Stability for line solitary waves of Zakharov-Kuznetsov equation
Stability for line solitary waves of Zakharov-Kuznetsov equation
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扎哈罗夫-库兹涅佐夫方程线孤立波的稳定性
DOI：
10.1016/j.jde.2017.01.006
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Yohei Yamazaki
中科院分区：
文献类型：
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作者：
Yohei Yamazaki
本文研究了二维Zakharov-Kuznetsov方程在R× TL上的线孤立波稳定性，该方程是Korteweg-de弗里斯方程的高维推广，其中TL是周期为2 π L的环面. Benjamin [2]、Pego和Weinstein [41]以及Martel和Merle [30]证明了能量空间上Korteweg-de弗里斯方程单孤子的轨道稳定性和渐近稳定性。我们把Korteweg-de弗里斯方程的单孤子看作是Zakharov-Kuznetsov方程在R× TL上的线孤波。本文利用Evans函数方法和Zakharov-Kuznetsov方程的参数，证明了Zakharov-Kuznetsov方程的线孤立波的稳定性和横向不稳定性。利用Martel和Merle [30]，[31]，[32]的论证和Liouville型定理，证明了Zakharov-Kuznetsov方程轨道稳定线孤波的渐近稳定性.如果L是线孤立波的临界周期，则线孤立波是轨道稳定的。然而，由于该线孤立波是定常方程的分歧点，定常方程的线性化算子是退化的。由于线性化算子的退化性，我们不能用通常的维里型估计来证明线孤立波的刘维尔型定理。为了证明线孤立波的Liouville型定理，我们修正了维里型估计。
In this paper, we consider the stability for line solitary waves of the two dimensional Zakharov–Kuznetsov equation on R× T L which is one of a high dimensional generalization of Korteweg–de Vries equation, where T L is the torus with the 2 π L period. The orbital and asymptotic stability of the one soliton of Korteweg–de Vries equation on the energy space was proved by Benjamin [2], Pego and Weinstein [41] and Martel and Merle [30]. We regard the one soliton of Korteweg–de Vries equation as a line solitary wave of Zakharov–Kuznetsov equation on R× T L. We prove the stability and the transverse instability of the line solitary waves of Zakharov–Kuznetsov equation by applying the method of Evans' function and the argument of Rousset and Tzvetkov [44]. Moreover, we prove the asymptotic stability for orbitally stable line solitary waves of Zakharov–Kuznetsov equation by using the argument of Martel and Merle [30],[31],[32] and a Liouville type theorem. If L is the critical period with respect to a line solitary wave, the line solitary wave is orbitally stable. However, since this line solitary wave is a bifurcation point of the stationary equation, the linearized operator of the stationary equation is degenerate. Because of the degeneracy of the linearized operator, we can not show the Liouville type theorem for the line solitary wave by using the usual virial type estimate. To show the Liouville type theorem for the line solitary wave, we modify a virial type estimate.
DOI：
10.1137/050648389
发表时间：
2006-11
期刊：
SIAM J. Math. Anal.
影响因子：
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作者：
Shu-Ming Chang;S. Gustafson;K. Nakanishi;Tai-Peng Tsai
通讯作者：
Shu-Ming Chang;S. Gustafson;K. Nakanishi;Tai-Peng Tsai