Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation

Affiliation(s)

^{1}
School of Science, Southwest University of Science and Technology, Mianyang, China.

^{2}
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, China.

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.

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.

KEYWORDS

Kadomtsev-Petviashvili Equation, Method of Variable Separation, Global Smooth Solution, Local Blow up Solution

Kadomtsev-Petviashvili Equation, Method of Variable Separation, Global Smooth Solution, Local Blow up Solution

Cite this paper

Song, L. and Shang, Y. (2015) Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation.*Advances in Pure Mathematics*, **5**, 121-126. doi: 10.4236/apm.2015.53014.

Song, L. and Shang, Y. (2015) Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation.

References

[1] Kadomtsev, B.B. and Petviashvili, V.I. (1970) On the Stability of Solitary Waves in Weakly Dispersive Media. Soviet Physics—Doklady, 15, 539-541.

[2] Wazwaz, A.-M. (2009) Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Higher Education Press, 620-626.

[3] 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

[4] 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.

[5] 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.

[6] Evans, C. (2002) Partial Differential Equations. American Mathematical Society Providence, Rhode Island, Graduate Studies in Mathematics, Vol. 19.

[7] 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

[8] 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

[9] 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.

[10] 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

[1] Kadomtsev, B.B. and Petviashvili, V.I. (1970) On the Stability of Solitary Waves in Weakly Dispersive Media. Soviet Physics—Doklady, 15, 539-541.

[2] Wazwaz, A.-M. (2009) Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Higher Education Press, 620-626.

[3] 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

[4] 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.

[5] 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.

[6] Evans, C. (2002) Partial Differential Equations. American Mathematical Society Providence, Rhode Island, Graduate Studies in Mathematics, Vol. 19.

[7] 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

[8] 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

[9] 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.

[10] 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