| Citation: |
Xing Lü, Xuejiao He. BÄCKLUND TRANSFORMATION TO SOLVE THE GENERALIZED (3+1)-DIMENSIONAL KP-YTSF EQUATION AND KINKY PERIODIC-WAVE, WRONSKIAN AND GRAMMIAN SOLUTIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 758-781. doi: 10.11948/20220110
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BÄCKLUND TRANSFORMATION TO SOLVE THE GENERALIZED (3+1)-DIMENSIONAL KP-YTSF EQUATION AND KINKY PERIODIC-WAVE, WRONSKIAN AND GRAMMIAN SOLUTIONS
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Abstract
The Kadomtsev-Petviashvili equation is considered to be a basic model describing nonlinear dispersive wave in fluids, which is an integrable equation with two spatial dimensions. The Yu-Toda-Sasa- Fukuyama equation plays a crucial role in fluid dynamics, plasma physics and weakly dispersive media. In this paper, we investigate a generalized (3+1)-dimensional Kadomtsev-Petviashvili-Yu-Toda-Sasa-Fukuyama equation, and multiple types of solutions are derived. With symbolic computation, a class of kinky periodic-wave solutions, determinant solutions and the bilinear Bäcklund transformation are constructed. We obtain two types of determinant solutions, that is, Wronskian and Grammian solutions. By choosing the appropriate matrix elements of determinants, many kinds of solutions are derived. In addition to the soliton solutions, the complexiton solutions and rational solutions are given. As illustrative examples, a few particular solutions are computed and plotted.
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References
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Xing Lü, Xuejiao He. BÄCKLUND TRANSFORMATION TO SOLVE THE GENERALIZED (3+1)-DIMENSIONAL KP-YTSF EQUATION AND KINKY PERIODIC-WAVE, WRONSKIAN AND GRAMMIAN SOLUTIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(2): 758-781. doi: 10.11948/20220110
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Figure 1.
The kinky periodic-wave solution via Eq. (2.4) with
$ c_{1}=-\frac{1}{4}, c_{2}=0, c_{3}=-\frac{3}{4} $ $ a_{2}=b_{1}=b_{2}=d_{2}=1 $ $ \delta_{1}=2 $ $ t=1 $ $ z=1 $ -
Figure 2.
Maya chart of Eq. (3.5): the Plücker relation.
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Figure 3.
Soliton solutions to Eq. (1.7) with
$ c_{1}=-1, c_{2}=1, c_{3}=2 $ $ z=1 $ $ t=1 $ $ \gamma=1 $ $ l_{1}=2 $ $ k_{1}=3 $ $ \xi_{1}^{0}=\zeta_{1}^{0}=0 $ $ l_{1}=1 $ $ l_{2}=-1.2 $ $ k_{1}=1.5 $ $ k_{2}=-1.8 $ $ \xi_{1}^{0}=\xi_{2}^{0}=\zeta_{1}^{0}=\zeta_{2}^{0}=0 $ $ l_{1}=1 $ $ l_{2}=-1.2 $ $ l_{3}=0.2 $ $ k_{1}=1.6 $ $ k_{2}=-1.8 $ $ k_{3}=0.6 $ $ \xi_{1}^{0}=\xi_{2}^{0}=\xi_{3}^{0}=\zeta_{1}^{0}=\zeta_{2}^{0}=\zeta_{3}^{0}=0 $ $ l_{1}=1 $ $ l_{2}=0.4 $ $ l_{3}=1.8 $ $ l_{4}=-1 $ $ k_{1}=1.5 $ $ k_{2}=-0.2 $ $ k_{3}=2 $ $ k_{4}=-1.5 $ $ \xi_{1}^{0}=\xi_{2}^{0}=\xi_{3}^{0}=\xi_{4}^{0}=\zeta_{1}^{0}=\zeta_{2}^{0}=\zeta_{3}^{0}=\zeta_{4}^{0}=0 $ -
Figure 4.
The complexiton solution to Eq. (1.7) with
$ c_{1}=1, c_{2}=2, c_{3}=3 $ $ t=1 $ $ z=1 $ $ \gamma=1 $ $ N=1 $ $ \alpha_{11}=2 $ $ \beta_{11}=1 $ $ N=2 $ $ \alpha_{11}=2, \alpha_{12}=1 $ $ \beta_{11}=1, \beta_{12}=2 $ -
Figure 5.
Maya chart of Eq. (4.7): the Jacobi identity.
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Figure 6.
Soliton solutions to Eq. (1.7) with
$ z=1 $ $ t=1 $ $ \gamma=1 $ -
Figure 7.
Plot of the rational solution and the contour via Eq. (5.11) with
$ c_{1}=1, c_{2}=-2 $ $ z=1 $ $ t=3 $