Wave Lengths of Periodic Waves for the Vakhnenko Equation
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摘要: 主要研究Vakhnenko方程的光滑周期行波解的波长.通过变量变换, Vakhnenko方程可以转化为一个平面多项式微分系统.利用动力系统的临界周期分支方法研究这个多项式微分系统,其主要结果是给出了周期函数T(h)或波长函数λ(a)的单调性质.与KdV方程比较, 波长函数λ(a)单调递减到一个有限的数,而不是单调递增到无穷.结果表明, 对于固定波速c, Vakhnenko方程不存在任意小或任意大波长的光滑周期行波解.
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关键词:
- Vakhnenko方程 /
- 周期波 /
- 周期函数 /
- 波长 /
- 单调性
Abstract: The wave lengths of smooth periodic traveling wave solutions to the Vakhnenko equation were studied. The Vakhnenko equation was reduced to a planar polynomial differential system through the transformation of variables. The polynomial differential system was treated with the critical period bifurcation method based on the dynamical system theory. The main results involve the monotonicity properties of periodic function T(h) (or wave length function λ(a)). In comparison with the wave length for the KdV equation, wave length function λ(a) monotonically decreases to a finite value rather than monotonically increases to infinity. This shows that, for fixed wave speed c, there exist no smooth periodic wave solutions with arbitrarily small wave lengths or arbitrarily large wave lengths, to the Vakhnenko equation.-
Key words:
- Vakhnenko equation /
- periodic wave /
- periodic function /
- wave length /
- monotonicity
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