Reducibility for the linearized two-component intermediate long-wave equation

Abstract

In this paper, we study the reducibility of a class of linear operators depending on time quasi-periodically, which arises from linearizing the two-component intermediate long-wave equation at a small amplitude quasi-periodic function, with a Diophantine frequency vector $\omega \in O_0\subset R^{\nu }$. We prove that there exists a Cantor-like set $O_{\infty }\subset O_0$ such that for every $\omega \in O_{\infty }$, these operators can be reduced to the ones with constant coefficients by some linear, real, reversibility-preserving and quasi-periodic time-dependent transformations. As a conclusion, the stability of the solutions of the linearized two-component intermediate long-wave equation is obtained. Besides the KAM reducibility transformation, these transformations are obtained from some pseudo-differential operators and the flows of some PDEs. The key difficulty lies in estimating the norms and the upper and lower bounds of the operators related to the integral operator $T_{\delta }$ due to their complicated structures. To overcome the difficulties, we treat these operators as the principal pseudo-differential operators up to the remainders of lower order and impose some constraints on the parameters.