Applied Mathematics, 2013, 4, 1599-1602
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412216
Open Access AM
Doubly and Triply Periodic Waves Solutions for the KdV
Equation*
Ying Huang, Dengguo Xu
Department of Mathematics, Chuxiong Normal University, Chuxiong, China
Email: huang11261001@163.cn
Received September 6, 2013; revised October 6, 2013; accepted October 13, 2013
Copyright © 2013 Ying Huang, Dengguo Xu. 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
Based on the arbitrary constant solution, a series of explicit doubly periodic solutions and triply periodic solutions for
the Korteweg-de Vries (KdV) equation are first constructed with the aid of the Darboux transformation method.
Keywords: KdV Equation; Doubly Periodic Solution; Triply Periodic Solution; Darboux Tr ansformation
1. Introduction
The famo us KdV equati on
6
txxxx
uuuu0 (1)
is a shallow water wave equation early derived by
Korteweg de and Vries, its first application was discov-
ered in the study of collision-free hydro-magnetic waves
in 1960. Subsequently, it has arisen in a number of phy-
sical contexts, such as stratified internal waves, ion-
acoustic waves, plasma physics, lattice dynamics and so
on. Following the further studies of these physical prob-
lems, its exact solutions have attracted much attention
and have been extensively studied [1-7]. However, in
contrast to solitary wave solutions, the analytic periodic
solutions represent only a small subclass of its known
solutions, and multi-periodic solutions are scarce. It is
always useful to seek more and various multi-periodic
solutions for recovering interactions among some simple
periodic waves in a nonl inear medium.
We know that the Darboux transformation method is
the main method to construct exact multi-soliton solu-
tions, and this method is scarcely used for solving multi-
periodic solutions [8-10]. In the paper, not only explicit
doubly periodic solutions are available, but also a group
of explicit triply periodic solutions is obtained by means
of the Darboux transformation method.
2. Doubly Periodic Solutions
According to [11], the linear system

01
,
0
42 ,
x
x
tx
u
uu
Au

 






(2)
is the Lax pair for Equation (1), with the Darboux matrix

2
1
,, ,
i
ii i
Dxt
 


 

(3)
where
42
x
x
A
uuu

 
i
u
, are
the spectral parameters. The monograph [11] further points
out, if is a known solution to Equati on (1), then

,0,1,
ii

2
i
2
122
iii
uu
 (4)
becomes new solution generated from , with
i
u







21 22
11 12
,,,, ,
,, ,,
ii
ii ii
iii
ii ii
axt axt
axt axt


(5)
where, i
and i
are arbitrary constants, but
ii
22

0
, and



22
,, ,,
i
ijk
xta xt
 u is the
fundamental solution matrix to the lax pair on .
i
Only solving the fundamental solution matrix of the
lax pair corresponding to constant solution 0, it is
possible to construct multi-periodic solutions to the KdV
Equation (1). Substituting into the system (2) yields
u
0
u
*This work was supported by the Chinese Natural Science Foundation
Grant (11261001) and Yunnan Provincial Department of Education
Research Foundation Grant (2012Y130).
Y. HUANG, D. G. XU
1600

0
00
01
,
0
01
=4 2.
0
x
t
u
uu

 

 

 


(6)
If setting
0
=42
x
ut

 , then we can assert that
both the system (6) and the following linear system
0
01
0u

 


have exactly the same solutions. Under the condition for
0
u
, by the eigenvalue method, we obtain the com-
plex-valued fundamental solution matrix to the above
system
ee
,
ee
ia ia
ia ia
ia ia



(7)
where

0
aa u
. Because the real and
imaginary parts of a complex-valued solution are also
solutions, we thus take

0
cos sin
,, sin cos
xt aa




(8)
as the fundamental solution matrix to the the system (6),
where

a
 
.
For simplicity, we setting
 
22
,, j
iiiiijij
aaaa i
 


,0,1,2ij
,
.
From (5), we have
000 0
00
0000
sincos ,
cos sin
a
 
 

in the above formula, choosing 00
1, 0
 and
00
0, 1
, respectively, we get
00
tan
ta0
 (9)
and
00 0
cot ,
ca
(10)
respectively, with (4), the periodic wave solutions
22
11 000
2secuua

and
22
12 000
2cscuua

are obtained.
Now we construct the doubly periodic solutions gen-
erated from , thanks to (4), we see that
1
u

2
21 0001
=2222 ,uu
2

 
(11)
we first give 1
, then substitute 0
and 1
into (11).
Again according to [11], we can obtain the fundamental
solution matrix to the lax pair associated with the known
periodic wave solution in the following manner
1
u
 
0
10
2
00 0
00
1
,, ,,
cossinsincos ,
xt xt
aa
PQ

 
 

 

 

 



(12)
where
2
00 0
cos sinPa
 
,
2
00 0
sin cosQa
 
 
11
1,0 . After combining (5)
and (12), choosing

, we get
2
10 0011
101 1
tan .
tan
t
a
a
 

 
 (13)
Substituting (9) and (13) into (11), we have new dou-
bly periodic solution

22 22
010011
21 02
0011
2sec sec
.
tan tan
aa
uu aa


 (14)
Again substituting (10) and (13) into (11), we obtain
another new doubly periodic solution

22 22
01 0011
22 02
0011
2cscsec .
cot tan
aa
uu aa


 (15)
Similarly, choosing 11
0, 1
, we have
2
10 0011
101 1
cot ,
cot
c
a
a
 

 
 (16)
which implies the doubly periodic solutions

22 22
01 0011
23 02
0011
2 seccsc
tancot
aa
uu aa


 (17)
and

22 22
01 0001
24 02
0011
2csccsc .
cot cot
aa
uu aa


 (18)
Specially, although 23
u
is a doubly periodic solu-
tion, its structure is very similar to a given two-soliton
solution in [1].
3. Triply Periodic Solutions
As shown in [11], the fundamental so lution matrix to the
lax pair associated with the doubly periodic wave solu-
tion can be given by
2
u
 
1
21
2
11 1
1
,,,, ,xt xt
 

 

 

(19)
substituting (12) into (19 ), in exactly the same manner as
in Section 2, we get
Open Access AM
Y. HUANG, D. G. XU 1601


 
12 022
21
20 01022
tan
tan
t
a
a
 
 


 
and


 
12 022
21
20 01022
cot .
cot
c
a
a
 
 


 
Owing to (4) and (11), we have
 
22
3000 2112
22uu


2
0
u
i
. (20)
Here, we set tan
ii
Fa
, cot ,0,1,2
ii i
Gai
.
Substituting 01
,
tt
2t
and
into (20), we obtain trip-
ly periodic solution






22
31 000
2
22 22
12 01001102
2
02 101002
2
22 22
12022 20 010
2
02 101002
2sec
2secsec
2secsec .
uua
aaFF
FF FF
aaFF
FF FF



 



 

 

Similarly, we have






22
32 000
2
22 22
12 01001102
2
02 101002
2
22 22
12 02220010
2
02 101002
2csc
2cscsec
2sec csc,
uua
aaGF
FGG F
aaFG
FGG F



 



 

 







22
33 000
2
22 22
12 01001102
2
021010 02
2
22 22
12 02220010
2
021010 02
2sec
2sec csc
2secsec ,
uua
aaFF
GF FF
aaGF
GF FF



 

 







22
34 000
2
22 22
12 01001102
2
02 101002
2
22 22
12 02220010
2
02 101002
2csc
2csccsc
2sec csc,
uua
aaGF
GGG F
aaGG
GGG F



 

 



 


 
22
35 000
2
22 22
12010 01 102
2
02 101002
2
22 22
12 02220010
2
02 101002
2sec
2secsec
2csc sec,
uua
aaFG
FF FG
aaFF
FF FG



 



 





 


 
22
36 000
2
22 22
12 01001102
2
02 101002
2
22 22
12022 20 010
2
02 101002
2csc
2csc sec
2csccsc ,
uua
aaGG
FG GG
aaFG
FG GG



 

 

 

 







22
37 000
2
22 22
12 01001102
2
021010 02
2
22 22
12 02220010
2
021010 02
2sec
2sec csc
2csc sec,
uua
aaFG
GF FG
aaGF
GF FG



 



 



and






22
38 000
2
22 22
12010 01 102
2
02 101002
2
22 22
12 02220010
2
02 101002
2csc
2csccsc
2csccsc .
uua
aaGG
GG GG
aaGG
GG GG



 



 



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Open Access AM
1602
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