| Citation: |
Jiao-Jiao Dong, Biao Li, Manwai Yuen. GENERAL HIGH-ORDER BREATHER SOLUTIONS, LUMP SOLUTIONS AND MIXED SOLUTIONS IN THE (2+1)-DIMENSIONAL BIDIRECTIONAL SAWADA-KOTERA EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 271-286. doi: 10.11948/20190361
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GENERAL HIGH-ORDER BREATHER SOLUTIONS, LUMP SOLUTIONS AND MIXED SOLUTIONS IN THE (2+1)-DIMENSIONAL BIDIRECTIONAL SAWADA-KOTERA EQUATION
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Abstract
In this paper, we investigate some interesting solution structures of the (2+1)-dimensional bidirectional Sawada-Kotera (bSK) equation. We obtain general high-order breather solutions by utilizing the Hirota's bilinear method united with the perturbation expansion technique. Taking a long-wave limit of the obtained breather solutions and then making particular parameter constraints, smooth rational solutions are generated, which include high-order lumps and mixed solutions consisting of lumps and stripe. In order to easily explore the dynamical behaviors, some plots are presented to analyse these solutions. -
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References
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Jiao-Jiao Dong, Biao Li, Manwai Yuen. GENERAL HIGH-ORDER BREATHER SOLUTIONS, LUMP SOLUTIONS AND MIXED SOLUTIONS IN THE (2+1)-DIMENSIONAL BIDIRECTIONAL SAWADA-KOTERA EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 271-286. doi: 10.11948/20190361
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Figure 1. (Color online) First-order breather solution
$ u $ of Eq. (1.1) with the parameters$ K_{1} = -i, K_{2} = i, W_{1} = 2-\frac{3}{2}i, W_{2} = 2+\frac{3}{2}i $ ; ($ a $ )3-D plot, ($ b $ )density plot, ($ c $ )contour plot -
Figure 2. (Color online) First-order lump solution
$ u $ of equation (1.1) with the parameters$ K_{1} = 1-i, K_{2} = 1+i, W_{1} = 1, W_{2} = 1 $ : ($ a $ ) 3-D plot; ($ b $ ) density plot; ($ c $ ) contour plot -
Figure 3. (Color online) Second-order breather solution
$ u $ of equation (1.1) with the parameters$ K_{1} = \frac{1}{7}+\frac{i}{9}, K_{2} = \frac{1}{7}-\frac{i}{9}, W_{1} = \frac{2}{9}, W_{2} = \frac{2}{9}, K_{3} = -\frac{1}{8}+\frac{i}{4}, K_{4} = -\frac{1}{8}-\frac{i}{9}, W_{3} = \frac{1}{10}-\frac{2}{9}i, W_{4} = \frac{1}{10}+\frac{2}{9}i, \phi_{1} = 0, \phi_{2} = 0, \phi_{3} = 4, \phi_{4} = 4 $ : ($ a $ ) the 3-D plot; ($ b $ ) density plot; ($ c $ ) contour plot -
Figure 4. (Color online) Second-order lump solution
$ u $ of equation (1.1) with the parameters$ K_{1} = 2-\frac{3}{2}i, K_{2} = 2+\frac{3}{2}i, K_{3} = 2-\frac{3}{2}i, K_{4} = 2+\frac{3}{2}i, W_{1} = 2, W_{2} = 2, W_{3} = 3, W_{4} = 3 $ : ($ a $ ) 3-D plot; ($ b $ ) density plot -
Figure 5. (Color online) Time evolution of three-order lump solution
$ u $ of equation (1.1): ($ a $ )($ b $ )($ c $ ): 3-D plot; ($ d $ ):density plot with$ t = -14 $ , ($ e $ ):density plot with$ t = 0 $ , ($ f $ ):density plot with$ t = 14 $ -
Figure 6. (Color online) Profiles of mixed solution between first-order lump and a stripe with the parameters
$ K_{1} = 3+3i, K_{1} = 3-3i, W_{1} = 5, W_{2} = 5 $ at times ($ a $ )$ t = -100 $ ; ($ b $ )$ t = 0 $ ; ($ c $ )$ t = 100 $ ; ($ d $ ) contour plot at$ t = -100 $ ; ($ e $ ) contour plot at$ t = 0 $ ; ($ f $ ) contour plot at$ t = 100 $ -
Figure 7. (Color online) Profiles of mixed solution between second-order lump and a stripe with the parameters
$ K_{1} = \frac{2}{3}, K_{2} = \frac{2}{3}, K_{3} = -\frac{3}{4}+\frac{i}{3}, K_{4} = -\frac{3}{4}-\frac{i}{3}, k_{5} = 1, W_{1} = \frac{4}{3}-i, W_{2} = \frac{4}{3}+i, W_{3} = -\frac{4}{3}-\frac{i}{4}, W_{4} = -\frac{4}{3}+\frac{i}{4}, w_{5} = \frac{1}{2} $ ; ($ d $ ) contour plot at$ t = -15 $ ; ($ e $ ) contour plot at$ t = 0 $ ; ($ f $ ) contour plot at$ t = 15 $