| Citation: |
Jian-Guo Liu, Wen-Hui Zhu, Li Zhou. INTERACTION SOLUTIONS AND ABUNDANT EXACT SOLUTIONS FOR THE NEW (3+1)-DIMENSIONAL GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION IN FLUID MECHANICS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 960-971. doi: 10.11948/20190172
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INTERACTION SOLUTIONS AND ABUNDANT EXACT SOLUTIONS FOR THE NEW (3+1)-DIMENSIONAL GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION IN FLUID MECHANICS
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Abstract
In this work, we present the interaction solutions and abundant exact solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation based on the Hirota's bilinear form and a direct function. The obtained interaction solutions contain the interaction between the rational function and the $\tanh$ function and the interaction between the rational function and the $\cos$ function. The dynamical properties of these resulting solutions are analyzed and shown in three-dimensional plots, corresponding contour graphs and plane figures.-
Keywords:
- Interaction solutions /
- bilinear form /
- exact solutions /
- dynamical properties
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References
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Jian-Guo Liu, Wen-Hui Zhu, Li Zhou. INTERACTION SOLUTIONS AND ABUNDANT EXACT SOLUTIONS FOR THE NEW (3+1)-DIMENSIONAL GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION IN FLUID MECHANICS[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 960-971. doi: 10.11948/20190172
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Figure 1. Plots of the interaction solutions (2.6) for
$ \iota_1 = \iota_6 = 2 $ ,$ k = t = 1 $ ,$ \iota_3 = -1 $ ,$ \iota_5 = 5 $ ,$ \iota_{11} = -6 $ ,$ \iota_{10} = 3 $ ,$ j_3 = -2 $ , when$ x = -5 $ in (a) (d) (g),$ x = 0 $ in (b) (e) (h) and$ x = 5 $ in (c) (f) (i).$ y = -8 $ in (g) (h) and (i). -
Figure 2. Plots of the interaction solutions (3.3) for
$ \iota_1 = \iota_5 = j_1 = 2 $ ,$ \iota_2 = -1 $ ,$ z = \iota_6 = 0 $ ,$ \iota_{11} = -6 $ ,$ \iota_{10} = 3 $ ,$ j_3 = -2 $ , when$ x = -3 $ in (a) (d),$ x = 0 $ in (b) (e) and$ x = 3 $ in (c) (f). -
Figure 3. Plots of the interaction solutions (3.3) for
$ \iota_1 = \iota_5 = j_1 = 2 $ ,$ \iota_2 = -1 $ ,$ \iota_6 = 1 $ ,$ t = 20 $ $ \iota_{11} = -6 $ ,$ \iota_{10} = 3 $ ,$ j_3 = -2 $ , when$ z = -35 $ in (a) (d),$ z = 0 $ in (b) (e) and$ z = 35 $ in (c) (f). -
Figure 4. Solution (4.3) with
$ \Lambda _1 = -2 $ ,$ \Lambda _2 = 1 $ ,$ \Lambda _3 = 0 $ ,$ x = t = 0 $ , (a) three-dimensional graph (b) contour graph. -
Figure 5. Solution (4.3) with
$ \Lambda _1=0 $ ,$ \Lambda _2=1 $ ,$ \Lambda _3=1 $ ,$ x=t=0 $ , (a) three-dimensional graph (b) contour graph. -
Figure 6. Solution (4.3) with
$ \Lambda _1=-2 $ ,$ \Lambda _2=0 $ ,$ \Lambda _3=1 $ ,$ x=t=0 $ , (a) three-dimensional graph (b) contour graph. -
Figure 7. Solution (4.3) with
$ \Lambda _1 = -2 $ ,$ \Lambda _2 = 1 $ ,$ \Lambda _3 = 1 $ ,$ z = 0 $ , when$ t = -5 $ in (a) (d),$ t = 0 $ in (b) (e) and$ t = 5 $ in (c) (f). -
Figure 8. Solution (4.9) when
$ z = -10 $ in (a) (d),$ z = 0 $ in (b) (e) and$ z = 10 $ in (c) (f).