环面上的Kirchhoff-Pohozaev方程的可积性
On the integrability of the Kirchhoff-Pohozaev equation on tori
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- Università degli Studi di Milano（米兰大学）
- Università degli Studi RomaTre（罗马第三大学）
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中文总结 AI 辅助
本文研究n维环面上的Kirchhoff-Pohozaev方程，证明其哈密顿量为对合的，在n=1维时有限维约化完全可积，n>1时给出可积性充分条件，还证明其形式Birkhoff标准形在n=1维时可积。
中文摘要 AI 辅助
本文研究了文献[P2]中引入的Kirchhoff-Pohozaev方程及其运动常数（见文献BoitiManfrin2025、BoitiManfrin2026）在n维环面上的情况。我们证明这些哈密顿量均为对合的，且由无穷多个运动常数生成，这些常数在固定相空间上均有定义且为对合的。随后，我们研究限制在有限傅里叶支撑上的Kirchhoff-Pohozaev方程：在n=1维时，证明该有限维约化始终是完全可积的，并在原点邻域给出解析Birkhoff标准形；还给出n>1时可积性的充分条件；最后证明n=1时Kirchhoff-Pohozaev方程的形式Birkhoff标准形是可积的。
英文摘要
In this paper we study the Kirchhoff-Pohozaev equation introduced in \cite{P2} and its constants of motion (see \cite{BoitiManfrin2025}, \cite{BoitiManfrin2026}) on $n$ dimensional tori. We show that these Hamiltonians are all in involution and we prove that they are generated by an infinite list of constants of motion which are all defined and in involution on a fixed phase space. Then we study the Kirchhoff-Pohozaev equation restricted to a finite Fourier support. In dimension $n=1$ we show that such finite dimensional reduction is always completely integrable and provide an analytic Brikhoff normal form in a neighborhood of the origin. We also give sufficient conditions for integrability for $n>1$. We finally show that the formal Birkhoff Normal form of the Kirchhoff-Pohozaev equation is integrable for $n=1$.
