Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation
-
Wenguang Cheng
und
Biao Li
Abstract
The truncated Painlevé method is developed to obtain the nonlocal residual symmetry and the Bäcklund transformation for the (2+1)-dimensional KdV–mKdV equation. The residual symmetry is localised after embedding the (2+1)-dimensional KdV–mKdV equation to an enlarged one. The symmetry group transformation of the enlarged system is computed. Furthermore, the (2+1)-dimensional KdV–mKdV equation is proved to be consistent Riccati expansion (CRE) solvable. The soliton–cnoidal wave interaction solution in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral is obtained by using the consistent tanh expansion (CTE) method, which is a special form of CRE.
Acknowledgments:
This work is supported by the National Natural Science Foundation of China under Grant Nos 11271211 and 11435005, and K.C. Wong Magna Fund in Ningbo University.
References
[1] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, Berlin 1986.10.1007/978-1-4684-0274-2Suche in Google Scholar
[2] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer Verlag, New York 1989.10.1007/978-1-4757-4307-4Suche in Google Scholar
[3] G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York 2010.10.1007/978-0-387-68028-6Suche in Google Scholar
[4] J. Weiss, M. Taboe, and G. Carnevale, J. Math. Phys. 24, 522 (1983).10.1063/1.525721Suche in Google Scholar
[5] R. Conte, The Painlevé Property: One Century Later, Springer, New York 1999.10.1007/978-1-4612-1532-5Suche in Google Scholar
[6] G. A. Guthrie, Proc. R. Soc. Lond, Ser. A. 446, 107 (1994).10.1098/rspa.1994.0094Suche in Google Scholar
[7] S. Y. Lou, J. Math. Phys. 35, 2336 (1994).10.1063/1.530556Suche in Google Scholar
[8] S. Y. Lou and X. B. Hu, J. Phys. A Math. Gen. 30, L95 (1997).10.1088/0305-4470/30/5/004Suche in Google Scholar
[9] X. R. Hu, S. Y. Lou, and Y. Chen, Phys. Rev. E 85, 056607 (2012).Suche in Google Scholar
[10] J. C. Chen, X. P. Xin, and Y. Chen, J. Math. Phys. 55, 053508 (2014).10.1063/1.4871554Suche in Google Scholar
[11] J. C. Chen and Y. Chen, J. Nonlinear Math. Phys. 21, 454 (2014).10.1080/14029251.2014.936764Suche in Google Scholar
[12] S. Y. Lou, X. R. Hu, and Y. Chen, J. Phys. A Math. Theor. 45, 155209 (2012).10.1088/1751-8113/45/15/155209Suche in Google Scholar
[13] S. Y. Lou, J. Phys. A Math. Phys. 30, 4803 (1997).10.1088/0305-4470/30/13/028Suche in Google Scholar
[14] G. W. Bluman and Z. Y. Yan, Euro. J. Appl. Math. 16, 239 (2005).10.1017/S0956792505005838Suche in Google Scholar
[15] F. Galas, J. Phys. A Math. Gen. 25, L981 (1992).10.1088/0305-4470/25/15/014Suche in Google Scholar
[16] X. P. Xin and Y. Chen, Chin. Phys. Lett. 30, 100202 (2013).10.1088/0256-307X/30/10/100202Suche in Google Scholar
[17] S. Y. Lou, Residual symmetries and Bäcklund transformations, arXiv:1308.1140v1.Suche in Google Scholar
[18] X. N. Gao, S. Y. Lou, and X. Y. Tang, J. High Energy Phys. 05, 029 (2013).10.1007/JHEP05(2013)029Suche in Google Scholar
[19] S. Y. Lou, Stud. Appl. Math. 134, 372 (2015).10.1111/sapm.12072Suche in Google Scholar
[20] S. Y. Lou, X. P. Cheng, and X. Y. Tang, Chin. Phys. Lett. 31, 070201 (2014).10.1088/0256-307X/31/7/070201Suche in Google Scholar
[21] D. Yang, S. Y. Lou, and W. F. Yu, Commun. Theor. Phys. 60, 387 (2013).10.1088/0253-6102/60/4/01Suche in Google Scholar
[22] Y. H. Wang and H. Wang, Phys. Scr. 89, 125203 (2014).10.1088/0031-8949/89/12/125203Suche in Google Scholar
[23] W. G. Cheng, B. Li, and Y. Chen, Commun. Theor. Phys. 63, 549 (2015).10.1088/0253-6102/63/5/549Suche in Google Scholar
[24] X. R. Hu and Y. Q. Li, Appl. Math. Lett. 51, 20 (2016).Suche in Google Scholar
[25] X. R. Hu and Y. Chen, Z. Naturforsch. A 70, 729 (2015).10.1515/zna-2015-0254Suche in Google Scholar
[26] Y. Q. Liu, F. Duan, and C. Hu, J. Appl. Math. Phys. 3, 697 (2015).Suche in Google Scholar
[27] R. M. Miura, J. Math. Phys. 9, 1202 (1968).10.1063/1.1664700Suche in Google Scholar
[28] K. Konno and Y. H. Ichikawa, J. Phys. Soc. Jpn. 37, 1631 (1974).10.1143/JPSJ.37.1631Suche in Google Scholar
[29] J. F. Zhang, Int. J. Theor. Phys. 37, 1541 (1998).10.1023/A:1026615919186Suche in Google Scholar
[30] A. J. Keane, A. Mushtaq, and M. S. Wheatland, Phys. Rev. E 83, 066407 (2011).10.1103/PhysRevE.83.066407Suche in Google Scholar PubMed
©2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Robust Finite-Time Passivity for Discrete-Time Genetic Regulatory Networks with Markovian Jumping Parameters
- Multi-Soliton Solutions of the Generalized Sawada–Kotera Equation
- Electrical Conduction in Transition-Metal Salts
- Importance of Unit Cells in Accurate Evaluation of the Characteristics of Graphene
- Understanding the Formation Mechanism of Two-Dimensional Atomic Islands on Crystal Surfaces by the Condensing Potential Model
- The Thermodynamic Functions in Curved Space of Neutron Star
- Spanning Trees of the Generalised Union Jack Lattice
- Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids
- Negative Energies in the Dirac Equation
- Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation
- Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy
- Unidirectional Excitation of Graphene Plasmon in Attenuated Total Reflection (ATR) Configuration
- Completed Optimised Structure of Threonine Molecule by Fuzzy Logic Modelling
Artikel in diesem Heft
- Frontmatter
- Robust Finite-Time Passivity for Discrete-Time Genetic Regulatory Networks with Markovian Jumping Parameters
- Multi-Soliton Solutions of the Generalized Sawada–Kotera Equation
- Electrical Conduction in Transition-Metal Salts
- Importance of Unit Cells in Accurate Evaluation of the Characteristics of Graphene
- Understanding the Formation Mechanism of Two-Dimensional Atomic Islands on Crystal Surfaces by the Condensing Potential Model
- The Thermodynamic Functions in Curved Space of Neutron Star
- Spanning Trees of the Generalised Union Jack Lattice
- Prolongation Structure of a Generalised Inhomogeneous Gardner Equation in Plasmas and Fluids
- Negative Energies in the Dirac Equation
- Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation
- Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy
- Unidirectional Excitation of Graphene Plasmon in Attenuated Total Reflection (ATR) Configuration
- Completed Optimised Structure of Threonine Molecule by Fuzzy Logic Modelling
