Applied Mathematics, 2011, 2, 1313-1316
doi:10.4236/am.2011.211183 Published Online November 2011 (http://www.SciRP.org/journal/am)
Jacobi Elliptic Function Solutions for (2 + 1) Dimensional
Boussinesq and Kadomtsev-Petviashvili Equation
Chunhuan Xiang
College of Mathematics and Statistics, Chongqing University of Arts and Sciences, Chongqing, China
Received March 30, 2011; revised August 29, 2011; accepted Septe mber 7, 2011
Abstract
(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi
elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the
equation are reported. Numerical simulation results are shown. These new solutions may be important for the
explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function
method can be a useful tool in obtaining evolution solutions of nonlinear system.
Keywords: Jacobi Ellip tic Fu nct ion , T rav eli ng Wave Solution, Kadomtsev-Petviashvili Equation, Jacobi
Elliptic Function Expansion Method, Numerical Simulation
1. Introduction
It is well known that the nonlinear physical phenomena
are related to nonlinear partial differential equations,
which are employed in natural and applied science such
as fluid dynamics, plasma physics, biology, etc. Their
solution spaces are infnite-dimensional and contain di-
verse solution structures. In the past few years, wide va-
riety of the powerful and direct methods to find all kinds
of analysis solutions of nonlinear evolution equations
had been developed [1-13]. The basic purpose of them is
to construct new solitary wave solutions and periodic
solutions. (2 + 1) dimensional Boussinesq and Kdomtsev-
Petviashvili (BKP) equation is an important nonlinear
partial differential equation in mathematical physics,
which had been mentioned in literatures [14-16]. The
equation belongs to a symmetry and integrable system.
The aim of this paper is to app ly the Jacobi elliptic func-
tion expansion method [11] to solve (2 + 1) dimensional
BKP equation. The general BKP equation has the form
y
x
UW
x
y
VW
6() 6()
txxx yyyxy
WW WWUWV (1)
where , and are the
functions about
(,,)Uxyt ,(,,)Vxyt (,,)Wxyt
x
yt, and . ,,
y
xy
WUV , are the deriva-
tives of ,
x
y and , respectively. t
2. Transformed Boussinesq and
Kadomtsev-Petviashvili Equation and
Jacobi Elliptic Function Expansion
Method
In this section, we will apply the Jacobi elliptic function
expansion method to BKP equation.
Using a wave variable, we obtain the transformed
wave solutions as: (,,) ()Uxyt U

, (,,) ()Vxyt V


,
and (,,)WxytW()


, where
x
yct

 c, is non-
zero constant.
Plugging ()U

()V

, and ()W

and integrating (1)
once with respect to

and considering the constants of
integration to be zero, we obtain the transformed BKP
equation:
0UW

,
0VW

,
266WcWUWVW

0

. (2)
The above equations are an ordinary differential equa-
tion. Fu and Liu gave an example in Jacobi elliptic func-
tion expansion method [11] to find the solutions for the
ordinary differential equation. According to the method,
the ansatz solutions for (2) are supposed as
2
01 2
()Uaasn asn;

   (3a)
2
01 2
()Vbbsnbsn ;

  
2
(3b)
01 2
()Wccsn csn;

   (3c)

C. H. XIANG 1314
where
s
n

is Jacobi elliptic function, 01201
201
and 2 c are the expansion coefficients to be
determined later. Substituting (3) into (2) yields a system
of algebraic equations.
,,,,,aaabb
,,bc ,c
22
01201 2
() ()aa snasncc sncsn     0
0
2
(4a)
22
01 201 2
() ()bbsn bsnccsn csn     (4b)




2
012
2
012
2
012
2
012012
2()
()
6()
() ()
0
ccsncsn
cccsnc sn
ccsncsn
aasnasnbb snbsn








 
 
 


(4c)
Solving the above first two Equations (4a) and (4b),
we obtain .
000111222
Equating the coefficients of Jacobi elliptic function
;;abcabcabc  
s
n

(4c) for the to zero with the above results and the
follow relation
2222
()1();()1 ()cnsndnm sn2

    , (5)
we have
2
002
111 01
22
12222 02
2
1112
22
222
12 40;
22240;
12 844240;
2224 0;
48 120;
caa a
aca maaa
aacamamaaa
mamaa a
mam aa

 
 
 

(6)
where dn

,cn

and are different kinds
of Jacobi elliptic functions and modulus of Jacobi elliptic
function, respectively.
(01)mm
Solving (6), we obtain
2
01
223
2
844
;0;
24
2;443 2
3
cmm
aa
mm
acmm
 



4
;m
and
2
01
223
2
844
;0;
24
2;4432
3
cmm
aa
mm
acmm
 



4
.m
;
With the help of (5), we rewritten (3) as the follow
three kinds of Jacobi elliptic functions:
Case [I]
2
02
;
()
UVW
Uaasn


 (7)
Case [II]

2
02
;
1( );
UVW
Uaa cn




  (8)
Case [III]
2
02 2
;
1() .
UVW
dn
Uaa m







 (9)
3. Periodic Traveling Wave Solutions for
Boussinesq and Kadomtsev-Petviashvili
Equation
When the modulus of Jacobi elliptic function
and , Jacobi elliptic functions asymptotically
transformed into periodic trigonometric and hyperbolic
traveling wave solutions:
0m
1m
0m,
snsin;cos ;1cn dn

    (10)
1m
tanh;sech ;sechsncn dn

    (11)
With the relation (10) and (11), Equations (7)-(9) are
transformed into
2
02
0; ;
(sin) ;
mUVW
Uaa


 (12)
2
02
1; ;
(tanh);
mUVW
Uaa


 (13)
The simulation results of (7) and (13) are given with
the help of Mathematica software in Figures 1 and 2 for
some special local parameter. From the Figures 1 and 2,
we know that the amplitude of the wave is stable. These
wave solutions may further help us to find some new
physical phenomena.
4. Discussions and Conclusions
Some new analytical solutions of BKP equation are ob-
tained by successfully employing Jacobi ellip tic function
expansion method in this paper. When the modulus of
Jacobi elliptic function and , the Jacobi
elliptic functions asymptotically transformed into peri-
odic trigonometric and hyperbolic traveling wave solu-
tions. The results obtained in this pap er are new solution s
in the representation of Jacobi elliptic function and soli-
tary wave solutions. These new solutions may be impor-
tant for the explanation of some practical physical prob-
lems. These solutions may help us to learn more about
the complex nonlinear evolutions systems. The Jacobi
elliptic function expansion method in its present form is
a successful, direct and concise tool in obtaining a serials
0m1m

C. H. XIANG 1315
Figure 1. Jacobi elliptic func tion solutions Equation (7) is shown for U with x  (–2, 2), y  (–2, 2), m = 0.2, from left to right t
= –0.2, t = 0, t = 0.2, respectively.
Figure 2. Hyperbolic function solutions Equation (13) is shown for U with x  (–3, 3), y  (–3, 3), m = 1, from left to right t =
–0.2, t = 0, t = 0.2, respectively.
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[5] C. Xiang, “Analytical Solutions of KdV Equation with
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The support from Science Foundation of Chongqing Uni-
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C. H. XIANG 1316
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