Back
 JAMP  Vol.8 No.2 , February 2020
Exact Solutions for (2 + 1)-Dimensional KdV-Calogero-Bogoyavlenkskii-Schiff Equation via Symbolic Computation
Abstract: This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding density plots.
Cite this paper: Li, Y. and Chaolu, T. (2020) Exact Solutions for (2 + 1)-Dimensional KdV-Calogero-Bogoyavlenkskii-Schiff Equation via Symbolic Computation. Journal of Applied Mathematics and Physics, 8, 197-209. doi: 10.4236/jamp.2020.82015.
References

[1]   Gear, J.A. and Grimshaw, R. (1984) Weak and Strong Interactions between Internal Solitary Waves. Studies in Applied Mathematics, 70, 235-258.
https://doi.org/10.1002/sapm1984703235

[2]   Ruggieri, M. and Speciale, M.P. (2013) Similarity Reduction and Closed Form Solutions for a Model Derived from Two-Layer Fluids. Advances in Difference Equations, 1, 355-363.
https://doi.org/10.1186/1687-1847-2013-355

[3]   Christodoulides, R.G.P. (2010) Steady Gap Solitons in a Coupled Korteweg-de Vries System: A Dynamical Systems Approach. Physica D: Nonlinear Phenomena, 239, 635-639.
https://doi.org/10.1016/j.physd.2010.01.016

[4]   Weoss, J., Tabor, M. and Carnevale, G. (1983) Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526.
https://doi.org/10.1063/1.525721

[5]   Matveev, V.B. and Salle, M.A. (1991) Darboux Transformmations and Solitons. Springer, Berlin.
https://doi.org/10.1007/978-3-662-00922-2

[6]   Bluman, G.W., Checiakov, A.F. and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential Equations. Springer, New York.
https://doi.org/10.1007/978-0-387-68028-6

[7]   Hirota, R. (2004) The Direct Method in Soliton Theory. Spring, Berlin.
https://doi.org/10.1017/CBO9780511543043

[8]   Ablowitz, M.J. and Clarkkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambrige.
https://doi.org/10.1017/CBO9780511623998

[9]   Ma, W.X. (2015) Lump Solutions to the Kadomtsev-Petviashvili Equation. Physics Letters A, 379, 1975-1978.
https://doi.org/10.1016/j.physleta.2015.06.061

[10]   Ma, W.X., Zhou, Y. and Dougherty, R. (2016) Lump-Type Solutions to Nonlinear Differential Equantions Derived from Generalized Bilinear Equations. International Journal of Modern Physics B, 30, Artical ID: 1640018.
https://doi.org/10.1142/S021797921640018X

[11]   Zhang, H.Q. and Ma, W.X. (2016) Lump Solution to the (2 + 1)-Dimensional Sawada-Kotera Equation. Nonlinear Dynamics, 87, 2305-2310.
https://doi.org/10.1007/s11071-016-3190-6

[12]   Hua, Y.F., Guo, B.L. and Ma, W.X. (2016) Interaction Behavior Associated with a Generalized (2 + 1)-Dimensional Hirota Bilinear Equation for Nonlinear Waves. Applied Mathematical Modelling, 74, 184-198.
https://doi.org/10.1016/j.apm.2019.04.044

[13]   Zhang, H.Q. and Ma, W.X. (2017) Mixed Lump-Kink Solutions to the KP Equation. Computers and Mathematics with Applications, 74, 1399-1405.
https://doi.org/10.1016/j.camwa.2017.06.034

[14]   Yang, J.Y. and Ma, W.X. (2017) Abundant Lump-Type Solutions of the Jimbo Miwa Equation in (2 + 1)-Dimensions. Computers and Mathematics with Applications, 73, 220-225.
https://doi.org/10.1016/j.camwa.2016.11.007

[15]   Wang, Y.H., Wang, H., Zhang, H.S. and ChaoLu, T.M. (2017) Exact Interaction Solutions of an Extended (2 + 1)-Dimensinal Shallow Water Wave Equation. Commnications in Theoretical Physics, 68, 165.
https://doi.org/10.1088/0253-6102/68/2/165

[16]   Yang, J.Y., Ma, W.X. and Qin, Z.Y. (2017) Lump and Lump-Soliton Solutions to the (2 + 1) Dimensional Ito Equation. Analysis and Mathematical Physics, 8, 427-436.

[17]   Wang, H., Wang, Y.H. and Ma, W.X. (2018) Lump Solutions of a New Extended (2 + 1)-Dimensional Boussinesq Equation. Modern Physics Letters B, 32, Article ID: 1850376.
https://doi.org/10.1142/S0217984918503761

[18]   Ma, W.X. (2018) Abundant Lumps and Their Interaction Solutions of (2 + 1)-Dimensional Linear PDEs. Journal of Geometry and Physics, 133, 10-16.
https://doi.org/10.1016/j.geomphys.2018.07.003

[19]   Sumayah, B. and Ma, W.X. (2018) A Study of Lump-Type and Interaction Solutions to a (2 + 1)-Dimensional Jimbo-Miwa-Like Equation. Computers and Mathematics with Applications, 76, 1576-1582.
https://doi.org/10.1016/j.camwa.2018.07.008

[20]   Ma, W.X. and Zhou, Y. (2018) Lump Solutions to Nonlinear Partial Differential Equations via Hirota Bilinear Forms. Journal of Differential Equations, 264, 2633-2659.
https://doi.org/10.1016/j.jde.2017.10.033

[21]   Ma, W.X., Yong, X. and Zhang, H.Q. (2018) Diversity of Interaction Solutions to the (2 + 1)-Dimensinal Ito Equation. Computers and Mathematics with Applications, 75, 289-295.
https://doi.org/10.1016/j.camwa.2017.09.013

[22]   Huang, L.L., Yue, Y.F. and Chen, Y. (2018) Localized Waves and Interaction Solutions to a (2 + 1)-Dimensional Generalized KP Equation. Computers and Mathematics with Applications, 76, 831-844.
https://doi.org/10.1016/j.camwa.2018.05.023

[23]   Ma, W.X. (2019) Lump and Interaction Solutions to Linear (2+1)-Dimensional PDEs. Acta Mathematica Scientia, 39, 498-508.

[24]   Ma, W.X. (2019) Lump and Interaction Solutions of Linear PDEs in (2 + 1)-Dimensions. Aast Asian Journal on Applied Mathematics, 9, 185-194.
https://doi.org/10.4208/eajam.100218.300318

[25]   Ren, B., Ma, W.X. and Yu, J. (2019) Characteristics and Interactions of Solitary and Lump Waves of a (2 + 1)-Dimensional Coupled Nonlinear Partial Differential Equation. Nonlinear Dynamics, 96, 717-727.
https://doi.org/10.1007/s11071-019-04816-x

[26]   Zhou, Y., Manukure, S. and Ma,W.X. (2019) Lump and Lump-Soliton Solutions to the Hirota-Satsuma-Ito Equation. Communications in Nonlinear Science and Numerical Simulation, 68, 56-62.
https://doi.org/10.1016/j.cnsns.2018.07.038

[27]   Wang, Y.H. and Chen, Y. (2013) Binary Bell Polynomial Manipulations on the Integrability of a Generalized (2 + 1)-Dimensional Korteweg-de Vries Equation. Journal of Mathematical Analysis and Applications, 400, 624-634.
https://doi.org/10.1016/j.jmaa.2012.11.028

[28]   Peng, Y.Z. (2010) A New (2 + 1)-Dimensional KdV Equation and Its Localized Structures. Communications in Theoretical Physics, 54, 863-865.
https://doi.org/10.1088/0253-6102/54/5/17

[29]   Chen, J.C. and Ma, Z.Y. (2017) Consistent Riccati Expansion Solvability and Soliton-Cnoidal Wave Interaction Solution of a (2 + 1)-Dimensional Korteweg-de Vries Equation. Applied Mathematics Letters, 64, 87-93.
https://doi.org/10.1016/j.aml.2016.08.016

[30]   Toda, K. and Yu, S.J. (2000) The Investigation into the Schwarz-Korteweg-de Vries Equation and the Schwarz Derivative in (2 + 1)-Dimensional. Journal of Mathematical Physics, 41, 4747-4751.
https://doi.org/10.1063/1.533374

[31]   Zhang, Y. and Xu, G.Q. (2014) Intergrability and Exact Solutions for a (2 + 1)-Dimensional Variable-Coefficient KdV Equation. Applications of Mathematics, 9, 646-658.

[32]   Lü, X., Lin, F.H. and Qi, F.H. (2015) Analytical Study on a Two Dimensional Korteweg-de Vries Model with Bilinear Representation, Bäcklund Transformaation and Soliton Solutions. Applications of Mathematical Models, 39, 3221-3226.
https://doi.org/10.1016/j.apm.2014.10.046

[33]   Lü, X., Ma, W.X. and Khalique, C.M. (2015) A Direct Bilinear Bäcklund Transformaation of a (2 + 1)-Dimensional Korteweg-de Vries Equation. Applied Mathematics Letters, 50, 37-42.
https://doi.org/10.1016/j.aml.2015.06.003

[34]   Zhao, Z.L. and Han, B. (2017) The Riemann-Bäcklund Method to a Quasiperiodic Wave Solvable Generalized Variable-Coefficient (2 + 1)-Dimensional KdV Equation. Nonlinear Dynamics, 87, 2661-2676.
https://doi.org/10.1007/s11071-016-3219-x

[35]   Huang, L.L. and Chen, Y. (2018) Nonlocal Symmetry and Similarity Reductions for a (2 + 1)-Dimensional Korteweg-de Vries Equation. Nonlinear Dynamics, 92, 221-234.
https://doi.org/10.1007/s11071-018-4051-2

 
 
Top