Reducibility of Quasi-periodic Linear KdV Equation

Abstract

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

$$\begin{aligned} u_t+(1+ a_{1}(\omega t)) u_{xxx}+ a_{2}(\omega t,x) u_{xx}+ a_{3}(\omega t,x)u_{x} +a_{4}(\omega t,x)u=0 \end{aligned}$$

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.