Figure 1.
$ \alpha = 6,\beta = 1,\gamma = \delta = 3 $
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May
2021, 4(2): 105-115.
doi: 10.3934/mfc.2021006
New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation
| 1. | School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China |
| 2. | College of Information Science and Engineering, Shandong Agricultural University, Taian, Shandong, 271018, China |
In this paper, we investigate explicit exact traveling wave solutions of the generalized (3+1)-dimensional KP equation
| $ \begin{equation} \ (u_{t}+\alpha uu_{x}+\beta u_{xxx})_{x}+\gamma u_{yy}+\delta u_{zz} = 0, \ \ \ \ \beta>0 \;\;\;\;\;\;(1) \ \end{equation}$ |
describing the dynamics of solitons and nonlinear waves in the field of plasma physics and fluid dynamics, where
| $ \alpha, \beta, \gamma, \delta $ |
| $ \delta = 0 $ |
| $ \gamma = 0 $ |
Keywords:
(3+1)-dimensional KP equation,
exact traveling wave solution,
simplified homogeneous balance method,
extended tanh method,
simplest equation method.
Mathematics Subject Classification:
Primary: 35C07, 35C08; Secondary: 35A25.
Citation:
Yuanqing Xu, Xiaoxiao Zheng, Jie Xin. New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation. Mathematical Foundations of Computing,
2021, 4
(2)
: 105-115.
doi: 10.3934/mfc.2021006
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References:
| [1] |
T. Alagesan, A. Uthayakumar and K. Porsezian,
Painlevé analysis and Bäcklund transformation for a three-dimensional Kadomtsev-Petviashvili equation, Chaos Solitons Fractals, 8 (1997), 893-895.
doi: 10.1016/S0960-0779(96)00166-X. |
| [2] |
H. Chen and H. Zhang,
New multiple soliton-like solutions to the generalized $(2+1)$-dimensional KP equation, Appl. Math. Comput., 157 (2004), 765-773.
doi: 10.1016/j.amc.2003.08.072. |
| [3] |
L. Cheng, Y. Zhang, Z.-S. Tong and J.-Y. Ge,
Rational and complexion solutions of the $(3+1)$-dimensional KP equation, Nonlinear Dynam., 72 (2013), 605-613.
doi: 10.1007/s11071-012-0738-y. |
| [4] |
S. M. El-Sayed and D. Kaya, The decomposition method for solving (2 + 1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Appl. Math. Comput., 157 (2004), 523-534. |
| [5] |
X. Hao, Y. Liu, Z. Li and W.-X. Ma,
Painlevé analysis, soliton solutions and lump-type solutions of the $(3+1)$-dimensional generalized KP equation, Comput. Math. Appl., 77 (2019), 724-730.
doi: 10.1016/j.camwa.2018.10.007. |
| [6] |
L. He and Z. Zhao, Multiple lump solutions and dynamics of the generalized $(3+1)$-dimensional KP equation, Modern Phys. Lett. B, 34 (2020), 20pp.
doi: 10.1142/S0217984920501675. |
| [7] |
A. H. Khater, O. H. El-Kalaawy and M. A. Helal,
Two new classes of exact solutions for the KdV equation via Bäcklund transformations, Chaos Solitons Fractals, 8 (1997), 1901-1909.
doi: 10.1016/S0960-0779(97)00090-8. |
| [8] |
W.-X. Ma, A. Abdeljabbar and M. G. Asaad,
Wronskian and Grammian solutions to a $(3+1)$-dimensional generalized KP equation, Appl. Math. Comput., 217 (2011), 10016-10023.
doi: 10.1016/j.amc.2011.04.077. |
| [9] |
W.-X. Ma and Z. Zhu,
Solving the $(3+1)$-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218 (2012), 11871-11879.
doi: 10.1016/j.amc.2012.05.049. |
| [10] |
M. Wang, X. Li and J. Zhang,
Two-soliton solution to a generalized KP equation with general variable coefficients, Appl. Math. Lett., 76 (2018), 21-27.
doi: 10.1016/j.aml.2017.07.011. |
| [11] |
M. Wang, J. Zhang and X. Li,
Decay mode solutions to cylindrical KP equation, Appl. Math. Lett., 62 (2016), 29-34.
doi: 10.1016/j.aml.2016.06.012. |
| [12] |
A.-M. Wazwaz,
Multiple-soliton solutions for a $(3+1)$-dimensional generalized KP equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 491-495.
doi: 10.1016/j.cnsns.2011.05.025. |
| [13] |
A.-M. Wazwaz,
Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh-coth method, Appl. Math. Comput., 190 (2007), 633-640.
doi: 10.1016/j.amc.2007.01.056. |
| [14] |
Z. Yan,
Multiple solution profiles to the higher-dimensional Kadomtsev-Petviashvilli equations via Wronskian determinant, Chaos Solitons Fractals, 33 (2007), 951-957.
doi: 10.1016/j.chaos.2006.01.122. |
| [15] |
L. Yang, K. Yang and H. Luo,
Complex version KdV equation and the periods solution, Phys. Lett. A, 267 (2000), 331-334.
doi: 10.1016/S0375-9601(00)00128-6. |
| [16] |
H.-Y. Zhang and Y.-F. Zhang, Analysis on the $M$-rogue wave solutions of a generalized $(3+1)$-dimensional KP equation, Appl. Math. Lett., 102 (2020), 9pp.
doi: 10.1016/j.aml.2019.106145. |
show all references
References:
| [1] |
T. Alagesan, A. Uthayakumar and K. Porsezian,
Painlevé analysis and Bäcklund transformation for a three-dimensional Kadomtsev-Petviashvili equation, Chaos Solitons Fractals, 8 (1997), 893-895.
doi: 10.1016/S0960-0779(96)00166-X. |
| [2] |
H. Chen and H. Zhang,
New multiple soliton-like solutions to the generalized $(2+1)$-dimensional KP equation, Appl. Math. Comput., 157 (2004), 765-773.
doi: 10.1016/j.amc.2003.08.072. |
| [3] |
L. Cheng, Y. Zhang, Z.-S. Tong and J.-Y. Ge,
Rational and complexion solutions of the $(3+1)$-dimensional KP equation, Nonlinear Dynam., 72 (2013), 605-613.
doi: 10.1007/s11071-012-0738-y. |
| [4] |
S. M. El-Sayed and D. Kaya, The decomposition method for solving (2 + 1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Appl. Math. Comput., 157 (2004), 523-534. |
| [5] |
X. Hao, Y. Liu, Z. Li and W.-X. Ma,
Painlevé analysis, soliton solutions and lump-type solutions of the $(3+1)$-dimensional generalized KP equation, Comput. Math. Appl., 77 (2019), 724-730.
doi: 10.1016/j.camwa.2018.10.007. |
| [6] |
L. He and Z. Zhao, Multiple lump solutions and dynamics of the generalized $(3+1)$-dimensional KP equation, Modern Phys. Lett. B, 34 (2020), 20pp.
doi: 10.1142/S0217984920501675. |
| [7] |
A. H. Khater, O. H. El-Kalaawy and M. A. Helal,
Two new classes of exact solutions for the KdV equation via Bäcklund transformations, Chaos Solitons Fractals, 8 (1997), 1901-1909.
doi: 10.1016/S0960-0779(97)00090-8. |
| [8] |
W.-X. Ma, A. Abdeljabbar and M. G. Asaad,
Wronskian and Grammian solutions to a $(3+1)$-dimensional generalized KP equation, Appl. Math. Comput., 217 (2011), 10016-10023.
doi: 10.1016/j.amc.2011.04.077. |
| [9] |
W.-X. Ma and Z. Zhu,
Solving the $(3+1)$-dimensional generalized KP and BKP equations by the multiple exp-function algorithm, Appl. Math. Comput., 218 (2012), 11871-11879.
doi: 10.1016/j.amc.2012.05.049. |
| [10] |
M. Wang, X. Li and J. Zhang,
Two-soliton solution to a generalized KP equation with general variable coefficients, Appl. Math. Lett., 76 (2018), 21-27.
doi: 10.1016/j.aml.2017.07.011. |
| [11] |
M. Wang, J. Zhang and X. Li,
Decay mode solutions to cylindrical KP equation, Appl. Math. Lett., 62 (2016), 29-34.
doi: 10.1016/j.aml.2016.06.012. |
| [12] |
A.-M. Wazwaz,
Multiple-soliton solutions for a $(3+1)$-dimensional generalized KP equation, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 491-495.
doi: 10.1016/j.cnsns.2011.05.025. |
| [13] |
A.-M. Wazwaz,
Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh-coth method, Appl. Math. Comput., 190 (2007), 633-640.
doi: 10.1016/j.amc.2007.01.056. |
| [14] |
Z. Yan,
Multiple solution profiles to the higher-dimensional Kadomtsev-Petviashvilli equations via Wronskian determinant, Chaos Solitons Fractals, 33 (2007), 951-957.
doi: 10.1016/j.chaos.2006.01.122. |
| [15] |
L. Yang, K. Yang and H. Luo,
Complex version KdV equation and the periods solution, Phys. Lett. A, 267 (2000), 331-334.
doi: 10.1016/S0375-9601(00)00128-6. |
| [16] |
H.-Y. Zhang and Y.-F. Zhang, Analysis on the $M$-rogue wave solutions of a generalized $(3+1)$-dimensional KP equation, Appl. Math. Lett., 102 (2020), 9pp.
doi: 10.1016/j.aml.2019.106145. |
Figure 2.
$ \alpha = -6,\beta = 1,\gamma = \delta = 3 $
Figure 3.
$ \alpha = 6,\beta = \lambda = \mu = B = 1,\gamma = \delta = 3,c = y = z = 0 $
Figure 4.
$ \alpha = -6,\beta = \lambda = \mu = B = 1,\gamma = \delta = 3,c = y = z = 0 $
Figure 5.
$ u_{11} $ as $ \alpha = -6,\beta = 1,\gamma = 3 $ , $ \delta = 3 $ , $ \lambda = \mu = 1,k = 2 $ , $ y = z = 0 $
Figure 6.
$ u_{12} $ as $ \alpha = -6,\beta = 1,\gamma = 3,\delta = 3 $ , $ \lambda = \mu = 1,k = 2 $ , $ y = z = 0 $
Figure 7.
$ u_{2} $ as $ \alpha = 6,\beta = 1,\gamma = 3,\delta = 3 $ , $ k = 6,\lambda = \mu = 1, y = z = 0 $
Figure 8.
$ u_{2} $ as $ \alpha = -6,\beta = 1,\gamma = 3,\delta = 3 $ , $ k = 6,\lambda = \mu = 1, y = z = 0 $
Figure 9.
$ u_{31} $ as $ \alpha = -6,\beta = 1,\gamma = 3,\delta = 3 $ , $ \lambda = \mu = 1,k = 2, y = z = 0 $
Figure 10.
$ u_{32} $ as $ \alpha = -6,\beta = 1,\gamma = 3,\delta = 3 $ , $ \lambda = \mu = 1,k = 2, y = z = 0 $
Figure 11.
$ \alpha = 6,\beta = d = 1,\gamma = 3,\delta = 3 $ , $ \lambda = \mu = 1, k = 3, y = z = 0 $
Figure 12.
$ \alpha = -6 $ , $ \beta = d = 1,\gamma = 3,\delta = 3 $ , $ \lambda = \mu = 1, k = 3, y = z = 0 $
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