Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 767254, 5 pages
http://dx.doi.org/10.1155/2013/767254
Research Article
The Multisoliton Solutions for the (2 + 1)-Dimensional
Sawada-Kotera Equation
Zhenhui Xu,1 Hanlin Chen,2 and Wei Chen3
1
Applied Technology College, Southwest University of Science and Technology, Mianyang 621010, China
School of Science, Southwest University of Science and Technology, Mianyang 621010, China
3
School of Mathematics and Computer Science, Mianyang Normal University, Mianyang 621000, China
2
Correspondence should be addressed to Zhenhui Xu; xuzhenhui19@163.com
Received 13 November 2012; Revised 19 December 2012; Accepted 1 January 2013
Academic Editor: Lan Xu
Copyright © 2013 Zhenhui Xu et al. 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.
Applying bilinear form and extended three-wavetype of ansätz approach on the (2 + 1)-dimensional Sawada-Kotera equation, we
obtain new multisoliton solutions, including the double periodic-type three-wave solutions, the breather two-soliton solutions,
the double breather soliton solutions, and the three-solitary solutions. These results show that the high-dimensional nonlinear
evolution equation has rich dynamical behavior.
1. Introduction
As is well known that the exact solutions of nonlinear evolution equations play an important role in nonlinear science
field, especially in nonlinear physical science since they can
provide much physical information and more insight into the
physical aspects of the problem and thus lead to further applications. The search for exact solutions of nonlinear partial
differential equations has long been an interesting and hot
topic in nonlinear mathematical physics. Consequently, many
methods are available to look for exact solutions of nonlinear
evolution equations, such as the inverse scattering method,
the Lie group method, the mapping method, Exp-function
method, and ansätz technique [1–4]. Very recently, Wang
et al. [5] proposed a new technique called extended threewave approach to seek multiwave solutions for integrable
equations, and this method has been used to investigate
several equations [6, 7]. In this paper, we consider the
following Sawada-Kotera equation:
5
𝑢𝑡 = (𝑢𝑥𝑥𝑥𝑥 + 5𝑢𝑢𝑥𝑥 + 𝑢3 + 5𝑢𝑥𝑦 )
3
𝑥
− 5 ∫ (𝑢𝑦𝑦 ) 𝑑𝑥 + 5𝑢𝑢𝑦 + 5𝑢𝑥 ∫ (𝑢𝑦 ) 𝑑𝑥.
Equation (1) was derived by B. G. Konopelchenko and V. G.
Dubrovsky, and was called the Sawada-Kotera (SK) equation;
for example, see [8]. By means of the two-soliton method,
the exact periodic soliton solutions, N-soliton solutions, and
traveling wave solutions of the SK equation were found [8–
10].
In this paper, we discuss further the (2 + 1)-dimensional
SK equation, by using bilinear form and extended three-wave
type of ansätz approach, respectively [5, 11–15], and some new
multisoliton solutions are obtained.
2. The Multisoliton Solutions
We assume
𝑢 = −2(ln 𝑓)𝑥𝑥 ,
(2)
where 𝑓 = 𝑓(𝑥, 𝑦, 𝑡) is an unknown real function. Substituting (2) into (1), we can reduce (1) into the following equation
[8]:
(1)
(𝐷𝑥6 + 5𝐷𝑦 𝐷𝑥3 − 5𝐷𝑦2 + 𝐷𝑥 𝐷𝑡 ) 𝑓 ⋅ 𝑓 = 0,
(3)
2
Abstract and Applied Analysis
where the Hirota bilinear operator 𝐷 is defined by (𝑛, 𝑚 ≥ 0)
𝐷𝑥𝑚 𝐷𝑡𝑛 𝑓 (𝑥, 𝑡) ⋅ 𝑔 (𝑥, 𝑡)
𝑚
𝑛
𝜕
𝜕
𝜕
𝜕
−
) ( − 󸀠)
𝜕𝑥 𝜕𝑥󸀠
𝜕𝑡 𝜕𝑡
󵄨
× [𝑓 (𝑥, 𝑡) 𝑔 (𝑥󸀠 , 𝑡󸀠 )]󵄨󵄨󵄨󵄨𝑥󸀠 =𝑥,𝑡󸀠 =𝑡 .
=(
(4)
Substituting (7) into (2) yields the three-soliton solution of
SK equation as follows:
1
𝑢1 = − (2 [2√𝛿3 𝑎12 cosh (𝜉1 + ln (𝛿3 ))
2
−
𝛿1 𝑎12 𝑎32 cosh (𝜂1 )
])
𝑎12 −𝑎32
× (2√𝛿3 cosh (𝜉1 +
Now we suppose the solution of (3) as
1
ln (𝛿3 ))
2
−1
𝑓 = 𝑒−𝜉 + 𝛿1 cos (𝜂) + 𝛿2 cosh (𝛾) + 𝛿3 𝑒𝜉 ,
(5)
where 𝜉 = 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 𝑡, 𝜂 = 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 𝑡, 𝛾 = 𝑎3 𝑥 +
𝑏3 𝑦 + 𝑐3 𝑡, and 𝑎𝑖 , 𝑏𝑖 , and 𝑐𝑖 (𝑖 = 1, 2, 3) are some constants
to be determined later. Substituting (5) into (3) and equating
all the coefficients of different powers of 𝑒𝜉 , 𝑒−𝜉 , sin(𝜂), cos(𝜂),
sinh(𝛾), cosh(𝛾), and the constant term to zero, we can obtain
a set of algebraic equations for 𝑎𝑖 , 𝑏𝑖 , 𝑐𝑖 , and 𝛿𝑗 (𝑖 = 1, 2, 3;
𝑗 = 1, 2, 3). Solving the system with the aid of Maple, we get
the following results.
−
−
𝛿1 𝑎12 𝑎3 sinh (𝜂1 )
))
(𝑎12 −𝑎32 )
× (2√𝛿3 cosh (𝜉1 +
1
ln (𝛿3 ))
2
−1 2
1
𝑏1 = − 𝑎1 (4𝑎12 + 3𝑎32 ) ,
4
1
𝑏3 = − 𝑎3 (6𝑎12 + 𝑎32 ) ,
4
𝑏2 =
𝛿2 =
𝛿 𝑎2 cosh (𝜂1 )
− 𝛿1 cosh (𝛾1 )) ] ,
− 1 12
𝑎1 − 𝑎32
3 2
𝑖𝑎 𝑎 ,
2 1 3
where 𝜉1 = 𝑎1 𝑥 + 𝐾1 𝑦 + 𝐿 1 𝑡, 𝜂1 = 𝑎3 𝑥 − 𝐻1 𝑦 + 𝐽1 𝑡, and
𝛾1 = 𝑀1 𝑦 + 𝑁1 𝑡.
If taking 𝑎3 = 𝑖𝐴 3 in (7), then we have
𝛿 𝑎2
− 21 1 2,
𝑎1 − 𝑎3
𝛿3 = 𝛿3 ,
9
𝑐1 = 𝑎1 (5𝑎34 + 40𝑎12 𝑎32 + 16𝑎14 ) ,
16
𝑐3 =
(8)
1
+ [ (2 (2√𝛿3 𝑎1 sinh (𝜉1 + ln (𝛿3 ))
2
Case 1. If 𝑎2 = 0, then
𝑐2 = −
𝛿1 𝑎12 cosh (𝜂1 )
− 𝛿1 cosh (𝛾1 ))
𝑎12 − 𝑎32
(6)
−
9
𝑎 (𝑎4 + 20𝑎12 𝑎32 + 40𝑎14 ) ,
16 3 3
where 𝑎1 , 𝑎3 , 𝛿1 , and 𝛿3 are free real constants. Substituting
(6) into (5) and taking 𝛿3 > 0, we have
− 𝛿1 cosh (𝑀1 𝑦 + 𝑁1 𝑡) −
1
ln (𝛿3 ))
2
𝛿1 𝑎12
𝑎12 − 𝑎32
1
ln (𝛿3 ))
2
+ 𝛿1 cos (𝑀2 𝑦 + 𝑁2 𝑡)
45 2
𝑖𝑎 𝑎 (2𝑎12 + 𝑎32 ) ,
4 1 3
𝑓1 = 2√𝛿3 cosh (𝑎1 𝑥 + 𝐾1 𝑦 + 𝐿 1 𝑡 +
𝑓2 = 2√𝛿3 cosh (𝑎1 𝑥 + 𝐾2 𝑦 + 𝐿 2 𝑡 +
𝛿1 𝑎12 cos (𝐴 3 𝑥 − 𝐻2 𝑦 + 𝐽2 𝑡)
,
𝑎12 + 𝐴23
where 𝛿3 > 0, 𝐾2 = −(1/4)𝑎1 (4𝑎12 −3𝐴23 ), 𝐿 2 = (9/16)𝑎1 (5𝐴43 −
40𝑎12 𝐴23 + 16𝑎14 ), 𝑀2 = (3/2)𝑎12 𝐴 3 , 𝑁2 = −(45/4)𝑎12 𝐴 3 (2𝑎12 −
𝐴23 ), 𝐻2 = 𝐴 3 𝑥 − (1/4)𝐴 3 (6𝑎12 − 𝐴23 ), and 𝐽2 = (9/16)𝐴 3 (𝐴43 −
20𝑎12 𝐴23 + 40𝑎14 ). Substituting (9) into (2) yields the double
breather soliton solution of SK equation as follows:
𝑢2 = − (2 [2𝑎12 √𝛿3 cosh (𝜉2 +
(7)
× cosh (𝑎3 𝑥 − 𝐻1 𝑦 + 𝐽1 𝑡) ,
where 𝐾1 = (1/4)𝑎1 (4𝑎12 + 3𝑎32 ), 𝐿 1 = (9/16)𝑎1 (5𝑎34 + 40𝑎12 𝑎32 +
16𝑎14 ), 𝑀1 = −(3/2)𝑎12 𝑎3 , 𝑁1 = (45/4)𝑎12 𝑎3 (2𝑎12 + 𝑎32 ), 𝐻1 =
(1/4)𝑎3 (6𝑎12 + 𝑎32 ), and 𝐽1 = (9/16)𝑎3 (𝑎34 + 20𝑎12 𝑎32 + 40𝑎14 ).
(9)
+
1
ln (𝛿3 ))
2
𝛿1 𝑎12 𝐴23 cos (𝜂2 )
])
𝑎12 + 𝐴23
× (2√𝛿3 cosh (𝜉2 +
1
ln (𝛿3 ))
2
−1
+𝛿1 cos (𝛾2 ) −
𝛿1 𝑎12 cos (𝜂2 )
)
𝑎12 + 𝐴23
Abstract and Applied Analysis
3
1
ln (𝛿3 ))
2
+ 2 [(2𝑎1 √𝛿3 sinh (𝜉2 +
+
The expression (𝑢3 ) is the breather two-soliton solution of
SK equation which is a periodic wave in 𝑥, 𝑦 and meanwhile
is a two-soliton in 𝑥, 𝑦 (refer to Figure 1(b)).
𝛿1 𝑎12 𝐴 3 sin (𝜂2 )
)
𝑎12 +𝐴23
Case 3. If 𝑎2 = 𝑏1 = 0, then
1
× (2√𝛿3 cosh (𝜉2 + ln (𝛿3 ))
2
𝑎1 = 2𝑎3 ,
𝑐1 = −
−1 2
𝛿1 𝑎12 cos (𝜂2 )
) ],
+𝛿1 cos (𝛾2 )−
𝑎12 +𝐴23
where 𝜉2 = 𝑎1 𝑥 + 𝐾2 𝑦 + 𝐿 2 𝑡, 𝜂2 = 𝐴 3 𝑥 − 𝐻2 𝑦 + 𝐽2 𝑡, and
𝛾2 = 𝑀2 𝑦 + 𝑁2 𝑡.
Case 2. If 𝑎2 ≠ 0, then
𝑏1 = −𝑎13 ,
𝑏2 = 𝑎23 ,
𝛿1 = 𝛿1 ,
𝑏3 = −𝑎33 ,
𝛿2 = 𝛿2 ,
𝑐1 = 9𝑎15 ,
𝛿3 = 𝛿3 ,
𝑐2 = 9𝑎25 ,
(11)
𝑐3 = 9𝑎35 ,
𝑓3 = 2√𝛿3 cosh (𝑎1 𝑥 − 𝑎13 𝑦 + 9𝑎15 𝑡 +
Substituting (12) into (2) yields the breather two-soliton
solution of SK equation as follows:
1
ln (𝛿3 ))
2
−1
(13)
1
ln (𝛿3 ))
2
+
+ 𝛿1 cos (−√21𝑎33 𝑦 + 20√21𝑎35 𝑡)
(15)
1
ln (𝛿3 ))
2
𝜂3 = 𝑎2 𝑥 +
where (5/152)𝛿22 − (7/228)𝛿12 > 0. Substituting (15) into (2)
yields the breather two-soliton solution of SK equation as
follows:
𝑢4 = − (2 [8√𝐾4 𝑎32 cosh (𝜉4 −
1
ln (𝐾4 ))
2
1
ln (𝐾4 ))
2
−1
(16)
1
+ 2 [ (4√𝐾4 𝑎3 sinh (𝜉4 − ln (𝐾4 ))
2
× (2√𝐾4 cosh (𝜉4 −
1
ln (𝐾4 ))
2
−1 2
+𝛿1 cos (𝛾4 ) + 𝛿2 cosh (𝜂4 ) ) ] ,
−1 2
9𝑎15 𝑡,
169 5
1
7 2
5 2
𝑎3 𝑡 − ln (
𝛿2 −
𝛿 ))
2
2
152
228 1
+𝛿2 𝑎3 sinh (𝜂4 ) )
+𝛿1 cos (𝜂3 ) + 𝛿2 cosh (𝛾3 ) ) ] ,
where 𝜉3 = 𝑎1 𝑥 − 𝑎13 𝑦
𝛾3 = 𝑎3 𝑥 − 𝑎33 𝑦 + 9𝑎35 𝑡.
5 2
7 2
𝛿 −
𝛿
152 2 228 1
+𝛿1 cos (𝛾4 ) + 𝛿2 cosh (𝜂4 ) )
−𝛿1 𝑎2 sin (𝜂3 ) + 𝛿2 𝑎3 sinh (𝛾3 ) ))
× (2√𝛿3 cosh (𝜉3 +
𝑓4 = 2√
× (2√𝐾4 cosh (𝜉4 −
1
ln (𝛿3 ))
2
+ [ (2 (2√𝛿3 𝑎1 sinh (𝜉3 +
where 𝑎3 , 𝛿1 , and 𝛿2 are free real constants. Substituting (14)
into (5) and taking 𝛿3 > 0, we have
+𝛿2 𝑎32 cosh (𝜂4 ) ])
−𝛿1 𝑎22 cos (𝜂3 ) + 𝛿2 𝑎32 cosh (𝛾3 ) ])
+𝛿1 cos (𝜂3 ) + 𝛿2 cosh (𝛾3 ) )
(14)
5 2
7 2
𝛿2 −
𝛿,
152
228 1
(12)
+ 𝛿2 cosh (𝑎3 𝑥 − 𝑎33 𝑦 + 9𝑎35 𝑡) .
× (2√𝛿3 cosh (𝜉3 +
349 5
𝑐3 = −
𝑎,
4 3
3
349 5
+ 𝛿2 cosh (−𝑎3 𝑥 + 𝑎3 3 𝑦 +
𝑎 𝑡) ,
2
4 3
1
ln (𝛿3 ))
2
+ 𝛿1 cos (𝑎2 𝑥 + 𝑎23 𝑦 + 9𝑎25 𝑡)
𝑢3 = − (2 [2√𝛿3 𝑎12 cosh (𝜉3 +
𝛿3 =
× cosh (−2𝑎3 𝑥 +
where 𝑎1 , 𝑎2 , 𝑎3 , 𝛿1 , 𝛿2 , and 𝛿3 are free real constants.
Substituting (11) into (5) and taking 𝛿3 > 0, we have
169 5
𝑎,
2 3
𝑐2 = −20√21𝑎35 ,
(10)
3
𝑏3 = − 𝑎33 ,
2
𝑏2 = √21𝑎33 ,
𝑎23 𝑦
+
9𝑎25 𝑡,
and
where 𝐾4 = (5/152)𝛿22 − (7/228)𝛿12 , 𝜉4 = −2𝑎3 𝑥 + (169/2)𝑎35 𝑡,
𝜂4 = −𝑎3 𝑥 + (3/2)𝑎33 𝑦 + (349/4)𝑎35 𝑡, and 𝛾4 = −√21𝑎33 𝑦 +
20√21𝑎35 𝑡.
Abstract and Applied Analysis
−0.06
−0.04
−0.02
0
−20
0
−1
−2
𝑢3
𝑢2
4
−10
0
𝑦
10
20
−20
30 −30
−10
10
0
20
−3
−4
−5
−10
𝑥
−5
𝑦
0
(a)
0
4
𝑥
120000
−2
100000
−4
𝑢5
𝑢4
10 8
−8
(b)
0
−6
−8
−3
−2
5
−4
80000
60000
40000
20000
0
−1
𝑦
0
−4
1
2
3
2
4
0
−2
−4
−2
𝑥
𝑥
0
(c)
2
4
−4
−2
0
2
4
𝑦
(d)
Figure 1: (a) The figure of 𝑢2 as 𝛿1 = 1, 𝛿3 = 1, and 𝑡 = 1. (b) The figure of 𝑢3 as 𝛿1 = √2, 𝛿2 = 1, and 𝑡 = 0. (c) The figure of 𝑢4 as 𝛿1 = √2,
𝛿2 = √5, and 𝑡 = 0.005. (d) The figure of 𝑢5 as 𝛿1 = 1, 𝛿2 = 1, and 𝑡 = 0.
The expression (𝑢4 ) is the breather two-soliton solution of
SK equation which is a periodic wave in 𝑦-𝑡 and meanwhile
is a two-soliton in 𝑥, 𝑦 and in 𝑥-𝑡, respectively (refer to
Figure 1(c)).
Notice that 𝑢3 and 𝑢4 are also the breather two-soliton
solutions, but their structure is different, because the two
wave propagation directions are different in the 𝑢3 and 𝑢4 ,
respectively (refer to Figures 1(b) and 1(c)).
If taking 𝑎1 = 𝑖𝐴 1 , 𝑎3 = 𝑖𝐴 3 in (12), then we have
𝑓5 = 2 cos (𝐴 1 𝑥 +
𝐴31 𝑦
+
+ 𝛿2 cos (𝐴 3 𝑥 + 𝐴33 𝑦 + 9𝐴53 𝑡) ,
𝑢5 =
2 [2𝐴21 cos (𝜉5 ) + 𝛿1 𝑎22 cos (𝜂5 ) + 𝛿2 𝐴23 cos (𝛾5 )]
2 cos (𝜉5 ) + 𝛿1 cos (𝜂5 ) + 𝛿2 cos (𝛾5 )
2
+ 2[
2𝐴 1 sin (𝜉5 ) + 𝛿1 𝑎2 sin (𝜂5 ) + 𝛿2 𝐴 3 sin (𝛾5 )
],
2 cos (𝜉5 ) + 𝛿1 cos (𝜂5 ) + 𝛿2 cos (𝛾5 )
(18)
where 𝜉5 = 𝐴 1 𝑥 + 𝐴31 𝑦 + 9𝐴51 𝑡, 𝜂5 = 𝑎2 𝑥 + 𝑎23 𝑦 + 9𝑎25 𝑡, and
𝛾5 = 𝐴 3 𝑥 + 𝐴33 𝑦 + 9𝐴53 𝑡.
9𝐴51 𝑡)
+ 𝛿1 cos (𝑎2 𝑥 + 𝑎23 𝑦 + 9𝑎25 𝑡)
when 𝛿3 = 1. Substituting (17) into (2) gives the doubleperiodic three-wave solution of SK equation as follows:
(17)
3. Conclusion
By using bilinear form and extended three-wave type of
ansätz approach, we discuss further the (2 + 1)-dimensional
Abstract and Applied Analysis
Sawada-Kotera equation and find some new multisoliton
solutions. The result shows that the extended three-wave type
of ansätz approach may provide us with a straightforward and
effective mathematical tool for seeking multiwave solutions of
high-dimensional nonlinear evolution equations.
Acknowledgments
This work was supported by Chinese Natural Science Foundation Grant nos. 11061028, 10971169. Sichuan Educational
Science Foundation Grant no. 09zc008.
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