Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 230871, 3 pages
http://dx.doi.org/10.1155/2013/230871
Research Article
Analysis of Stability of Traveling Wave for
Kadomtsev-Petviashvili Equation
Jun Liu,1 Xi Liu,2 Gui Mu,1 Chunyan Zhu,1 and Jie Fu1
1
2
College of Mathematics and Information Science, Qujing Normal University, Qujing, Yunnan 655011, China
College of Information Science and Engineering, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Jun Liu; liujunxei@126.com
Received 31 January 2013; Accepted 4 February 2013
Academic Editor: de Dai
Copyright © 2013 Jun Liu 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.
This paper presents the boundedness and uniform boundedness of traveling wave solutions for the Kadomtsev-Petviashvili (KP)
equation. They are discussed by means of a traveling wave transformation and Lyapunov function.
1. Introduction
In general, we use the following system, which is equivalent to (2):
We consider the Kadomtsev-Petviashvili (KP) equation:
𝑢𝑡𝑥 + 6𝑢𝑥 𝑢𝑥𝑥 + 𝑢𝑥𝑥𝑥𝑥 + 𝑢𝑦𝑦 + 𝑐𝑢 = 0.
𝑢(4) + 𝑎𝑢󸀠󸀠󸀠 + 𝑓 (𝑡, 𝑢, 𝑢󸀠󸀠 ) + 𝑔 (𝑢󸀠 ) + 𝑑𝑢
(3)
= 𝑝 (𝑡, 𝑢, 𝑢󸀠 , 𝑢󸀠󸀠 , 𝑢󸀠󸀠󸀠 ) ,
(1)
where
It is well known that Kadomtsev-Petviashvili equation arises
in a number of remarkable nonlinear problems both in
physics and mathematics. By using various methods and
techniques, exact traveling wave solutions, solitary wave solutions, doubly periodic solutions, and some numerical solutions have been obtained in [1–6].
In this paper, (1) can be changed into an ordinary differential equation by using traveling wave transformation; the
boundedness and uniform boundedness of solution for the
resulting ordinary differential equation are discussed using
the method of Lyapunov function.
𝑓 (𝑡, 𝑢, 𝑢󸀠 ) = (
𝑝 (𝑡, 𝑢, 𝑢󸀠 , 𝑢󸀠󸀠 , 𝑢󸀠󸀠󸀠 ) = −𝑎𝑢󸀠󸀠󸀠 ,
𝑑=
𝑐
.
𝛼4
6 󸀠2
𝑢 ,
𝛼2
(4)
We consider the following system, which is equivalent to
𝑥1󸀠 = 𝑥2 ,
𝑥2󸀠 = 𝑥3 ,
𝑥3󸀠 = 𝑥4 ,
𝑥4󸀠 = − 𝑎𝑥4 − 𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) − 𝑔 (𝑥2 ) − 𝑑𝑥1
(5)
+ 𝑝 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) .
Taking a traveling wave transformation 𝜉 = 𝛼𝑥 + 𝛽𝑦 + 𝛾𝑡 in
(1), then (1) can be transformed into the following form:
𝛾
𝛽2
6
6 2 𝑐
+ 4 + 2 𝑢) 𝑢󸀠󸀠 + 2 𝑢󸀠 + 4 𝑢 = 0.
3
𝛼
𝛼
𝛼
𝛼
𝛼
𝑔 (𝑢󸀠 ) =
(3):
2. The Boundedness
𝑢(4) + (
𝛾
𝛽2
6
+
+ 𝑢) 𝑢󸀠󸀠 ,
𝛼3 𝛼4 𝛼2
Theorem 1. If the following conditions hold for the system (5):
(i) there are positive constants 𝑎, 𝑏, 𝑑, 𝛿, 𝑘, and 𝜆 such that
𝑘 ≤ 𝑏2 𝜆,
(2)
2
𝑎𝑏
𝑔 (𝑥2 )
𝑔 (𝑥2 )
−[
] − 𝑎2 𝑑 ≥ 𝛿,
𝑥2
𝑥2
(𝑥2 ≠ 0) .
(6)
2
Abstract and Applied Analysis
(ii) 𝑓(𝑡, 𝑥1 , 𝑥2 , 0) = 0, 0 ≤ 𝑓(𝑡, 𝑥1 , 𝑥2 , 𝑥3 )/𝑥3 − 𝑏 ≤ 2𝛿𝜆/
𝑘 (𝑥2 ≠ 0).
(iii) 𝑥3 𝑓t󸀠 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )+𝑥2 𝑥3 𝑓𝑥󸀠1 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )+𝑥32 𝑓𝑥󸀠2 (𝑡, 𝑥1 ,
𝑥2 , 𝑥3 ) ≤ 0.
𝑞(𝑡)(𝑥12
𝑥22
Taking the total derivative of (7) with respect to 𝑡 along the
trajectory of (5), we obtain
2
1
𝑑𝑉
= − 2𝑎𝑏2 [𝑥4 + 𝑔(𝑥2 )]
𝑑𝑡
𝑎
− 2𝑏3 𝑥2 𝑥3 [
1/2
𝑥42 ) ,
𝑥32
(iv) |𝑝(𝑡, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 )| ≤
+ + +
where 𝑞(𝑡) is a nonnegative continuous function and
∞
∫0 𝑞(𝑡)𝑑𝑡 < ∞.
− 2𝑎𝑏2 [
× [𝑎𝑏
Then, all the solutions of system (5) are bounded.
Proof. We first construct the Lyapunov function 𝑉 = 𝑉(𝑡, 𝑥1 ,
𝑥2 , 𝑥3 , 𝑥4 ) defined by
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏]
𝑥3
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
2𝑏2
− 𝑏] 𝑥32 −
𝑥3
𝑎
𝑔 (𝑥2 ) 𝑔2 (𝑥2 )
−
− 𝑎2 𝑑] 𝑥22
𝑥2
𝑥22
𝑥3
+ 4𝑏2 ∫ 𝑓𝑡󸀠 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) 𝑑𝑥3
0
𝑥3
2
2
+ 4𝑏2 𝑥2 ∫ 𝑓𝑥󸀠1 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) 𝑑𝑥3
2
𝑉 = 𝑏 (2𝑥4 + 𝑎𝑥3 + 𝑏𝑥2 ) + 2𝑏𝑑(2𝑥3 + 𝑎𝑥2 + 𝑏𝑥1 )
0
𝑥3
2
+ (𝑏2 − 4𝑑) (𝑎𝑥4 + 𝑏𝑥2 ) + 4𝑎𝑏2
+ 4𝑏2 𝑥3 ∫ 𝑓𝑥󸀠2 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 ) 𝑑𝑥3
0
𝑥2
𝑔 (𝑥2 ) 𝑎𝑑
×∫ [
−
] 𝑥2 𝑑𝑥2
𝑥2
𝑏
0
2
2
+ [2𝑏 (𝑏 − 4𝑑) + 4𝑎
𝑥3
+ 8𝑏2 ∫ [
0
2
+ 2𝑏 (𝑏𝑥2 + 𝑎𝑥3 + 2𝑥4 ) 𝑝 (𝑡, 𝑥, 𝑥2 , 𝑥3 , 𝑥4 ) .
(7)
(11)
𝑑] 𝑥32
By using conditions (i) and (iii), it follows that
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏] 𝑥3 𝑑𝑥3 .
𝑥3
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
𝑑𝑉
2𝑏2 𝛿 2
− 𝑏]
≤−
𝑥2 − 2𝑏3 𝑥2 𝑥3 [
𝑑𝑡
𝑎
𝑥3
It follows from conditions (i) and (ii) that
− 2𝑎𝑏2 [
𝑏2 − 4𝑑 ≥ 0,
𝑥2
0
𝑥3
𝑎 (𝑏 − 𝑑) 2
𝑔 (𝑥2 ) 𝑎𝑑
−
] 𝑥2 𝑑𝑥2 ≤
𝑥2 ,
𝑥2
𝑏
2𝑏
0≤∫ [
0
(8)
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
𝛿𝜆 2
− 𝑏] 𝑥3 𝑑𝑥3 ≤
𝑥.
𝑥3
𝑘 3
𝑉 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) ≥
+
𝑥22
+
𝑥32
+
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏]
𝑥3
+ 2𝑎𝑏2 [
(9)
Thus, we deduce that the function 𝑉(𝑡, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 )
defined in (7) is a positive definite function which has infinite
inferior limit and infinitesimal upper limit. Hence, there
exsits a positive constant 𝜀(> 0) such that
𝜀 (𝑥12
According to (ii), we have
2𝑏3 𝑥2 𝑥3 [
Summing up the above discussions, we get
𝑉 ≥ 2𝑏 (𝑏2 − 4𝑑) 𝑥32 + 4𝑎2 𝑑𝑥32 .
𝑥42 ) .
(12)
+ 2𝑏2 (𝑏𝑥2 + 𝑎𝑥3 + 2𝑥4 ) 𝑝 (𝑡, 𝑥, 𝑥2 , 𝑥3 , 𝑥4 ) .
2
0≤∫ [
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏] 𝑥32
𝑥3
(10)
=−
𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏] 𝑥32
𝑥3
𝑏4 𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏] 𝑥22
[
2𝑎
𝑥3
2
2𝑏2 𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
𝑏
+
− 𝑏] ⋅ (𝑎𝑥3 + 𝑥2 )
[
𝑎
𝑥3
2
≥−
𝑏4 𝑓 (𝑡, 𝑥1 , 𝑥2 , 𝑥3 )
− 𝑏] 𝑥22
[
2𝑎
𝑥3
=−
𝑏4 2𝛿𝜆 2
𝑏4 𝛿𝜆 2
⋅
𝑥2 = −
𝑥.
2𝑎 𝑘
𝑎𝑘 2
(13)
Abstract and Applied Analysis
3
Hence,
2
[5] D. Kaya and S. M. El-Sayed, “Numerical soliton-like solutions of
the potential Kadomtsev-Petviashvili equation by the decomposition method,” Physics Letters A, vol. 320, no. 2-3, pp. 192–199,
2003.
[6] Z. Li, Z. Dai, and J. Liu, “New periodic solitary-wave solutions
to the 3 + 1-dimensional Kadomtsev-Petviashvili equation,”
Mathematical & Computational Applications, vol. 15, no. 5, pp.
877–882, 2010.
[7] C. Tunç, “On the uniform boundedness of solutions of some
non-autonomous differential equations of the fourth order,”
Applied Mathematics and Mechanics, vol. 20, no. 6, pp. 622–628,
1999.
4
𝑑𝑉
2𝑏 𝛿 2 𝑏 𝛿𝜆 2
≤−
𝑥 +
𝑥
𝑑𝑡
𝑎 2
𝑎𝑘 2
+ 2𝑏2 (𝑏𝑥2 + 𝑎𝑥3 + 2𝑥4 ) 𝑝 (𝑡, 𝑥, 𝑥2 , 𝑥3 , 𝑥4 )
=−
𝑏2 𝛿 2 𝑏2 𝛿 2
𝑥 +
(𝑏 𝜆 − 𝑘) 𝑥22
𝑎 2 𝑎𝑘
1/2
+ 2𝑏2 (4 + 𝑎2 + 𝑏2 )
× (𝑥22 + 𝑥32 + 𝑥42 )
1/2
(𝑥22 + 𝑥32 + 𝑥42 )
1/2
𝑞 (𝑡)
≤−
1/2
𝑏2 𝛿 2
𝑥2 + 2𝑏2 (4 + 𝑎2 + 𝑏2 ) (𝑥22 + 𝑥32 + 𝑥42 ) 𝑞 (𝑡)
𝑎
≤−
1/2
𝑏2 𝛿 2
𝑉
𝑥 + 2𝑏2 (4 + 𝑎2 + 𝑏2 ) ⋅ 𝑞 (𝑡) ⋅
𝑎 2
𝜀
≤ 2𝑏2 (4 + 𝑎2 + 𝑏2 )
1/2
⋅
𝑞 (𝑡)
⋅ 𝑉 ≡ 𝜑 (𝑉, 𝑡) .
𝜀
(14)
Thus, all the solutions of system (5) are bounded.
Theorem 2. Let conditions (i)–(iv) of Theorem 1 be satisfied for
the system (5), and let the following condition hold:
(4 + 𝑎2 + 𝑏2 )
1/2
⋅
𝑞 (𝑡)
𝛿
⋅ 𝑉 − 𝑥22 ≤ 0.
𝜀
𝑎
(15)
Then, all the solutions of system (5) are uniformly bounded.
Proof. It is clear that the function 𝑉(𝑡, 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) defined
in (7) satisfies the conditions (15), therefore, all the solutions
of system (5); are uniformly bounded [7].
Acknowledgments
This work was financially supported by the Chinese Natural
Science Foundation (11061028) and Yunnan Natural Science
Foundation (2010CD086).
References
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equation,” Communications in Nonlinear Science and Numerical
Simulation, vol. 15, no. 9, pp. 2331–2336, 2010.
[2] D.-s. Li and H.-q. Zhang, “New soliton-like solutions to the
potential Kadomstev-Petviashvili (PKP) equation,” Applied
Mathematics and Computation, vol. 146, no. 2-3, pp. 381–384,
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[3] X. Zeng, Z. Dai, D. Li, S. Han, and H. Zhou, “Some exact
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