关于Zakharov-Rubenchik系统的周期解
On Periodic Solutions of the Zakharov-Rubenchik System
中文总结 AI 辅助
本文研究一维Zakharov-Rubenchik系统的周期解，利用半群理论证明线性方程的整体适定性，并通过变量替换和Banach不动点定理获得修正系统的局部适定性。
中文摘要 AI 辅助
我们研究具有周期初始条件的一维Zakharov-Rubenchik系统。该系统模拟了不同物理现象中低频波与高频波的相互作用。利用半群理论，我们证明了相关线性方程在$H^q(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$（$q,s \in \mathbb{R}$）中的整体适定性。随后，我们考虑修正的Zakharov-Rubenchik系统，通过变量替换和Banach不动点定理，在空间$H^s(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$（$s>3/2$）中获得局部适定性。
英文摘要
We study the one-dimensional Zakharov-Rubenchik system with periodic initial conditions. This system models the interaction of low-frequency waves with high-frequency waves in different physical phenomena. Using semigroup theory, we prove the global well-posedness of the associated linear equation in $H^q(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $q,s \in \mathbb{R}$. Subsequently, we consider the modified Zakharov-Rubenchik system and, by means of a change of variables and the Banach Fixed Point Theorem, we obtain local well-posedness in the space $H^s(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $s>3/2$.
