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# New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation

• * Corresponding author: Jie Xin

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

$\begin{equation} \ (u_{t}+\alpha uu_{x}+\beta u_{xxx})_{x}+\gamma u_{yy}+\delta u_{zz} = 0, \ \ \ \ \beta>0 \;\;\;\;\;\;(1) \ \end{equation}$

describing the dynamics of solitons and nonlinear waves in the field of plasma physics and fluid dynamics, where $\alpha, \beta, \gamma, \delta$ 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.

Mathematics Subject Classification: Primary: 35C07, 35C08; Secondary: 35A25.

 Citation: • • Figure 1.  $\alpha = 6,\beta = 1,\gamma = \delta = 3$

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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