Exact Solutions of Mix Modified KdVs’ Equation by Exp function Method

2019 International Conference on Applied Mathematics, Modeling, Simulation and Optimization (AMMSO 2019) ISBN: 978-1-60595-631-2

Exact Solutions of Mix Modified KdVs’ Equation by Exp-function Method

Xiong-ping XIA

College of Science, Guilin University of Technology, Guilin 541004, China *Corresponding author

Keywords: Exp-function method, KdV equation, Soliton solution.

Abstract. This paper applies the Exp-function method to obtain generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computation method. The mix modified KdVs’ equation is used as an example to illustrate the effectiveness of this method. The solution process is straightforward and concise, and its applications are promising for other nonlinear evolution equations.

Introduction

KdV equation (u_t6uu_x u_xxx 0) is very an important nonlinear equation of described motion of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice and it is an important mathematical model with wide applications in quantum mechanics and nonlinear optics. Typical examples are widely used in various fields such as solid state physics, plasma physics, fluid physics and quantum field theory. A great deal of research work has been invested during the past decades for the study of KdV equation [1-5] and modified KdV equation [6-10].

The investigation of exact solutions of KdV equation and modified KdV equation plays an important role in the study of some nonlinear physical phenomena. Recently, many new methods have been proposed to solve the KdV equation and improved KdV equation. Such as, variational iteration method, tanh-function method, F-expansion method, and A do-mian decomposition method [8-14].

He and Abdou [15] proposed first a new method of solving some nonlinear equation in 2006, which is the Exp-function method. As it is a straightforward and concise method, was successfully applied to obtain generalized solitary solutions and periodic solutions of some nonlinear evolution equation arising in mathematical physics and differential difference equations. The solution procedure of this method, with the aid of Maple, is of utter simplicity and this method can be easily extended to other kinds of nonlinear evolution equations. Yokus et al.[16] applied the exp-function method to study the two-component second order KdV evolutionary system. Bibi et al.[17] researched exact solutions of (3+1)-dimensional KdV-ZK equations, Mohyud-Din and Irshad[18] studied KdV-ZK equation, Sanchez et al.[19] researched Rosenau-KdV-RLW Equation, Inc and Kilic [20] analyzed time-fractional fifth-order KdV-like equation.

The present paper aims to extend the Exp-function method to find new solitary solutions, compact-like solutions as well as periodic solutions for nonlinear evolution equations in mathematical physics. To illustrate the basic idea of the Exp-function method, we consider the mix modified KdVs’ equation.

1

6 ( ) 0

N i

t x xxx

i

u u u u



 



When i1, it is normative KdV equation: u_t6uu_xu_xxx 0.

Analysis of the Method and Its Application on the Mix Modified KdVs’ Equation

3

1

6 ( ) 0

N i

t x xxx

i

u k u u k u





 



Where prime denotes the differential with respect to . The present we only discuss its form of the mix modified KdVs’ equation when i2,3

I.i2, equation is u_t   6 (u u2) u_x u_xxx 0 (3)

II.i3, equation is u_t  6 (u u2u3) u_x u_xxx0 (4)

Case i2

In view of the Exp-function method [15], the solution of Eq. (3) can be expressed as:

exp( ) ( , ) ( )

exp( ) d

n n c q

m m p

a n

u x t u

b m

 







 





Where c d p q, , , are positive integers which are unknown to be determined later, a_n and b_m are unknown constant. Eq. (5) can be re-written in an alternative form as follows:

exp( ) exp( )

( )

exp( ) exp( )

c d

p q

a c a d

u

b p b q

 



 





  



   (6)

In order to determine the values of c and p, we balance the linear term of the highest order u'''

with the highest order nonlinear term u u2 ' in Eq. (6), we have

1

2

exp[( 3 ) ] '''

exp[4 ]

c c p

u

c p

 

 



 (7)

2 3

4

exp[(3 ) ] '

exp[4 ]

c c p

u u

c p

 

 



 (8)

Here c_i is a coefficient for simplicity. By balancing highest order of Exp-functionin in Eqs. (7) and (8), we have 3c  p c 3p, which leads to the limit pc. Proceeding in the same manner as illustrated above, we can determine values of d and q. Balancing the linear term of lowest order in Eq. (6)

1

2

exp[ ( 3 ) ] '''

exp[ 4 ]

d q d

u

d q

 

  



  (9)

2 3

2

exp[ (3 ) ] '

exp[ 4 ]

d q d

u u

d q

 

  



  (10)

Here d_i are determined coefficients only for simplicity, By balancing highest order of Exp-functionin in Eqs. (9) and (10), we have  (q 3 )d  (3q d ), lead to the limit qd.

Case (1) p c 1 and d  q 1. We can freely choose the values of c and d , but we will illustrate that the final solution does not strongly depend upon the choice of values of c and d. For simplicity, we set p c 1 and d  q 1, the trial function, Eq. (6) becomes

1exp( ) 0 1exp( )

( ) a a a

Substituting Eq. (11) with Eq. (3), using the Maple, and equating to zero the coefficients of all powers of exp(n) yields a set of algebraic equations for a₀, b₀, a₁, a_1, b_1 and  as

3 2

1 1 0 1 0

(k   6ka 6ka )( a a b )0 (12)

3 2 2 2 3

1 1 0 1 1 0 1 1 1 1 1 1 1

2 3 3 2 3 2 2 2 2

1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0

2( 4 3 6 6 6 4

6 6 2 6 2 3 ) 0

k a a ka ka a ka a ka a k a b a b

ka b ka b k a b a b ka a b k a b a b ka b

 

 

     

 

        

        (13)

2 3 3 2 2

1 1 1 1 0 1 1 1 1 1 1 0 0 0 0

4( a_ a b_ )( 3ka_ 6ka 6ka a_ 8k b_ b_ 3ka b_ 6ka b k b b ) 0

             (14)

2 3 2 2

1 1 1 1 1 0 1 1 0

(6ka_ 6ka b_ k b_ b_ )(a b_ a b_ )0 (15) Solving this system with the aid of Maple, we obtain the following coefficients results:

Case a: 1 1 1 a a b  

 , 0 1

0 1 a b b a    , 2

3 1 1

2

1 1

6ka 6ka k b b          , 2

3 1 1

2

1 1

6 6

( ka ka )

kx k t

b b

  

 

    . (16)

Case b: a1a1,

2

1 0 1 0 1

0 2 2 2

1 1 1 1

2 2

a a b a b

b

a a b k b

  

   

 

  ,  ,   (17)

Where a₀, a_1 and b_1 are free parameters. Inserting Eqs.(16) and (17) into Eq. (11) admits the following generalized solitonary solution of Eq. (3):

1

1 0 1 1

1 1

0 1

1 1 1

1

exp[ ( )] exp[ ( )]

( , ) ( )

exp[ ( )] exp[ ( )]

a

k x t a a k x t

b

u x t u

a b

k x t b k x t

a                        (18) 1

1 0 1 1

1

2 2

1 0 1 0 1

1 2 2 2 1 1

1 1 1 1

exp[ ( )] exp[ ( )]

( , ) ( )

2

exp[ ( )] exp[ ( )]

2

a

k x t a a k x t

b

u x t u

a a b a b

k x t b k x t

a a b k b

                               (19)

Here 2 2 2

1 /k k 6a1/b1 6a1/b1

     _{ }  _{ } . In case k is an imaginary number, the obtained solitonary

solution Eqs.(18) and (19) reduces to the periodic solution or compact-like solution. We write kiK. Using the transformation:

1 1 1

exp[iK x( t)]cos[ (K xt)]isin[ (K xt)] (20) We have the following new periodic solution of Eq. (18) and (19)

1 0 1 1 1 0 1 1 1 1 1 1 1 1 1

1 1 1 1

1 1 1 1

{cos[ ( )] sin[ ( )]} {cos[ ( )] sin[ ( )]} ( , )

{cos[ ( )] sin[ ( )]} {cos[ ( )] sin[ ( )]}

( ) cos[ ( )] ( ) sin[ (

a

K x t i K x t a a K x t i K x t

b u x t

a b

K x t i K x t b K x t i K x t

a

a a

a K x t i a K x

b b                                            0 0 1 1 1 1 )]

(1 ) cos[ ( )] (1 ) sin[ ( )]

t a

a b

b K x t i b K x t

1 1

1 0 1

1 1

2 2

1 0 1 0 1

1 2 2 2 1

1 1 1 1

1 1

1 1

( ) cos[ ( )] ( ) sin[ ( )] ( , )

2

(1 ) cos[ ( )] (1 ) sin[ ( )] 2

a a

a K x t a i a K x t

b b

u x t

a a b a b

b K x t i b K x t

a a b k b

 

 

 

 

 

  

 

   

     





     

 

(22)

If we search for a periodic solution or compact-like solution, the imaginary part in the denominator of Eqs. (21) and (22) must be zero, which requires that: On the one hand, 1b_1 0 b_1 1, on the other hand, lead to too a₁/b₁a₁0, we obtain:

1 0 1

1 0

2

1 1

1

2 cos[ ( )] ( , ) ( )

2 cos[ ( )]

a K x t a a

u x t u

a K x t a

 







  

 

  (23)

1 0

2

1 0 0

2 2

1 1

1

1

2 cos[ ( )] ( , ) ( )

2 2 cos[ ( )]

2

a K x t a

u x t u

a a a

K x t

a a k

 







 

 

 



 

 

(24)

For simplicity, we only take Eqs. (23) and (24) as example here. At last, we will give out some interesting figures of Eqs. (23) and (24). Figure 1 (a) and (b) show the single generalized solitonary solution for the real part of Eqs. (23) and (24).

(a) (b)

Figure 1. The single generlized solitonary solution for the real part Equation (27) at a_1 1.0,a₀ 2.4,K1.6, 1.0

for (a); Equation (28) at a_12.0,a₀ 2.0,K1.7, 1.0 for (b).

Case (2) p=c= 2 and d=q=1. We consider the case p=c=2 and d=q=1, Eq. (3) can be expressed as: kx t

   .

2 1 0 1

1 0 1

exp(2 ) exp( ) exp( )

( , ) ( )

exp(2 ) exp( ) exp( )

a a a a

u x t u

b b b

  



  





   

 

    (25)

Applied the above same method, and attain the coefficients a₂a_1/b_1 , b₁a b_{1 1}/a_1 ,

0 0 1/ 1

b a b_ a_ and   k3 6ka_2₁/b_2₁6ka_1/b_1. a₁, a₀, a_1 and b_1 are free parameters and substitute into Eq. (25), we have

1

1 0 1

1

0 1

1 1

1

1 1

exp(2 ) exp( ) exp( )

( )

exp(2 ) exp( ) exp( )

a

a a a

b u

a b a b

b

a a

  



  



 

 



 

   



   

Here 3 2 2

1 1 1 1

( 6 / 6 / )

kx k ka b ka b t

    _{ }  _{ } . In case k is an imaginary number, the obtained solitonary solution (26) reduces to the periodic solution or compact-like solution. We write k iK. Using exp[iK x( ₂t)]cos[ (K x₂t)]isin[ (K x₂t)]. Then Eq. (26) becomes:

1 1

1 1 1 1 0 1 1 1 1

1 1

0 1

1 1 1 1

1 1 1 1 1 1

1 1 1

cos(2 ) ( ) cos (2 cos ) sin

( , ) ( )

cos(2 ) ( ) cos (2 cos ) sin

a a

a a a i a a

b b

u x t u

a b

a b a b

b i b

a a a

   



   

 

 

 



 

 

  

     

 

     

(27)

here ₁K x( ₂t), 2 2 2

2 /k (k 6a1/b1 6a1/b1)

     _{ }  _{ } .

Case i3

There we only discuss p c 1 and d q 1

1 0 1

0 1

exp( ) exp( )

( )

exp( ) exp( )

a a a

u

b b

 



 





  



   (28)

The coefficientsb_1a b_{1 0}/a₀, a₁a₀/b₀, 3 2 2 3 3 3

0 0 0 0 0 0 0

2(3ka 3ka b ka b 2k b ) /b

     . Where a₀, a_1

and b₀ are free parameters, and substituting Eq. (28) yields

0

0 1

0

1 0 0

0

exp( ) exp( )

( )

exp( ) exp( )

a

a a

b u

a b b

a

 



 





  



  

, here

3 2 2 3 3

0 0 0 0 0 0

3 0

2(3ka 3ka b ka b 2k b )

kx t

b

      (29)

In case k is an imaginary number, the obtained solitonary solution reduces to the periodic solution or compact-like solution. Using the above transformation, then Eq. (28) becomes:

0 0

1 4 0 1 4

0 0

1 0 1 0

4 0 4

0 0

( ) cos[ ( )] ( ) sin[ ( )]

( , )

(1 ) cos[ ( )] (1 ) sin[ ( )]

a a

a K x t a i a K x t

b b

u x t

a b a b

K x t b i K x t

a a

 

 

 

 

     



     

(30)

here ₄ /k (6a₀36a b_{0 0}2 2a b_{0 0}24k b2 ₀3) /b₀3.

Conclusion

The Exp-function method with a computerized symbolic computation has been successfully used to obtain the generalized solitonary solutions and periodic solutions to nonlinear evolution equations arising in mathematical physics. The validity of this method has been tested by applying it to the mix modified KdVs’ equation. Finally, it is worthwhile to mention that the Exp-function method is straightforward, concise. Therefore, it is a promising and a powerful new method for other nonlinear evolution equations in mathematical physics. Its applications are worth further studying.

Acknowledgement

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