# American Institute of Mathematical Sciences

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

* Corresponding author: Jie Xin

Received  August 2020 Revised  March 2021 Published  May 2021 Early access  April 2021

Fund Project: The second author is supported by the Natural Science Foundation of Shandong Province (No. ZR2018BA016) and the third author is supported by the National Natural Science Foundation of China (No. 11371183)

In this paper, we investigate explicit exact traveling wave solutions of the generalized (3+1)-dimensional KP equation
 $$$\ (u_{t}+\alpha uu_{x}+\beta u_{xxx})_{x}+\gamma u_{yy}+\delta u_{zz} = 0, \ \ \ \ \beta>0 \;\;\;\;\;\;(1) \$$$
describing the dynamics of solitons and nonlinear waves in the field of plasma physics and fluid dynamics, where
 $\alpha, \beta, \gamma, \delta$
are nonzero constants. By using the simplified homogeneous balance method, we get one single soliton solution and one double soliton solution of (1). Moreover, we use the extended tanh method with a Riccati equation and the simplest equation method with Bernoulli equation to obtain seven sets of explicit exact traveling wave solutions. When
 $\delta = 0$
or
 $\gamma = 0$
, (1) reduces to (2+1)-dimensional KP equation. Therefore, we can get some exact traveling wave solutions of (2+1)-dimensional KP equation.
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
##### References:
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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.
$\alpha = 6,\beta = 1,\gamma = \delta = 3$
$\alpha = -6,\beta = 1,\gamma = \delta = 3$
$\alpha = 6,\beta = \lambda = \mu = B = 1,\gamma = \delta = 3,c = y = z = 0$
$\alpha = -6,\beta = \lambda = \mu = B = 1,\gamma = \delta = 3,c = y = z = 0$
$u_{11}$ as $\alpha = -6,\beta = 1,\gamma = 3$, $\delta = 3$, $\lambda = \mu = 1,k = 2$, $y = z = 0$
$u_{12}$ as $\alpha = -6,\beta = 1,\gamma = 3,\delta = 3$, $\lambda = \mu = 1,k = 2$, $y = z = 0$
$u_{2}$ as $\alpha = 6,\beta = 1,\gamma = 3,\delta = 3$, $k = 6,\lambda = \mu = 1, y = z = 0$
$u_{2}$ as $\alpha = -6,\beta = 1,\gamma = 3,\delta = 3$, $k = 6,\lambda = \mu = 1, y = z = 0$
$u_{31}$ as $\alpha = -6,\beta = 1,\gamma = 3,\delta = 3$, $\lambda = \mu = 1,k = 2, y = z = 0$
$u_{32}$ as $\alpha = -6,\beta = 1,\gamma = 3,\delta = 3$, $\lambda = \mu = 1,k = 2, y = z = 0$
$\alpha = 6,\beta = d = 1,\gamma = 3,\delta = 3$, $\lambda = \mu = 1, k = 3, y = z = 0$
$\alpha = -6$, $\beta = d = 1,\gamma = 3,\delta = 3$, $\lambda = \mu = 1, k = 3, y = z = 0$
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