Research ArticleOpen AccessGoogle Scholar indexed
Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation
School of Science, Southwest University of Science and Technology, Mianyang, China
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, China
- 1 School of Science, Southwest University of Science and Technology, Mianyang, China
- 2 School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, China
Advances in Pure Mathematics·Volume 05 (2015)·Pages 121–126·Published 12 March 2015·DOI10.4236/apm.2015.53014
Copy link · social · email
Abstract
In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.
KeywordsKadomtsev-Petviashvili EquationMethod of Variable SeparationGlobal Smooth SolutionLocal Blow up Solution
- Kadomtsev, B.B. and Petviashvili, V.I. (1970) On the Stability of Solitary Waves in Weakly Dispersive Media. Soviet Physics—Doklady, 15, 539-541.
- Wazwaz, A.-M. (2009) Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Higher Education Press, 620-626.
- Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and inverse Scattering. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511623998
- Liu, X.Q., Song, J. and Han, Y.Q. (2001) New Explicit Solutions to the n-Dimensional Landau-Lifshitz Equations. Physics Letter A, 281, 324-326.
- Zhong, P.H., Yang, R.H. and Yang, G.S. (2008) Exact Periodic and Blow up Solutions for 2D Ginzburg-Landau Equation. Physics Letters A, 373, 19-22.
- Evans, C. (2002) Partial Differential Equations. American Mathematical Society Providence, Rhode Island, Graduate Studies in Mathematics, Vol. 19.
- Qu, C.Z., Zhang, S.L. and Liu, R.C. (2000) Separation of Variables and Exact Solutions to the Quasilinear Diffusion Equations with Nonlinear Source. Physica D: Nonlinear Phenomena, 144, 97-123. http://dx.doi.org/10.1016/S0167-2789(00)00069-5
- Estevez, P.G., Qu, C.Z. and Zhang, S.L. (2002) Separation of Variables of a Generalized Porous Medium Equation with Nonlinear Source. Journal of Mathematical Analysis and Applications, 275, 44-59. http://dx.doi.org/10.1016/S0022-247X(02)00214-7
- Shang, Y.D. (2013) Exact Solutions for the Nonlinear Diffusion Equations with Power Law Diffusivity. Journal of Guangzhou University (Nature Science Edition), 12, 1-8.
- Zhang, S.L., Lou, S.Y. and Qu, C.Z. (2002) Variable Separation and Exact Solutions to Generalized Nonlinear Diffusion Equations. Chinese Physics Letters, 19, 1741-1744. http://dx.doi.org/10.1088/0256-307X/19/12/301