| Citation: |
Xiaoxue Zhang, Chuanjian Wang, Changzhao Li, Lirong Wang. DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1090-1103. doi: 10.11948/20220137
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DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION
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Abstract
The degeneration of lump-type localized waves in the (2+1)-dimen-sional Ito equation is investigated through the parallel relationship of wave numbers. These lump-type localized waves can degenerate into three different kinds of localized wave solutions: singular lump-type localized wave, periodic variable amplitude localized wave, rogue wave. In the process of propagation, the lump-type localized waves keep the same waveform structure and amplitude. However, the periodic variable amplitude localized wave demonstrates three different kinds of waveform structures, which presents an interesting emit-absorb interaction phenomenon. By an emitting and absorbing interaction, the localized wave realizes the energy exchange from one localized wave to another, and keeps the original waveform structure. Rogue wave is a rational growing-and-decaying localized wave which is localized in both space and time.
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Keywords:
- (2+1)-dimensional Ito equation /
- lump-type localized wave /
- degeneration /
- rogue wave
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References
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Xiaoxue Zhang, Chuanjian Wang, Changzhao Li, Lirong Wang. DEGENERATION OF LUMP-TYPE LOCALIZED WAVES IN THE (2+1)-DIMENSIONAL ITO EQUATION[J]. Journal of Applied Analysis & Computation, 2022, 12(3): 1090-1103. doi: 10.11948/20220137
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Figure 1. (a) The spatial localized structure and periodic features of the lump-type localized wave chain. (b) Sectional view in period unit showing peak and two troughs. The parameter values are
$ (\alpha,\beta,\gamma,a_1,b_1,d_1,l_1)=(1,1,1,1,1,1.5,-3) $ . -
Figure 2. (a) The spatial structure structure of the singular lump-type localized wave. (b) Contour plot showing its symmetry. The parameter values are
$ (\alpha,\beta,\gamma,a_1,b_1,d_1,l_1)=(1,1,\frac{5}{8},1,1,1.5,-3) $ . -
Figure 3. The waveform structures of solution (3.6) at different points of time in one period. From left to right, the point in time takes
$ t=-1, -0.4, -0.1, 0.3 , 0.8 $ , respectively. The parameter values are$ (\alpha,\beta,b_1,d_1,a_1,l_1)=(1,1,1,1,2,-3) $ . -
Figure 4. The waveform structures of solution (3.6) at different points of time in one period. From right to left, the point in time takes
$ t=-0.8, -0.2, -0.07, 0.1 , 0.63 $ , respectively. The parameter values are$ (\alpha,\beta,b_1,d_1,a_1,l_1)=(1,1,1,1,1,-3) $ . -
Figure 5. The oscillation phenomena of localized wave: (a)
$ \delta<0 $ ; (b)$ \delta>0 $ . The parameter values are$ (\alpha,\beta,d_1, a_1,b_1,l_1 )=(1,1,1,0.2, 2.5,-0.15) $ and$ (\alpha,\beta,d_1, a_1,b_1,l_1 )=(1,1,1,0.2, 2.5,-0.3) $ , respectively. -
Figure 6. The time evolution plots of the maximum and minimum values: (a)
$ u_{max}(t) $ and (b)$ u_{min}(t) $ . The parameter values are$ (\alpha,\beta,a_1,b_1,d_1,l_1)=(1,1,1,1,1.5,-3) $ . -
Figure 7. The spatiotemporal evolution plots of the solution (3.10): (a)
$ t=-5 $ , (b)$ t=0 $ and (c)$ t=5 $ . The parameter values are$ (\alpha,\beta,a_1,b_1,d_1,l_1)=(1,1,1,1,1.5,-3) $ .