s

3

A

¹⁽⁵

B

^2;

A

³⁾⁽¹⁵

B

^2;8

A

³⁾

(11

B

^2;6

A

³⁾

;

27 25

(15

B

^2;8

A

³⁾

A

¹

(11

B

^2;6

A

³⁾

A

²

;

⁽⁸³⁾

where

is an arbitrary solution of the

⁵model given in Sect. 3, or more generally by (14) with

C

^{1 =24(15}

B

^2;8

A

³⁾

A

²1

25(11

B

^2;6

A

³⁾²

;

⁽⁸⁴⁾

v

³⁼

p

3

A

¹⁽⁵

B

^2;

A

³⁾⁽¹⁵

B

^2;8

A

³⁾

5

p

(11

B

^2;6

A

³⁾

;

v

^{4= 4}

B

^2;8₅

A

³

;