| Citation: |
Lijuan Shi, Zhenshu Wen. SEVERAL TYPES OF PERIODIC WAVE SOLUTIONS AND THEIR RELATIONS OF A FUJIMOTO-WATANABE EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1193-1203. doi: 10.11948/2156-907X.20180084
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SEVERAL TYPES OF PERIODIC WAVE SOLUTIONS AND THEIR RELATIONS OF A FUJIMOTO-WATANABE EQUATION
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Abstract
In this paper, we study periodic wave solutions of a Fujimoto–Watanabe equation by exploiting the bifurcation method of dynamical systems. We obtain all possible bifurcations of phase portraits of the system in different regions of the parametric space, and then give the sufficient conditions to guarantee the existence of several types of periodic wave solutions. What's more, we present their exact expressions and reveal their inside relations as well as their relations with solitary wave solutions.
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References
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Lijuan Shi, Zhenshu Wen. SEVERAL TYPES OF PERIODIC WAVE SOLUTIONS AND THEIR RELATIONS OF A FUJIMOTO-WATANABE EQUATION[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1193-1203. doi: 10.11948/2156-907X.20180084
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Figure 1. The graphics of the function
$ f(\varphi) $ -
Figure 2. The phase portraits of system (2.3) when
$ \alpha>0 $ and$ c>0 $ -
Figure 3. The phase portraits of system (2.3) when
$ \alpha>0 $ and$ c<0 $ -
Figure 4. The phase portraits of system (2.3) when
$ \alpha<0 $ and$ c>0 $ -
Figure 5. The phase portraits of system (2.3) when
$ \alpha<0 $ and$ c<0 $ - Figure 6. The profiles of periodic wave solutions (3.1), (3.2), (3.3), (3.4), (3.6), and (3.7), respectively
- Figure 7. The process of periodic wave solutions (3.22) converging to solitary wave solution (3.21)