Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation
Affiliation(s)
1 Department of Mathematics, Zhejiang Lishui University, Lishui, China. 2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China.
1 Department of Mathematics, Zhejiang Lishui University, Lishui, China. 2 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China.
ABSTRACT
Utilizing the Clarkson-Kruskal direct method, the symmetry of the (2 + 1)-dimensional dispersive long wave equation is derived. From which, through solving the characteristic equations, four types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, several soliton excitations are depicted from one of the solutions.
Utilizing the Clarkson-Kruskal direct method, the symmetry of the (2 + 1)-dimensional dispersive long wave equation is derived. From which, through solving the characteristic equations, four types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, several soliton excitations are depicted from one of the solutions.
Cite this paper
Ma, Z. , Fei, J. and Chen, Y. (2014) Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation. Applied Mathematics, 5, 3264-3269. doi: 10.4236/am.2014.520304.
Ma, Z. , Fei, J. and Chen, Y. (2014) Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation. Applied Mathematics, 5, 3264-3269. doi: 10.4236/am.2014.520304.
References
[1] Bluman, G.W. and Cole, J.D. (1974) Similarity Methods for Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-1-4612-6394-4 [2] Bluman, G.W. and Kumei, S. (1989) Symmetries and Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-1-4757-4307-4 [3] Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-4350-2 [4] Clarkson, P.A. and Kruskal, M.D. (1989) New Similarity Reductions of the Boussinesq Equation. Journal of Mathematical Physics, 30, 2201-2213.
http://dx.doi.org/10.1063/1.528613 [5] Clarkson, P.A. and Mansfield, E.L. (1994) Algorithms for the Nonclassical Method of Symmetry Reductions. SIAM Journal on Applied Mathematics, 54, 1693-1719.
http://dx.doi.org/10.1137/S0036139993251846 [6] Boiti, M., Leon, J.J.P. and Pempinelli, F. (1987) Integrable Two-Dimensional Generalisation of the Sine and Sinh- Gordon Equations. Inverse Problems, 3, 37-50.
http://dx.doi.org/10.1088/0266-5611/3/1/009 [7] Paquin, G. and Winternitz, P. (1990) Group Theoretical Analysis of Dispersive Long Wave Equations in Two Space Dimensions. Physica D, 46, 122-138.
http://dx.doi.org/10.1016/0167-2789(90)90115-6 [8] Lou, S.Y. (1994) Symmetries and Algebras of the Integrable Dispersive Long Wave Equations in 2+1-Dimensional Spaces. Journal of Physics A: Mathematical and General, 27, 3235-3243.
http://dx.doi.org/10.1088/0305-4470/27/9/033 [9] Lou, S.Y. (1995) Similarity Solutions of Dispersive Long Wave Equations in Two Space Dimensions. Mathematical Methods in the Applied Sciences, 18, 789-802.
http://dx.doi.org/10.1002/mma.1670181004 [10] Lou, S.Y. (1993) Painlevé Test for the Integrable Dispersive Long Wave Equations in Two Space Dimensions. Physics Letters A, 176, 96-100.
http://dx.doi.org/10.1016/0375-9601(93)90322-Q [11] Tang, X.Y. and Lou, S.Y. (2003) Extended Multilinear Variable Separation Approach and Multivalued Localized Excitations for Some (2 + 1)-Dimensional Integrable Systems. Journal of Mathematical Physics, 44, 4000-4025.
http://dx.doi.org/10.1063/1.1598619 [12] Tang, X.Y., Lou, S.Y. and Zhang, Y. (2002) Localized Exicitations in (2 + 1)-Dimensional Systems. Physical Review E, 66, Article ID: 046601.
http://dx.doi.org/10.1103/PhysRevE.66.046601 [13] Ma, Z.Y., Liu, Y.L., Lu, Z.M. and Zheng, C.L. (2006) Solitons and Waves in (2 + 1)-Dimensional Dispersive Long-Wave Equation. Communications in Theoretical Physics, 46, 799-803.
http://dx.doi.org/10.1088/0253-6102/46/5/006 [14] Ma, Z.Y. and Hu, Y.H. (2007) Solitons, Chaos and Fractals in the (2 + 1)-Dimensional Dispersive Long Wave Equation. Chaos, Solitons & Fractals, 34, 1667-1676.
http://dx.doi.org/10.1016/j.chaos.2006.04.073 [15] Ma, Z.Y. (2007) The Projective Riccati Equation Expansion Method and Variable Separation Solutions for the Nonlinear Physical Differential Equation in Physics. Chinese Physics B, 16, 1848-1854.
http://dx.doi.org/10.1088/1009-1963/16/7/007 [16] Ruan, H.Y. (2001) Study of Solitons Interaction in Integrable Models. Acta Physica Sinica, 50, 369-376.
[1] Bluman, G.W. and Cole, J.D. (1974) Similarity Methods for Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-1-4612-6394-4 [2] Bluman, G.W. and Kumei, S. (1989) Symmetries and Differential Equations. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-1-4757-4307-4 [3] Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York.
http://dx.doi.org/10.1007/978-1-4612-4350-2 [4] Clarkson, P.A. and Kruskal, M.D. (1989) New Similarity Reductions of the Boussinesq Equation. Journal of Mathematical Physics, 30, 2201-2213.
http://dx.doi.org/10.1063/1.528613 [5] Clarkson, P.A. and Mansfield, E.L. (1994) Algorithms for the Nonclassical Method of Symmetry Reductions. SIAM Journal on Applied Mathematics, 54, 1693-1719.
http://dx.doi.org/10.1137/S0036139993251846 [6] Boiti, M., Leon, J.J.P. and Pempinelli, F. (1987) Integrable Two-Dimensional Generalisation of the Sine and Sinh- Gordon Equations. Inverse Problems, 3, 37-50.
http://dx.doi.org/10.1088/0266-5611/3/1/009 [7] Paquin, G. and Winternitz, P. (1990) Group Theoretical Analysis of Dispersive Long Wave Equations in Two Space Dimensions. Physica D, 46, 122-138.
http://dx.doi.org/10.1016/0167-2789(90)90115-6 [8] Lou, S.Y. (1994) Symmetries and Algebras of the Integrable Dispersive Long Wave Equations in 2+1-Dimensional Spaces. Journal of Physics A: Mathematical and General, 27, 3235-3243.
http://dx.doi.org/10.1088/0305-4470/27/9/033 [9] Lou, S.Y. (1995) Similarity Solutions of Dispersive Long Wave Equations in Two Space Dimensions. Mathematical Methods in the Applied Sciences, 18, 789-802.
http://dx.doi.org/10.1002/mma.1670181004 [10] Lou, S.Y. (1993) Painlevé Test for the Integrable Dispersive Long Wave Equations in Two Space Dimensions. Physics Letters A, 176, 96-100.
http://dx.doi.org/10.1016/0375-9601(93)90322-Q [11] Tang, X.Y. and Lou, S.Y. (2003) Extended Multilinear Variable Separation Approach and Multivalued Localized Excitations for Some (2 + 1)-Dimensional Integrable Systems. Journal of Mathematical Physics, 44, 4000-4025.
http://dx.doi.org/10.1063/1.1598619 [12] Tang, X.Y., Lou, S.Y. and Zhang, Y. (2002) Localized Exicitations in (2 + 1)-Dimensional Systems. Physical Review E, 66, Article ID: 046601.
http://dx.doi.org/10.1103/PhysRevE.66.046601 [13] Ma, Z.Y., Liu, Y.L., Lu, Z.M. and Zheng, C.L. (2006) Solitons and Waves in (2 + 1)-Dimensional Dispersive Long-Wave Equation. Communications in Theoretical Physics, 46, 799-803.
http://dx.doi.org/10.1088/0253-6102/46/5/006 [14] Ma, Z.Y. and Hu, Y.H. (2007) Solitons, Chaos and Fractals in the (2 + 1)-Dimensional Dispersive Long Wave Equation. Chaos, Solitons & Fractals, 34, 1667-1676.
http://dx.doi.org/10.1016/j.chaos.2006.04.073 [15] Ma, Z.Y. (2007) The Projective Riccati Equation Expansion Method and Variable Separation Solutions for the Nonlinear Physical Differential Equation in Physics. Chinese Physics B, 16, 1848-1854.
http://dx.doi.org/10.1088/1009-1963/16/7/007 [16] Ruan, H.Y. (2001) Study of Solitons Interaction in Integrable Models. Acta Physica Sinica, 50, 369-376.