Quasi-Periodic Solution and Its Asymptotic Behavior for Camassa-Holm Equation
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摘要: 近20年来,浅水波模型Camassa-Holm(CH)方程受到诸多研究者关注。在之前的工作中,通过Hirota双线性方法得到了CH方程的单周期解.基于此,该文将对N=2时CH方程的拟周期解及其渐近行为进行研究.首先,回顾了坐标变换,扩展的双线性形式和Riemann(黎曼)θ-函数等内容,并在此基础上利用Hirota双线性方法构造了在N=2时CH方程的含有多个参数的拟周期解,并且此拟周期解是由Riemannθ-函数表示的。其次,发现了此拟周期解渐近行为的一个特点,即CH方程的此拟周期解可以退化为其二孤子解.
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关键词:
- Camassa-Holm方程 /
- 双线性形式 /
- 拟周期解 /
- Riemannθ-函数
Abstract: Many researchers have paid attention to the shallow water wave model Camassa-Holm (CH) equation over the last two decades. The one-periodic solution of CH equation based on the Hirota bilinear method had been presented in our previous work. In this paper, we offer quasi-periodic solution in genus two and its asymptotic behavior. First, we have rearranged the parameters appeared in the bilinear equation system, such as the coordinate transformation, extended bilinear form and Riemman theta function and so on. Then quasi-periodic solution of CH equation is presented, which is expressed by Riemann theta function in genus two. Second, asymptotic behavior of the quasi-periodic solution is discussed. It can be shown that this solution will degenerate into its two-soliton solution.-
Key words:
- Camassa-Holm equation /
- bilinear form /
- quasi-periodic solution /
- Riemman theta function
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