Applied Mathematics, 2010, 1, 406-410
doi:10.4236/am.2010.15053 Published Online November 2010 (http://www.SciRP.org/journal/am)
Series New Exact Solutions to Nonlinear
Nizhnik-Novikov-Veselov System
Jiping Hao, Weihui Zhong, Xinli Yan
Xi’an University of Architecture and Technology, Xi’an, China
Received August 27, 2010; revised September 20, 2010; accepted September 24, 2010
Abstract
One new solving expression is built for Nizhnik-Novikov-Veselov system in the paper. Through corre-
sponding auxiliary equation arrangement, more than 150 analytical solutions of elementary and Jacobi ellip-
tic functions are obtained so that the NNV system has a wider range of physical meaning. At the same time,
the existence and uniqueness of this systematic solution are discussed by fixed point theory of partially or-
dered space. The expression of the unique solution could be gained if making use of the technique of com-
puter.
Keywords: Nonlinear Nizhnik-Novikov-Veselov System, Solving Expression, Auxiliary Equation,
Analytical Solution, Fixed Point Theory of Partially Ordered Space
1. Introduction
(2+1) dimensional nonlinear NizhniK-Novikov-Veselov
system is:
33 0
txxx xx
xy
uuvu vu
uv
 




 (1)
One group double-period solutions to (1) are obtained in
[1], and the interaction between its various forms is dis-
cussed in [2]. One new exact solution is gained in [3],
constructed a peak soliton structure with different shape,
and the fractal phenomenon of soliton is studied. Refer-
ence [4] makes use of (GG
) expansion method, and
three types of new exact solutions (hyperbolic function
solution, trigonometric function solution and rational
function solution) to (1) are obtained. Through one new
solving expression and corresponding auxiliary equation
arrangement, more than 150 analytical solutions of ele-
mentary and Jacobi elliptic functions could be obtained,
which is only given some representative but not all in the
paper. So, it makes the NNV system has a wider range of
physical meaning. At the same time, the existence and
uniqueness of the systematic solution are discussed by
fixed point theory of partially ordered space. The expres-
sion of unique solution could be gained if making use of
the computer technique.
Similar to [4], make traveling-wave transform:






,,, ,,,uxytuvxytvx lyst 


(2)
where: l, s are non-zero constants. Substitute them into
(1):

30su uuv
ulv

 

 

 

(3)
Integrate the second equation of (3) and get integral
constant c1, we have:
1
ulvc

 (4)
Substitute (4) into the first equation of (1) and integrate
(get integral constant c2):


2
12
330lvcsl vlvc


 (5)
2. Analytical Solution
Let:
212
01 212
va afafbfbf

  (6)
where: f satisfies
2234
01 234
f
eefefef ef
  (7)
and 0
a,1 a,2 a,1 b,2 b,


ii 0,1,2,3,4e are constants to
be determined. Substitute (6) and (7) into (5):

J. P. HAO ET AL. 407

2 0132 411132 4
1
23 38
2
laebebe aebebe

 




2
10011222
3322c slalaababc 
 
2112110 121
336lae aec slalaa abf





22
2213120 21
3
433
2
laeaecslalaa af



 


 
34
23141224 2
526 32laeaeaaflae af 
  
1
12231101 12
33 6lbebecslblab abf


 



22
112 2120 21
343 32
2
lbe becslblabb f




 



34
10211220 2
256 320lbebebbflbebf

 
Make the coefficients of the power of f to be zero, and
9 algebraic equations are obtained. Solve them:
13
ae (8.1)
24
2ae (8.2)
11
be (8.3)
20
2be (8.4)
 
12 0311
0
1
63
6
leeeec sle
ale

while 10e (8.5a)
 
14 2313
0
3
63
6
leeeecsle
ale

while 30e (8.5b)
2
102100
0
3
812
62 3
lsl
ceeeae
e




while 00e (8.6a)
2
124304
4
3
812
62 3
lsl
ceeeae
e




while 40e (8.6b)



2201324111324
2
1001122
23 38
2
3322
l
claebeb eaebeb e
cslalaab ab

 
(8.7)
22
031 4
ee ee (from two equations) (8.8)
Solve (7):
1) If 2
0
eC, 12eBC,

2
22eACB
,
32eAB, 2
4
eA then

2
234 2
01 234
eefefefefAfBfC 
Through separating variables for (7), integrate and ar-
range:
(a) While 24BAC, then


20
20
4
2
4
41
221
BAC
BAC
BACe
B
fAAe


 
 




 



(9.1)
(b) While 24BAC, then

22
0
44
tg
222
A
CB ACBB
f
A
A



 


(9.2)
(c) While 24BAC, then

0
12
2
f
B
A 









 (9.3)
2) While


ii 0,1,2,3,4e for different values, equa-
tion (7) has different solutions：
(a) 01e

,14er

,2
216er ,


2
3212err ,


22
41er r

1
sec
f
hr 


(10.1)
(b) 01e

,14er

,

2
216er ,


2
3212er r,


22
41er r

1
sec
f
r 


or

1
csc r 

 (10.2)
(c) 01e

,14er

,

2
214er ,


2
3212er r ,


22
41er r

1
csc
f
hr 


(10.3)
(d) 2
0
em,2
14erm ,

222
216emrm ,


222
3212ermrm ,
 
222
411errm 

1
sn
f
r 

 (10.4)
(e) 2
0
em

,2
14erm,

222
212 6emrm ,


222
3212ermrm ,
 
2222
411ermrm 

1
cn
f
r 

 (10.5)
(f) 01e

,14er

,2
24erm ,

22
3222er rm,




222
411errm

1
dn
f
r 


(10.6)
(g) 01 0ee

, 2
2
eq, 2
32erq ,
22
22
42
cb
eqr a

 





 
1
ch sh
f
abqcqar 





 (10.7)
(h) 01
0,ee

2
2,eq 2 32,erq
22
22
42
cb
eqr a



 





J. P. HAO ET AL. 408
 
1
cos sin
f
abqc qar 

 

(10.8)
Above, “m” in (d), (e) and (f) represents module of
Jacobi elliptic function, and the solutions of (g) and (h)
can be refer to [5].
3) When 13
0ee and 0 e, 2 e, 4 e are different
values, over 100 kinds of solutions are given by [6] and
[7]. For example: while 01e

, 22e , 41e

, then
tgf

; while 01e,

2
21em , 2 4
em, then
snf

; while 2
04em,

2
222em ,
2
44em, then

sn1 dnfm

 .
4) When 04
0ee and 1 e, 2 e, 3 e are different
values, 40 kinds of solutions are given by [8]. For exam-
ple: while 14e,


2
241em , 2 34em, then
2
snf

; while

2
2
11em ,


2
221em,
31e, then

2
cn dnfm

.
When coefficients 0
e, 2 e, 3 e, 4 e in auxiliary equa-
tion (7) satisfy (8.8), and c2 takes value from (8.7). Equa-
tion (1) has traveling-wave solution according to (4) and
(6):
1
212
01 212
ulvc
va afafbfbf






 (11)
where: l represents longitudinal wave number, integral
constant c1 takes value from (8.6); according to (8), 0 a,
1
a, 2 a, 1 b, 2 b are functions of ei. Based on the above
discussion, there are more than 150 kinds sampling me-
thods, which are not all listed. So, the following are only
given a few representative solutions:
1) From this section (b) of (i), while 2
0
eC,
12eBC,

2
22eACB, 32eAB, 2
4
eA and
24BAC, then (9.2) is satisfied. Equation (8.8) is:

22 2
22 220ABCAC BBCACB 
So, Equation (1) has traveling-wave solution accord-
ing to (8) and (11):


1
2
1
2212 2
43
6
22 22
ulvc
lBACc sl
vl
A
BfAfBCfCf







 

where: f takes value from (9.2).
2) From this section (f) of (ii), while 01e, 14er

,
2
24erm,

22
3222er rm,
 
222
411errm , then (10.6) is satisfied and

22 22
342 0ermrm is hold based on (8.8). Equa-
tion (1) has traveling-wave solution according to (8) and
(11):

 
1
22 22
1
2222 12
16464 222
62
21 142
ulvc
c
vrr msrrmf
l
rrmfrff






 

where: f takes value from (10.6).
3) From this Section 3, while 13 0ee, 01e

,


2
21em , 2
4
em, then snf

, and (8.8) is
auto-satisfied. So, the solution to (1) is:

1
22 2
2sn sn
ulvc
vm









4) From this section (iv), while 04
0ee,

2
2
11em ,


2
221em, 31e , then

2
cn dnfm

, and (8.8) is auto-satisfied. Because
00e

from (8.6), we know that:
1
2
01cn sec,dn sec
4sec
emh h
fh

 


 

So, the solution to (1) is:

1
2
1
1344sec
6
ulvc
vcsllh
l







3. Existence and Uniqueness of the Solution
This paper makes use of fixed point theory of partially
ordered space [9-11] (author has not found any results
better than this from Chinese and foreign literatures for
the last decade), and the expression of exact solution to
(5) could be obtained combining with the computer tech-
nique.
Lemma [9-11] suppose E is real Banach space having
normal cone, 00
uv

,


00
,uv E, then binary opera-
tor


,
A
uv is mixed monotone (A increases with the
first argument and decreases with the second argument),
and satisfies:
1)


000
,uAuv,


00 0
,
A
vu v;
2) ,uv

, 00 uuvv

 ,

0,1   , make




.,
A
vuAuvvu 

.
So the operator A has a unique fixed point u
(


,
A
uu u

). For arbitrary


000
,wuv, let


nn-1n-1
,wAww(n1,2...

), there is n
n
limuw

, and
the error estimation is: n00
n
uwv u   .
Integrate (5), and transfer multiple integral formula
into single one:

J. P. HAO ET AL. 409
  
2
1
0
2
234
133vtcslvtlvtdt
l
ccc




 


 (12.1)
where: 2
c is different from that in (5) for one factor. If
,0ls and 13csl , let
  
2
1
0
2
234
1
,3()3
A
uvtcsl vtlutdt
l
ccc




 


 (12.2)
then binary operator A increases with u and decreases
with v, so A is mixed monotone. Here it makes


0,1EC, because


0,1C is real Banach space hav-
ing normal cone. Make the equivalent norm as

01
max Mt
t
ueut


 (0,M



0,1 ,ut C


00
,uv



0,1 ), so
 
 
00
2
1234
0
,0,1
13
Au vA
tcsldtccc
l





 

00
2
234
0
,1,0
13
Av uA
ltdtc cc
l





Make integral constant 1
2
3
2
csl
cl




, 30c,
423
1
2
ccc




, then

000000
0, ,1uAuvAvu v
So, operator A defined by (12.1) satisfies the condition
(i) of Lemma.
 
 

 
1
0
,,
133
AvuAuv
tlvtut cslvtutdt
l



  
  
 

 
1
0
133
Mt Mt
eetlvtutc sl
l
vtut dt









 
 
2
0
,,
162
Mt Mt
Avu Auv
eetllcvtutdt
l










2
20
23
23 1
Mt M
c
cvu edtuve
M


 

Make


2
23 0, 1 c
M


, then
 
,,
A
vuAuvv u  
Therefore, the binary operator A defined by (12.1) sat-
isfies all conditions of Lemma. Based on Lemma, there
is one unique continuous function

u

for the unique
solution to (5) while integral constants 1 c, 2 c, 3 c, 4 c
are regular selected. Get


00,1w randomly, let
  
n1n-1n-1
0
2
234
133 1wtcsllwt wtdt
l
ccc







then




n
n
limuw



, where 1 c, 2 c, 3 c, 4 c are
given their values according to the requirements of this
section.
4. Conclusions
Accompanied by the corresponding auxiliary equation,
this solving expression is built to obtain traveling-wave
analytical solution to nonlinear NNV system for the first
time. This method for some nonlinear evolution equa-
tions is also useful. Some articles solving nonlinear or-
dinary differential equation by fixed point theory can
often be found in various of mathematical journals. In
fact, this theory is also very useful for traveling-wave
solution to nonlinear evolution equation.
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J. P. HAO ET AL. 410
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