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Analysis of Stability of Traveling Wave for Kadomtsev-Petviashvili Equation
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.
We consider the Kadomtsev-Petviashvili (KP) equation: 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.
2. The Boundedness
In general, we use the following system, which is equivalent to (2): where
We consider the following system, which is equivalent to (3):
Theorem 1. If the following conditions hold for the system (5):(i) there are positive constants , , , , , and such that
(ii) , .(iii) .(iv) , where is a nonnegative continuous function and .
Then, all the solutions of system (5) are bounded.
Proof. We first construct the Lyapunov function defined by
It follows from conditions (i) and (ii) that Summing up the above discussions, we get
Thus, we deduce that the function defined in (7) is a positive definite function which has infinite inferior limit and infinitesimal upper limit. Hence, there exsits a positive constant such that Taking the total derivative of (7) with respect to along the trajectory of (5), we obtain By using conditions (i) and (iii), it follows that According to (ii), we have Hence,
Thus, all the solutions of system (5) are bounded.
This work was financially supported by the Chinese Natural Science Foundation (11061028) and Yunnan Natural Science Foundation (2010CD086).
- Z. Dai, J. Liu, and Z. Liu, “Exact periodic kink-wave and degenerative soliton solutions for potential Kadomtsev-Petviashvili equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2331–2336, 2010.
- 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, 2003.
- X. Zeng, Z. Dai, D. Li, S. Han, and H. Zhou, “Some exact periodic soliton solution and resonance for the potential Kadomtsrv-Petviashvili equation,” Chaos, Solitons & Fractals, vol. 42, pp. 657–661, 2009.
- I. E. Inan and D. Kaya, “Some exact solutions to the potential Kadomtsev-Petviashvili equation and to a system of shallow water wave equations,” Physics Letters A, vol. 355, no. 4-5, pp. 314–318, 2006.
- 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.
- Z. Li, Z. Dai, and J. Liu, “New periodic solitary-wave solutions to the -dimensional Kadomtsev-Petviashvili equation,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 877–882, 2010.
- 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.
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