| Citation: |
Hui Wang, Shou-Fu Tian, Tian-Tian Zhang, Yi Chen. THE BREATHER WAVE SOLUTIONS, M-LUMP SOLUTIONS AND SEMI-RATIONAL SOLUTIONS TO A (2+1)-DIMENSIONAL GENERALIZED KORTEWEG-DE VRIES EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 118-130. doi: 10.11948/20190011
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THE BREATHER WAVE SOLUTIONS, M-LUMP SOLUTIONS AND SEMI-RATIONAL SOLUTIONS TO A (2+1)-DIMENSIONAL GENERALIZED KORTEWEG-DE VRIES EQUATION
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Abstract
Under investigation in this work is a (2+1)-dimensional generalized Korteweg-de Vries equation, which can be used to describe many nonlinear phenomena in plasma physics. By using the properties of Bell's polynomial, we obtain the bilinear formalism of this equation. The expression of $ N $-soliton solution is established in terms of the Hirota's bilinear method. Based on the resulting $ N $-soliton solutions, we succinctly show its breather wave solutions. Furthermore, with the aid of the corresponding soliton solutions, the $ M $-lump solutions are well presented by taking a long wave limit. Two types of hybrid solutions are also represented in detail. Finally, some graphic analysis are provided in order to better understand the propagation characteristics of the obtained solutions. -
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References
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Hui Wang, Shou-Fu Tian, Tian-Tian Zhang, Yi Chen. THE BREATHER WAVE SOLUTIONS, M-LUMP SOLUTIONS AND SEMI-RATIONAL SOLUTIONS TO A (2+1)-DIMENSIONAL GENERALIZED KORTEWEG-DE VRIES EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 118-130. doi: 10.11948/20190011
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Figure 1. (Color online) The first-order breather wave solution of Eq.(1.1) with the corresponding parameters selection in (2.8). (a) Three dimensional plot. (b) Density plot. (c) The wave propagation pattern along
$ x $ axis. -
Figure 2. (Color online) Evolution plots of 1-lump solution for Eq.(1.1) by choosing suitable parameters
$ a = 1,b = 1,\alpha = 1,\beta = 1 $ . (a)$ t = -10 $ . (b)$ t = 0 $ . (c)$ t = 10 $ . -
Figure 3. (Color online) Evolution plots of 2-lump solution for Eq.(1.1) by choosing suitable parameters
$ p_{R} = 1,p_{I} = 1,\lambda_{R} = 1.4,\lambda_{I} = 0.5,\alpha = -1,\beta = 1 $ . (a)$ t = -10 $ . (b)$ t = 0 $ . (c)$ t = 10 $ . -
Figure 4. (Color online) Evolution plots of 3-lump solution for Eq.(1.1) by choosing suitable parameters
$ p_{R} = 1,p_{I} = 1,h_{R} = 1.4,h_{I} = 1.5,q_{R} = 1.5,q_{I} = 1.6,\alpha = -1,\beta = 1 $ . (a)$ t = -2 $ . (b)$ t = 0 $ . (c)$ t = 2 $ . -
Figure 5. (Color online)Evolution plots of hybrid solution composed of 1-lump and 1-soliton for Eq.(1.1) by choosing suitable parameters
$ a = 1, b = 1, k_{3} = -1, p_{3} = 0.5, \xi_{3}^{(0)} = 0, \alpha = 1,\beta = -1 $ . (a)$ t = -8 $ . (b)$ t = 0 $ . (c)$ t = 8 $ . -
Figure 6. (Color online) Hybrid of lump solution and breather wave solution for Eq.(1.1) with the corresponding parameters selection in (4.7). (a) Three dimensional plot (
$ t = 0 $ ). (b) Density plot ($ t = 0 $ ).