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arXiv 2006.01305math.AP
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一类Klein-Gordon型方程奇周期波的存在性和稳定性

Odd Periodic Waves for some Klein-Gordon Type Equations: Existence and Stability

Fábio Natali, Guilherme de Loreno

摘要:

本文建立了Klein-Gordon型方程奇周期波的存在性和稳定性,其中包括著名的$ϕ^4$和$ϕ^6$模型.利用常微分方程的一般平面理论确定了周期波的存在性。利用周期映射的单调性,结合标准Floquet理论的改进,建立了相应线性化算子的谱分析.利用周期映射的单调性证明了在能量空间的奇数扇区中的轨道稳定性。利用\cite{grillakis1}中的抽象方法给出了$ϕ^4$和$ϕ^6$模型显式解的轨道不稳定性。

英文摘要:

In this paper, we establish the existence and stability properties of odd periodic waves related to the Klein-Gordon type equations, which include the well known $ϕ^4$ and $ϕ^6$ models. Existence of periodic waves is determined by using a general planar theory of ODE. The spectral analysis for the corresponding linearized operator is established using the monotonicity of the period map combined with an improvement of the standard Floquet theory. Orbital stability in the odd sector of the energy space is proved using exclusively the monotonicity of the period map. The orbital instability of explicit solutions for the $ϕ^4$ and $ϕ^6$ models is presented using the abstract approach in \cite{grillakis1}.

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