| Citation: |
Shoufu Tian, Ding Guo, Xiubin Wang, Tiantian Zhang. TRAVELING WAVE, LUMP WAVE, ROGUE WAVE, MULTI-KINK SOLITARY WAVE AND INTERACTION SOLUTIONS IN A (3+1)-DIMENSIONAL KADOMTSEV - PETVIASHVILI EQUATION WITH BÄCKLUND TRANSFORMATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 45-58. doi: 10.11948/20190086
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TRAVELING WAVE, LUMP WAVE, ROGUE WAVE, MULTI-KINK SOLITARY WAVE AND INTERACTION SOLUTIONS IN A (3+1)-DIMENSIONAL KADOMTSEV - PETVIASHVILI EQUATION WITH BÄCKLUND TRANSFORMATION
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Abstract
We consider a (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation. By using the Hirota bilinear operators, we construct the bilinear form of the equation. Based on the resulting bilinear form, we further derive the Bäcklund transformation and the traveling wave solutions of the equation. Furthermore, lump solutions are constructed by searching the positive function from the Hirota bilinear formalism. Meanwhile, we also obtain the interaction solutions between lump solutions and the stripe solitons. We discuss the influences of each parameters on these exact solutions by using several graphics. Finally, we successfully construct its rogue wave solutions and multi-kink solitary wave solutions. It is hoped that our results can be used to enrich the dynamical behavior of the (3+1)-dimensional KP-type equations. -
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References
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Shoufu Tian, Ding Guo, Xiubin Wang, Tiantian Zhang. TRAVELING WAVE, LUMP WAVE, ROGUE WAVE, MULTI-KINK SOLITARY WAVE AND INTERACTION SOLUTIONS IN A (3+1)-DIMENSIONAL KADOMTSEV - PETVIASHVILI EQUATION WITH BÄCKLUND TRANSFORMATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 45-58. doi: 10.11948/20190086
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Figure 1. (Color online) The traveling wave solution (3.11) by choosing suitable parameters:
$ \eta = 2,\psi_{1} = 2,\psi_{2} = 4,\psi_{3} = \frac{44}{5},\psi_{5} = -6 $ . (a) Plot3d perspective view of the real part of the wave($ z = 0,y = 2 $ ). (b)The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ ) -
Figure 2. (Color online) The rational one-soliton solution (3.13) by choosing suitable parameters:
$ \psi_{1} = -8,\psi_{2} = -3,\psi_{3} = -5,\psi_{5} = \frac-{71}{9} $ . (a) Plot3d perspective view of the real part of the wave(y = 2). (b)The wave propagation pattern of the wave along the x axis (t = −4; t = 0; t = 4). -
Figure 3. (Color online) The lump solution (28) by choosing suitable parameters:
$ b_{1} = 1,b_{3} = 2,b_{6} = 3,b_{7} = 2 $ . (a) Plot3d perspective view of the real part of the wave($ y = 0 $ ). (b)The overhead view of the wave. (c) The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ ) -
Figure 4. (Color online) The interaction solution (5.5) by choosing suitable parameters:
$ k = 2,k_{2} = 2,b_{2} = 3,b_{6} = 2 $ . (a) Plot3d perspective view of the real part of the wave($ y = 0 $ ). (b)The overhead view of the wave. (c)The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ ) -
Figure 5. (Color online) The breather wave solution (6.3) by choosing suitable parameters:
$ n_{1} = 2,n_{7} = 3 $ . (a)Plot3d perspective view of the real part of the wave($ y = 0,z = 0 $ ). (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ ) -
Figure 6. (Color online)The rogue wave solution (6.4) by choosing suitable parameters:
$ n_{1} = 2,n_{7} = 3 $ . (a)Plot3d perspective view of the real part of the wave($ y = 0,z = 0 $ ). (b)The overhead view of the wave. (c) The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ ) -
Figure 7. (Color online)The double kink soliton solution (7.7) by choosing suitable parameters:
$ k_{1} = -1,k_{2} = 1,l_{1} = 1,l_{2} = -1,\alpha_{1} = 1,\alpha_{2} = -1,\xi_{1} = 1,\xi_{2} = 1 $ . (a) Plot3d perspective view of the real part of the wave($ y = 0 $ ). (b) The overhead view of the wave. (c) The wave propagation pattern of the wave along the$ x $ axis ($ t = -4,t = 0,t = 4 $ )