Solitary Wave Solution of the Two-Dimensional Regularized Long-Wave and Davey-Stewartson Equations in Fluids and Plasmas

Omar H. El-Kalaawy, Rafat S. Ibrahim

Department of Mathematics, Faculty of Science, Beni-Suef University Beni-Suef 62511, Egypt.

DOI: 10.4236/am.2012.38124                

Abstract

This paper investigates the solitary wave solutions of the (2+1)-dimensional regularized long-wave (2DRLG) equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas and (2+1) dimensional Davey-Stewartson (DS) equation which is governing the dynamics of weakly nonlinear modulation of a lattice wave packet in a multidimensional lattice. By using extended mapping method technique, we have shown that the 2DRLG-2DDS equations can be reduced to the elliptic-like equation. Then, the extended mapping method is used to obtain a series of solutions including the single and the combined non degenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear partial differential equations (NLPDEs).

Keywords

Exact Solitary Solutions; Extended Mapping Method; Two Dimension Regularized Long Wave and Da Vey-Stewartson Equations; Jacobi Elliptic Functions

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. Hirota, “Exact Solution of the Korteweg-de VriesEquation for Multiple Collisions of Solitons,” Physical Review Letters, Vol. 27, No. 18, 1971, pp. 1192-1194. Hdoi:10.1103/PhysRevLett.27.1192
[2]	C. Rogers and W. Shadwick, “B?cklund Transformations and Their Applications,” Academic Press, New York, 1982.
[3]	M. J. Ablowitz and P. A. Clarkson, “Solitons, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, Cambridge, 1991. Hdoi:10.1017/CBO9780511623998
[4]	J. Weiss, “The Panlevé Property for Partial Differential Equations. II: B?cklund Transformation, Lax Pairs, and the Schwarzian Derivative,” Journal of Mathematical Physics, Vol. 24, No. 6, 1983, pp. 1405-1413. Hdoi:10.1063/1.525875
[5]	Z. Y. Yan, “An Improved Algebra Method and Its Applications in Nonlinear Wave Equations,” Chaos, Solitons & Fractals, Vol. 21, No. 4, 2004, pp. 1013-1021.
[6]	G. T. Liu and T. Y. Fan, “New Applications of Developed Jacobi Elliptic Function Expansion Methods,” Physics Letters A, Vol. 345, No. 1-3, 2005, pp. 161-166. Hdoi:10.1016/j.physleta.2005.07.034
[7]	O. H. El-Kalaawy, “Exact Solitary Solution of Schamel Equation in Plasmas with Negative Ions,” Physics Plasmas, Vol. 18, No. 11, 2011, pp. 112302-112309. Hdoi:10.1063/1.3657422
[8]	M. L. Wang, “Exact Solutions for a Compound KdV-Burgers Equation,” Physics Letters A, Vol. 213, No. 5-6, 1996, pp. 279-287. Hdoi:10.1016/0375-9601(96)00103-X
[9]	H. J. S. Dorren, “On the Integrability of Nonlinear Partial Differential Equations,” Journal of Mathematical Physics, Vol. 40, No. 4, 1999, pp. 1966-1976. Hdoi:10.1063/1.532843
[10]	O. H. EL-Kalaawy, “Exact Soliton Solutions for Some Nonlinear Partial Differential Equations,” Chaos, Solitons & Fractals, Vol. 14, No. 4, 2002, pp. 547-552. Hdoi:10.1016/S0960-0779(01)00217-X
[11]	O. H. El-Kalaawy and R. S. Ibrahim, “Exact Solutions for Nonlinear Propagation of Slow Ion Acoustic Monotonic Double Layers and a Solitary Hole in a Semirelativistic Plasma,” Physics Plasmas, Vol. 15, No. 7, 2008, Article ID: 072303. Hdoi:10.1063/1.2956336
[12]	M. A. Abdou, “Further Improved F-Expansion and New Exact Solutions for Nonlinear Evolution Equations,” Nonlinear Dynamics, Vol. 52, No. 3, 2008, pp. 277-288. Hdoi:10.1007/s11071-007-9277-3
[13]	Y. J. Ren, S. T. Liu and H. Q. Zhang, “On a Generalized Improved F-Expansion Method,” Communications in Theoretical Physics, Vol. 45, No. 1, 2006, pp. 15-28. Hdoi:10.1088/0253-6102/45/1/003
[14]	E. G. Fan, “Uniformly Constructing a Series of Explicit Exact Solutions to Nonlinear Equations in Mathematical Physics,” Chaos, Solitons & Fractals, Vol. 16, No. 5, 2005, pp. 819-839. Hdoi:10.1016/S0960-0779(02)00472-1
[15]	E. Yomba, “The Extended Fans Sub-Equation Method and Its Application to KdV-MKdV, BKK and Variant Boussinesq Equations,” Physics Letters A, Vol. 336, No. 6, 2005, pp. 463-476.
[16]	D. H. Feng and G. X. Luo, “The Improved Fan Sub Equation Method and Its Application to the SK Equation,” Applied Mathematics and Computation, Vol. 215, No. 5, 2009, pp. 1949-1967. Hdoi:10.1016/j.amc.2009.07.045
[17]	S. Zhang and H. Q. Zhang, “Fan Sub-Equation Method for Wick-Type Stochastic Partial Differential Equations,” Physics Letters A, Vol. 374, No. 41, 2010, pp. 4180-4187. Hdoi:10.1016/j.physleta.2010.08.023
[18]	A. Borhanifar, M. M. Kabir and L. M. Vahdat, “New Periodic and Soliton Wave Solutions for the Generalized Zakharov System and (2+1)Dimensional Nizhink Novikov-Veselov System,” Chaos, Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1646-1654. Hdoi:10.1016/j.chaos.2009.03.064
[19]	W. Malfliet, “Solitary Wave Solutions of Nonlinear Wave Equations,” American Journal of Physics, Vol. 60, No. 7, 1992, pp. 650-654. Hdoi:10.1119/1.17120
[20]	E. Fan, “Extended tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-218. Hdoi:10.1016/S0375-9601(00)00725-8
[21]	E. Fan and Y. C. Hon, “Applications of Extended tanh Method to Special Types of Nonlinear Equations,” Applied Mathematics and Computation, Vol. 141, No. 2-3, 2003, pp. 351-358. Hdoi:10.1016/S0096-3003(02)00260-6
[22]	W. Malfiet and W. Hereman, “The tanh Method I: Exact Solutions of Nonlinear Evolution and Wave Equations,” Physica Scripta, Vol. 54, No. 6, 1996, pp. 563-568. Hdoi:10.1088/0031-8949/54/6/003
[23]	A. M. Wazwaz, “The tanh and the Sinecosine Methods for the Complex Modified KdV and the Generalized KdV Equation,” Computers & Mathematics with Applications, Vol. 49, No. 7-8, 2005, pp. 1101-1112. Hdoi:10.1016/j.camwa.2004.08.013
[24]	Q. M. Al-Mdallal and M. I. Syam, “Sinecosine Method for Finding the Soliton Solutions of the Generalized Fifth-Order Nonlinear Equation,” Chaos, Solitons & Fractals, Vol. 33, No. 5, 2007, pp. 1610-1617. Hdoi:10.1016/j.chaos.2006.03.039
[25]	J. H. He and X. H. Wu, “Variational Iteration Method: New Development and Applications,” Computers & Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 881-894. Hdoi:10.1016/j.camwa.2006.12.083
[26]	J. Biazar and H. Ghazvini, “Homotopy Perturbation Transform Method for Solving Hyperbolic Partial Differential Equations,” Computers & Mathematics with Applications, Vol. 56, No. 2, 2008, pp. 453-458. Hdoi:10.1016/j.camwa.2007.10.032
[27]	M. Wang, X. Li and J. Zhang, “The Expansion Method and Traveling Wave Solutions of nonlinear evolution Equations in Mathematical Physics,” Physics Letters A, Vol. 372, No. 4, 2008, pp. 417-423. Hdoi:10.1016/j.physleta.2007.07.051
[28]	J. Zhang, X. Wei and Y. J. Lu, “A Generalized -Expansion Method and Its Applications,” Physics Letters A, Vol. 372, 2008, pp. 36-53.
[29]	B. Zheng, “Travelling Wave Solutions of Two Nonlinear Evolution Equations by Using the -Expansion Method,” Applied Mathematics and Computation, 2010. Hdoi:10.1016/j.mac.2010.12.052
[30]	A. Borhanifar and M. M. Kabir, “New Periodicand Soliton Solutions by Application of Exp-Function Method for Linear Evolution Equations,” Journal of Computational and Applied Mathematics, Vol. 229, No. 1, 2009, pp. 158-167. Hdoi:10.1016/j.cam.2008.10.052
[31]	S. A. El-Wakil, M. A. Abdou and A. Hendi, “New Periodic Wave Solutions via Exp-Function Method,” Physics Letters A, Vol. 372, No. 6, 2008, pp. 830-840. Hdoi:10.1016/j.physleta.2007.08.033
[32]	H. Zhao and C. Bai, “New Doubly Periodic and Multiple Soliton Solutions of the Generalized (3+1)Dimensional Kadomtsev-Petviashvilli Equation with Variable Coefficients,” Chaos, Solitons & Fractals, Vol. 30, No. 1, 2006, pp. 217-226. Hdoi:10.1016/j.chaos.2005.08.148
[33]	M. A. Abdou and A. Elhanbaly, “Construction of Periodic and Solitary Wave Solutions by the Extended Jacobi Elliptic Function Expansion Method,” Communications in Nonlinear Science and Numerical Simulation, Vol. 12, No. 7, 2007, pp. 1229-1241. Hdoi:10.1016/j.cnsns.2006.01.013
[34]	Z. Yan, “Abudant Families of Jacobi Elliptic Function Solutions of the (2+1)-Dimensional Integrable Davey-Stewartson Equation via a New Method,” Chaos, Solitons & Fractals, Vol. 18, No. 2, 2003, pp. 299-309. Hdoi:10.1016/S0960-0779(02)00653-7
[35]	W. X. Ma and B. Fuchssteiner, “Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov Equation,” International Journal of Non-Linear Mechanics, Vol. 31, No. 3, 1996, pp. 329-338. Hdoi:10.1016/0020-7462(95)00064-X
[36]	W. X. Ma, “Travelling Wave Solutions to a Seventh Order Generalized KdV Equation,” Physics Letters A, Vol. 180, No. 3, 1993, pp. 221-224. Hdoi:10.1016/0375-9601(93)90699-Z
[37]	W. X. Ma, H. Y. Wu and J. S. He, “Partial Differential Equations Possessing Frobenius Integrable Decompositions,” Physics Letters A, Vol. 364, No. 1, 2007, pp. 29-32. Hdoi:10.1016/j.physleta.2006.11.048
[38]	W. X. Ma and J. H. Lee, “A Transformed Rational Function Method and Exact Solutions to the (3+1) Dimensional Jimbo-Miwa Equation,” Chaos, Solitons & Fractals, Vol. 42, No. 3, 2009, pp. 1356-1363. Hdoi:10.1016/j.chaos.2009.03.043
[39]	W. X. Ma, T. W. Huang and Y. Zhang, “A Multiple exp-Function Method for Nonlinear Differential Equations and Its Application,” Physica Scripta, Vol. 82, No. 6, 2010, Article ID: 065003. Hdoi:10.1088/0031-8949/82/06/065003
[40]	W. X. Ma, Y. Zhang, Y. Tang and J. Tu, “Hirota Bilinear Equations with Subspaces of Solutions,” Applied Mathematics and Computation, Vol. 218, No. 13, 2012, pp. 7174-7183. Hdoi:10.1016/j.amc.2011.12.085
[41]	F. C. You, T. C. Xia and J. Zhang, “Frobenius Integrable Decompositions for Two Classes of Nonlinear Evolution Equations with Variable Coefficients,” Modern Physics Letters B, Vol. 23, No. 12, 2009, pp. 1519-1524. Hdoi:10.1142/S0217984909019764
[42]	Z. Huang, “On Cauchy Problems for the RLW Equation in Two Space Dimensions,” Applied Mathematics and Mechanics, Vol. 23, 2002, pp. 159-164.
[43]	Y. Shang and P. Niu, “Explicit Exact Solutions for the RLW Equation and the SRLW Equation in Two Space Dimensions,” Applied Mathematics, Vol. 11, No. 3, 1988, pp. 1-5.
[44]	T. Kawahara, K. Araki and S. Toh, “Interactions of Two-Dimensionally Localized Pulses of the Regularized-Long-Wave Equation,” Physica D: Nonlinear Phenomena, Vol. 59, No. 1-3, 1992, pp. 79-89. Hdoi:10.1016/0167-2789(92)90207-4
[45]	A. Davey and K. Stewartson, “On Three-Dimensional Packets of Surface Waves,” Proceedings of the Royal Society A, Vol. 338, No. 1613, 1974, pp. 101-110. Hdoi:10.1098/rspa.1974.0076