In this paper, we consider the following one-dimensional, quasi-periodically forced, linear KdV equations

under the periodic boundary condition \(u(t,x+2\pi )=u(t,x)\), where \(\omega \)’s are frequency vectors lying in a bounded closed region \(\Pi _*\subset {\mathbb {R}}^b\) for some \(b>1\), \(a_1:{\mathbb {T}}^b\rightarrow {\mathbb {R}}\), \(a_i: {\mathbb {T}}^b\times {\mathbb {T}}\rightarrow {\mathbb {R}}\), \(i=2,3,4\), are real analytic, bounded from the above by a small parameter \(\epsilon _*>0\) under a suitable norm, and \(a_1,a_3\) are even, \(a_2,a_4\) are odd. Under the real analyticity assumption of the coefficients, we re-visit a result of Baldi et al. (Math Ann 359(1–2):471–536, 2014) by showing that there exists a Cantor set \(\Pi _{\epsilon _*}\subset \Pi _*\) with \(|\Pi _*\setminus \Pi _{\epsilon _*}|=O(\epsilon _*^{\frac{1}{100}})\) such that for each \(\omega \in \Pi _{\epsilon _*}\), the corresponding equation is smoothly reducible to a constant-coefficient one. Our main result removes a condition originally assumed in Baldi et al. (2014) and thus can yield general existence and linear stability results for quasi-periodic solutions of a reversible, quasi-periodically forced, nonlinear KdV equation with much less restrictions on the nonlinearity. The proof of our reducibility result makes use of some special structures of the equations and is based on a refined Kuksin’s estimate for solutions of homological equations with variable coefficients.

在本文中，我们考虑以下一维，准周期强迫线性KdV方程

在周期边界条件\（u（t，x + 2 \ pi）= u（t，x）\）下，其中\（\ omega \）是位于有界封闭区域\（\ Pi _中的频率向量* \ subset {\ mathbb {R}} ^ b \）的某些\（b> 1 \），\（a_1：{\ mathbb {T}} ^ b \ rightarrow {\ mathbb {R}} \），\ （a_i：{\ mathbb {T}} ^ b \ times {\ mathbb {T}} \ rightarrow {\ mathbb {R}} \），\（i = 2,3,4 \）是实数分析，有界从上面通过一个小参数\（\ epsilon _ *> 0 \）在适当的范数下，而\（a_1，a_3 \）是偶数，\（a_2，a_4 \）很奇怪 在系数的真实解析假设下，我们重新考察了Baldi等人的结果。（Math Ann 359（1-2）：471-536，2014），显示存在一个Cantor集\（\ Pi _ {\ epsilon _ *} \ subset \ Pi _ * \）和\（| \ Pi _ * \ setminus \ Pi _ {\ epsilon _ *} | = O（\ epsilon _ * ^ {\ frac {1} {100}}）\）这样，对于每个\（\ omega \ in \ Pi _ {\ epsilon _ *} \），相应的方程可以平滑地简化为一个常数系数。我们的主要结果消除了Baldi等人最初假定的条件。（2014年），因此可以产生一个可逆的，拟周期强迫的非线性KdV方程的拟周期解的一般存在性和线性稳定性结果，其对非线性的限制要小得多。我们的可约性结果的证明是利用方程的某些特殊结构，并且是基于经过改进的Kuksin估计的变系数同构方程的解。