耦合调制Korteweg-de Vries系统适定性理论中的丢番图条件
Diophantine conditions in well-posedness theory for a coupled modulated Korteweg-de Vries system
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中文总结 AI 辅助
研究圆环上耦合调制KdV系统适定性，耦合参数为1时已证全局适定，对于不同耦合用丢番图条件刻画共振，证明在一定条件下对任意\(s\in \mathbb{R}\)该系统在\(H^s(\mathbb{T})\times H^s(\mathbb{T})\)中全局适定，与未调制情况不同。
中文摘要 AI 辅助
我们研究了圆环上具有作用于线性色散项的时间非齐次调制的耦合调制Korteweg-de Vries（KdV）系统的适定性理论。当耦合参数等于1时，最近已证明，给定任何\(s\in \mathbb{R}\)，所得的调制KdV系统在\(H^s(\mathbb{T})\times H^s(\mathbb{T})\)中是全局适定的，具有足够不规则的调制。对于不同于1的耦合，我们使用丢番图条件来刻画共振，并证明（在对耦合常数的进一步限制下）对于任何\(s\in \mathbb{R}\)，耦合调制KdV系统在\(H^s(\mathbb{T})\times H^s(\mathbb{T})\)中是全局适定的。该结果与其未调制的对应结果不同，在未调制的情况下，已知对于\(s\geq s_*\in (5/7,1]\)全局适定性成立。
英文摘要
We study the well-posedness theory of a coupled modulated Korteweg-de Vries (KdV) system on the circle with a time non-homogeneous modulation acting on the linear dispersion term. When the coupling parameter is equal to one, it has been recently proved that given any $s\in \mathbb{R}$, the resulting modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$, with a sufficiently irregular modulation. For couplings different from one, we use Diophantine conditions to characterize the resonances and prove that (under further restrictions on the coupling constant) for any $s\in \mathbb{R}$ the coupled modulated KdV system is globally well-posed in $H^s(\mathbb{T})\times H^s(\mathbb{T})$. This result differs from its unmodulated counterpart where it is known that global well-posedness holds for $s\ge s_*\in (5/7,1]$.
