| Citation: |
Jianguo Liu, Wenhui Zhu, Li Zhou, Yan He. EXPLICIT AND EXACT NON-TRAVELING WAVE SOLUTIONS OF (3+1)-DIMENSIONAL GENERALIZED SHALLOW WATER EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2381-2388. doi: 10.11948/20190112
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EXPLICIT AND EXACT NON-TRAVELING WAVE SOLUTIONS OF (3+1)-DIMENSIONAL GENERALIZED SHALLOW WATER EQUATION
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Abstract
In this paper, a (3+1)-dimensional generalized shallow water equation is considered. New exact solutions in forms of the hyperbolic functions and the trigonometric functions are obtained based on an extended $(G'/G)$-expansion method and the variable separation method, which contain traveling wave solutions and non-traveling wave solutions. The particular localized excitations and the interactions between two solitary waves for these obtained exact solutions are shown in some three-dimensional graphics. -
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References
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Jianguo Liu, Wenhui Zhu, Li Zhou, Yan He. EXPLICIT AND EXACT NON-TRAVELING WAVE SOLUTIONS OF (3+1)-DIMENSIONAL GENERALIZED SHALLOW WATER EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(6): 2381-2388. doi: 10.11948/20190112
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Figure 1. Solution (2.7) at
$ f = 1+sech^2(y+c z) $ ,$ W(z, t) = sech(z^2+t^2) $ ,$ C_1 = 1 $ ,$ C_2 = 2 $ ,$ \varrho = 3 $ ,$ \rho = 0.1 $ ,$ c = 2 $ ,$ x = 1 $ ,$ a = A = 1 $ ,$ B = 0 $ . -
Figure 2. Solution (2.7) at
$ W(z, t)=\cos \sqrt{z^2-t^2} $ ,$ B=0 $ ,$ C_1=1 $ ,$ C_2=2 $ ,$ f=\coth^2(y+c z)+csch^2(y+c z) $ ,$ \varrho=3 $ ,$ \rho=0.1 $ ,$ c=2 $ ,$ t=1 $ ,$ a=A=1 $ . -
Figure 3. Solution (2.7) at
$ c=2 $ ,$ f= \frac{1}{1+0.2 sech^2(y+c z)+0.2 \tanh^2(y+c z)}-sech^2(y+c z) $ ,$ \varrho=3 $ ,$ \rho=0.1 $ ,$ C_2=2 $ ,$ a=A=1 $ ,$ B=0 $ ,$ W(z, t) = \tanh(z^2+t^2)+sech(t^2-z^2) $ ,$ x=1 $ . -
Figure 4. Solution (2.7) at
$ c = 2 $ ,$ f = \frac{1}{1+0.2 sech^2(y+c z)+0.2 tanh^2(y+c z)}-sech^2(y+c z) $ ,$ \varrho = 3 $ ,$ \rho = 0.1 $ ,$ C_2 = 2 $ ,$ a = A = 1 $ ,$ B = 0 $ ,$ W(z, t) = \tanh(z^2+t^2)+sech(t^2-z^2) $ ,$ y = 1 $ . -
Figure 5. Ssolution (2.10) at
$ W(z, t) = \cos \sqrt{z^2-t^2} $ ,$ C_1 = 1 $ ,$ \varrho = 3 $ ,$ \rho = -1 $ ,$ c = 2 $ ,$ f = \frac{1}{\coth^2(y+c z)+csch^2(y+c z)} $ ,$ C_2 = 2 $ ,$ x = 1 $ ,$ B = 0 $ ,$ a = A = 1 $ . -
Figure 6. Solution (2.10) at
$ \varrho=3 $ ,$ f= \frac{1}{1+0.2 sech^2(y+c z)+0.2 \tanh^2(y+c z)}-sech^2(y+c z) $ ,$ \rho=-1 $ ,$ c=2 $ ,$ C_2=2 $ ,$ a=A=1 $ ,$ B=0 $ ,$ W(z, t) = \tanh(z^2+t^2)+sech(t^2-z^2) $ ,$ x=1 $ . -
Figure 7. Solution (2.10) at
$ \varrho = 3 $ ,$ f = \frac{1}{1+0.2 sech^2(y+c z)+0.2 \tanh^2(y+c z)}-sech^2(y+c z) $ ,$ \rho = 0.1 $ ,$ c = 2 $ ,$ C_2 = 2 $ ,$ a = A = 1 $ ,$ B = 0 $ ,$ W(z, t) = \tanh(z^2+t^2)+sech(t^2-z^2) $ ,$ y = 1 $ .