Journal of Applied Mathematics and Physics, 2013, 1, 18-24
Published Online November 2013 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2013.15004
Open Access JAMP
Wronskian Determinant Solutions for the
(3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli
Equation
Hongcai Ma, Yongbin Bai
Department of Applied Mathematics, Donghua University, Shanghai, China
Received August 4, 2013; revised September 4, 2013; accepted October 1, 2013
Copyright © 2013 Hongcai Ma, Yongbin Bai. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this paper, we consider (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equ ation. Based on the bilinear form, we
derive exact solutions of (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation b y using the Wronskian
technique, which include rational solutions, soliton solutions, positons and negatons.
Keywords: (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equ ation; The Wronskian Technique; Soliton; Negaton;
Positon
1. Introduction
The Wronskian technique is introduced by Freeman and
Nimmo [1]. After that, many researches are based on the
Wronskian technique.
The (2 + 1)-dimensional BLMP equation was first de-
rived in [2]:
33
ytxxxyxx yx y
uuuu uu 0

(1)
where and subscripts represent partial
differentiation with respect to the given variable. This
equation was used to describe the (2 + 1)-dimensional
interaction of the Riemann wave propagated along the
y-axis with a long wave propaga ted along the x-ax is. The
Painlevé analysis, Lax pairs, Bäcklund transformation,
symmetry, similarity reductions and new exact solutions
of the (2 + 1)-dimensional BLMP equation are given in
[2-4]. In [5], based on the binary Bell polynomials, the
bilinear form for the BLMP equation is obtained. New
solutions of (2 + 1)-dimensional BLMP equation from
Wronskian formalism and the Hirota method are ob-
tained in [6,7].

,,uuxyt
The (3 + 1)-dimensional BLMP equation


3
30
y
t ztxxxy xxxzxxy xz
xx yz
uuuuuuu
uu u
 
 (2)
which was introduced in [8] has the bilinear form


33 0
yt ztyxzx
DDDDDDDDf f

  (3)
just by substituting



2ln, ,,
x
ufxyz tinto equation
(2), where the bilinear differential operator D is defined
by Hirota [9] as





0, 0
,,
,,
mn
tx
mn
sy
mn
DDatx btx
at sxybt sx y
sy






2. Wronskian Formulation
Solutions determined by

2ln
x
u f to the Equation
(2) are called Wronskian solutions, where


 
 
 
12
01 1
11 1
01 1
22 2
01 1
,,, 1
,1
N
N
N
N
NN N
fW N
N
 
 
 
 







,











(4)
and
 
2
0,,1,1
j
iii i
jji
x
 
N


. (5)

H. C. MA, Y. B. BAI 19
Lemma 1
,,,,,,,,,,,, 0DabDcdDacDbdDad Dbc
, (6)
w matrix, and are
n-dimensional columors. -dimensiona
real consta
bu hen we have
here D is

2NN
n vect,,,abcd
Lemma 2 Set to be an nl
column vector, and

1, ,
j
rj n to be ant

1, ,
j
bj n
t not to be zero. T
12
,,,, ,,
NN
i jN
rbbrba

, (7)
12
,,
N
b bb
where
Lemma 3 The following equalities hold:
11 ij 

T
112 2
,,,
jjjNNj
rbrbr brb.
 
2
11 1
11 1
Nn N
ii iiii
ii j
NN N  
 
 

 

 

 
. (8)
Proposition. Assuming that

,,,
ii

x
yzt  
,N
(where
) has continuous de- 0,, ,,1,2,txyzi
rivative up to any order and satisfies the following linear
differential conditions
,,, ,,,
1
4,,,
N
itixxxixxijjiyixizix
i,

 

 
 (9)
then

1fN

 defined by Equation (4) solves the bil-
inear Equation (2).
Proof. Using the conditions (9), we get that

2, ,
zyx
f
ffNN


3,1,2,1,
xz xy xx
fffNNNNN 



4,2, 1,23, 1, 12,2
xxz xxyxxx
f ffNNNNNNNNN


  
5,3,,2,, 1,34,2, 1, 1
33,1,223,,1 2,3
xxxz xxxy xxxx
fffNN NNNNNNN
NNNNNN NN

 





44,2,1,43,1,142,2
t
fNNNNNNN NN  ,





45,3,2,1,43,,142,3
yt zt xt
fff NNNNNNNN.N     
Hence, we have
N

 

 
 

 
 

33
2
5,3,,2,,1,
4,2,1,123,,1
3, 1,22,3
243, 1, 12,
63,1,2,1.
yt zt yx zx
DD DD DDDDff
NNNN
NNNN NNN
NNNNN
NNNNN
NNNNN
 


 


(10)
With the help of Lemma 2 and Lemma 3, we obtain
61NN



 
 

 

2
61 5,3,,2,,1,
4,2,1,1 23,,1
3, 1,22,3
63,1, 2,1.
NNNN NN
NNNN NNN
NNNNN
NNNNN

 
 
 
(11)
Substituting Equation (11) into Equation (10) and us-
ing lemma 1, we get


 

 
 

33
243,2, 13,1,
3,2, 13,1,
3,2,3,1,1 0.
yt ztyxzx
DD DD DDDDff
NNNNNN
NNNNNN
NNNNNN
 
  
   

 

1fN Therefore, we have shown that solves
lution of Equation (2) is
Equation (4) under the linear differential conditions (9),
The corresponding so


2,
221
xNN
f
ufN

  .
(12)
3. Wronskian Solutions
In what follows, according to [10-12], we would like to
present a few special Wronskian solutions to the (3 +
Open Access JAMP

H. C. MA, Y. B. BAI 20
1) pinelli
by-dimensional Boiti-Leon-Manna-Pemequation solving the linear conditions (9).
It is well known that the correspond ing Jordan form of
a real marix



1
2
0
1,
1
m
0
N
N
J
J
A
J













(13)
2)

(15)
where


have the following two type of blocks:
1)

,
i
J 


 (14)
0
1
i
i





01
ii
ikk





2
2
2
0
,
10
,,
01
ii
i
i
i
ill
ii
iii
A
IA
J
AI












 
 

0IA


,
ii

 and i

have thare all real constants type of e real eigenvalue. The first blocks i

with alge-
braic multiplicity and the second type
of blocks have the complex eigenvalue

1,
n
ii
i
kkN


1
iii


with algebraic multiplicity
3.1. Rational Solutions
Suppose A have the first type of Jordan blocks
i
l.
1
1
1
0
1,
01
N
N
A











 (16)
the eigenvalue In this case, if 10

, corresponding
to the following form:
00
10 ,
010
N
N


A 






(17)
from th e condition ( 9), we get

,,,,,,
0,4 ,,
ixxitixxxiy ixiz,
, 1.
ix
i

  (18)
where are all polynomials in

1
ii
 ,,
x
yz
(3 + 1)
and
and onskian solution to the dim
si
(19)
is called a rational Wronskian solution.
From Equation (18), we get
t,
en-a general Wr
onal Boiti-Leon-Manna-Pempinelli Equation (2)
2u 

1
12
ln,,,,
xk
W   
1,1,,1,1,1,1,
0,4 ,,.
x
xti xxxyxzx


 (20)
ing Maple, we get the fol- Solving Equation (20) by us
lowing formulas:


.Cxyz C  
11 2
Similarly, by solving
1,1, 1,
0,4 ,
i xxiti xxx



1,
(21)
1,1,1,1,
,,
iy ixiz ix
i  
 


(22)
-order are ob-
o be zero.
1) Zero-order: Taking
corresponding Wronskian dete
ra
then two special rational solution of lower
ta l constants tined after setting some integra

11
Cxyz C  
rminant and the as2
, the
sociated
tional Wronskian solution of zero-order read


111 2
,
f
WCxyz   C (23)
 
1
1
2l n
x
uW
  
12
2, C
Cxyz C
   (24)
where are arbitrary constants.
2) der: Taking
12
,CC
First-or

11 2
,Cxy  

z C we
can have




2
32 3
133 3243
C

2
2
22
223423
11 1
66
26 2
2
23 2
223
611
36
66
26
.
x
zyxyzx tz  zy
zy yCxCzCCyx 
Cy CzCy CCz 
Then, the corresponding Wronskian determinant and
Wroner are
Cz
(25)
rationalskian solution of first-ord





12
12
22
1
2
12 2
,,fW P  
22
2l
n ,
222
2,
x
uW
Cy zxyxzxyz
P
CC xy zC
P
  


 

where



22 222223
11
2PCxyxzxz xyzy zyxyzx




33 12
22 2 2
22314
3
114222
33
yztCCxyyzxy
xyzCxyzCCCC

 


 
Open Access JAMP

H. C. MA, Y. B. BAI 21
and are arbitrary real constants. Similarly,
e higher order rational Wronskian so-
lutions.
3.2. Solitons, Negatons and Positons
3.2.1. Solitons
If
1234
,,,CCCC
we can obtain mor
A
becomes to the following form
1
2
0
,
0N
N
N
A










 (26)
where the eigenvaluce 0.
i


nSubstituting the form of expression (26) into Equatio (9), the following system
of differential equations is obtained














,4,
iiiiiiiii
t xxx
i
 


(27)








,,
iii iiii
yxzx
 
 


By solving system (27), we get the n-soliton solution

(28)
with
of Equation (2)

12
2ln,,,
xN
uW    ,
i

being defined by
3
2
3
2
cosh4 ,
sinh4,
ii iii
iiiii
x
yztiodd
x
y









(29)
zti even     



where 12 0
N

  are arbitrary constants.
We and 2-soliton solutions present the 1-soliton
3
2
11111
2
1111
2lncosh4
anh 4
x
uxyzt
xyzt




 






 



3
1
2t 




3
2
2
2ln cosh
x
uW



1
111
3
2
2222
12
4 ,
sinh 4
2
xyzt
xyzt
PQ




 














w here
3
2
2222
3
2
11111
coth 4
tanh 4
Pxyz
Qxyz












2
t
t

Similarly, we can obtain 3-soliton, 4-soliton solution
and n-soliton.
3.2.2. Negatons and Positons
If the eigenvalue

11
0, J

  becomes to the foll-
owing form
(30)
We start from the eigenfuction

11
1
1
1
1
0
1,
01
kk
J













11
,

 which is dete-
rmined by













       
11111 1111
1111 1111
,4
,,
tx
yxzx

 


,
xx
(31)
General solution to this system in two cases of 10


are and 10





3
2
11 11111
cosh 4Cxyzt  




3
2
3
2
31 111
3
4111
4, 0,
cos 4
sin 4
zt
xyzt
Cxyz

  
 








21 11
sinhCx
y


 


11


11
()C  
2
11
, 0,t 




(32)
respectively, where and are arbitrary
re

123
,,CCC 4 C
al constants. When 10

, wegaton solution get ne
and when 10


, we solu
To construct Wronskian solutions corresponding to
Jordan blocks of higher-order, we use the basic
developed for the KdV equation [10,11].
get positon tions.
idea
Differentiating (9) with respect to 1

, we can findt
the vector function tha


  
1
11
111
T
1
11 1111
1
11
,,,
1!1 !
k
k


 



 





,
(33
satisfies
)
11
1
1
1, 1
1
0
1,
01
xxx
kk












 (34)
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H. C. MA, Y. B. BAI
Open Access JAMP
22
1, 1,1,1,
4 ,
txx
  (35)
where 1
1, 1,
,
xxxy z
 

 denotes the tive with respect to 1 deriva

and
k is an ar nonnegative integer. There
1bitraryfore, through
this set of eigenfunctions and Equation (12), a Wronskian
solution of order 11k to Equation (2) is presented as:
  
1
11
1
1111
2ln
xW


 

11 11
1
, ,,,
1!1!
k
k

 

 



(36)
orr
a n
where
which cesponds to the first type of Jordan blocks with
onzero real eigenvalue.
In what follows, several exact solutions of lower-order
are presented to the (3 + 1)-dimensional Boiti-Leon-
Manna-Pempinelli equation as
u


3
2
1111 1
4.
x
yzt

 ,

3
2
11 111
4.
x
yz t    


3
2
11 1
4yzt 


 


1-negaton
3
2
11111
3
2
1-positon1 111
3
2
11111
2-negaton
2lncosh
2tanh4 ,
2lncos4
2ta4,
2ln
x
x
x
u
xyzt
uxyzt
yz t
uW
 
 



 




 





 




 
 
 

1x






nx





 
 


 
1
1
11
11
3
2
111111
2-positon1 1
11
3
2
11 11111
cosh, cosh
4cosh,
cosh sinh12
2lncos, cos
4cosh ,
cos sin12
x
x
yzt
uW
xyzt







 



 


3.3. Interaction Solutions
We are now presenting examples of Wronskian interac-
tion solutions among different kinds of Wronskian so-
lutions to the (3 + 1)-dimensional Boiti-Leon-Manna-
Pempinelli equation.
Let us assume that there are two sets of eigenfunctions



12 12
,,,;,,
kl
,

  
(37)
associated with two different eigenvalues

and

,
ution respectively. A Wronskian sol









  
12 1
2
2ln, ,, ;
,,
xk
l
uW
2

 
 
 

(38)
is
two solutions determined by the two sets of eigenfunc-
tions in (37). In fact, we ca have more general Wron-
skian interaction solutions among three or more kinds of
solutions such as rational solutions, positons, solitons,
negatons, breathers and complexitons.
In what follows, we would like to show a few special
Wronskian interaction solutions depending on rational
solution, positons and solitons. Firstly, we choose three
different sets of special eigenfunctions:
n

rational
3
2
soliton111 1
3
2
positon2 222
,
cosh4 ,
cos4 ,
xyz
xyzt
x
yz













t
where 10

, 20


are constants.
Three Wronskian interaction determinants between
any two of a rational solution, a single soliton and a
single p os i ton are obtained as
said to be a Wronskian interaction solution between

H. C. MA, Y. B. BAI 23


 

 

  
22 2
soliton positon
21 2121
sin cos
,
cosh sinhcossinh ,
xyz
W
 

rational soliton
111
rational positon
,
sinhcosh ,
,
W
xyz
W






 
 
 
where
3
2
1111 1
4
x
yzt

 

3
2
21 1 11
4.
x
yz    t  
Further, the corresponding Wronskian interaction
solutions are


 
rational soliton
11
11
2ln,
2cosh
,
1
osh
rs x
uW
xyz
x


sinh cyz

  
 




 
rational positon
22
2ln ,
2cos
rp x
uW
xyz
xyz


22
2
sin cos



 

 


 
soliton positon
2112
21 212
2ln ,
2coshcos,
cosh sinhcossinh
sp x
uW 
 
1


 

 
where
3
2
1111 1
4
x
yzt

 ,

3
2
22 222
4.
x
yz     t
The following is one Wronskian interaction determi-
nant and solution involving the three eigenfunctions. The
Wronskian determinant is

  

 

rational solitonpositon
,,W
xy

21 1 2
12 21
sinh cos
sin cosh,
z


 


so that its corresponding Wronskian solution reads as


3
rational soliton positon3
2
2ln,,,
rsp x
q
uW p 

 
  

 


 
32112
12 21
312121
11122212
sinh cos
sin cosh,
sinh sin
sinh coscoshsin
pxyz
qxyz
 
 
 
2

  


 

with
3
2
1111 1
4
x
yzt

  ,
where

3
2
22 222
4.
x
yzt    
4. Conclusion
In this paper, by using the Wronskian technique, we have
derived the Wronskian determinant solution for the (3 +
1)-dimensional Boiti-Leon-Manna-Pempinelli equation
which describes the fluid propagating and can be consid-
ered as a model for an incompressible fluid. Moreover,
we obtained some rational solutions, soliton solutions,
the
resultant systems of linear partial differential equations
which guarantee that the Wronskian determinant solves
the equation in the bilinear form. The presented solutions
remarkable richness of the solution space of the
ensional Boiti-Leon-Manna-Pempinelli equa-
tion.
5. Acknowledgements
The work is supported by National Natural Science
Foundation of China (project No. 11371086), the Fund of
Science and Technology Commission of Shanghai Mu-
ty (project No. ZX201307000014) and the Fun-
damental Research Funds for the Central Universities.
REFERENCES
[1] N. C. Freeman and J. J. C. Nimmo, “Soliton Solutions of
e Vries and Kadomtsev-Petviashvili Equa-
nskian Technique,” Physics Letters A, Vol.
positons and negatons of this equation by solving
show the
(3 + 1)-dim
nicipali
the Korteweg-d
tions: The Wro
95, No. 1, 1983, pp. 1-3.
http://dx.doi.org/10.1016/0375-9601(83)90764-8
[2] M. Boiti, J. J.-P. Leon and F. Pempinelli, “On the Spec-
tral Transform of a Korteweg-de Vries Equation in Two
Spatial Dimensions,” Inverse Problems, Vol. 2, No. 3,
1986, pp. 271-279.
http://dx.doi.org/10.1088/0266-5611/2/3/005
[3] C.-J. Bai and H. Zhao, “New Solitary Wave and Jacobi
Periodic Wave Excitations in (2+1)-Dimensional Boiti-
Leon-Manna-Pempinelli System,” International Journal
of Modern Physics B, Vol. 22, No. 15, 2008, pp. 2407-
2420. http://dx.doi.org/10.1142/S021797920803954X
[4] Y. Li and D. Li, “New Exact Solutions for the (2+1)-
Dimensional Boiti-Leon-Manna-Pempinelli Equation,” Ap-
Open Access JAMP

H. C. MA, Y. B. BAI
Open Access JAMP
24
es, Vol. 6, No. 12, 2012, pp. plied Mathematical Scienc
579-587.
[5] L. Luo, “New Exact Solutions and Bäcklund Transforma-
tion for Boiti-Leon-Manna-Pempinelli Equation,” Physics
Letters A, Vol. 375, No. 7, 2011, pp. 1059-1063.
http://dx.doi.org/10.1016/j.physleta.2011.01.009
[6] L. Delisle and M. Mosaddeghi, “Classical and SUSY
Solutions of the Boiti-Leon-Manna-Pempinelli Equation,”
Journal of Phyd Theoretical, Vol.
46, No. 11, 20
sics A: Mathematical an
13, Article ID: 115203.
http://dx.doi.org/10.1088/1751-8113/46/11/115203
[7] M. Najafi and S. Arbabi, “Wronskian Determinant Solu-
tions of the (2+1)-Dimensional Boiti-Leon-Manna-Pem-
pinelli Equation,” Internati
thematical Sciences, Vol. 1, No. 1, 2013, pp. 8-11.
onal Journal of Advanced Ma-
ications in Theoretical Physics
[8] M. Darvishi, M. Najafi, L. Kavitha and M. Venkatesh,
“Stair and Step Soliton Solutions of the Integrable (2+1)
and (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli
Equations,” Commun , Vol.
58, No. 6, 2012, pp. 785-794.
http://dx.doi.org/10.1088/0253-6102/58/6/01
[9] R. Hirota, “The Direct Method in Soliton Theory,” Cam-
bridge University Press, Cambridge, 2004.
[10] W. X. Ma, “Wronskians, Generalized Wronskians and
Solutions to the Korteweg-de Vries Equation,” C haos,
Solitons and Fractals, Vol. 19, No. 1, 2004, pp. 163-170.
http://dx.doi.org/10.1016/S0960-0779(03)00087-0
[11] W.-X. Ma and Y. You, “Solving the Korteweg-de Vries
Equation by Its Bilinear Form: Wronskian Solutions,”
Transactions of the American Mathematical Society, Vol.
357, No. 5, 2005, pp. 1753-1778.
http://dx.doi.org/10.1090/S0002-9947-04-03726-2
[12] C.-X. Li, W.-X. Ma, X.-J. Liu and Y.-B. Zeng, “Wron-
skian Solutions of the Boussinesq Equatio
Negatons, Positons and Complexitons,” Invern—Solitons, se Problems,
5
Vol. 23, No. 1, 2007, pp. 279-296.
http://dx.doi.org/10.1088/0266-5611/23/1/01