Response Solutions for KdV Equations with Liouvillean Frequency

Abstract

In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations $${u_t} + {\partial _{xxx}}u - \varepsilon {\partial _x}f(\omega t,x,u) = 0,$$ where $$\omega = \xi \bar \omega ,\,\,\bar \omega = (1,\alpha)$$ is Liouvillean forced frequency and f is real analytic. We obtain a C∞ smooth response solution under zero mean-value periodic boundary conditions. The proof is based on a modified infinite dimensional KAM theory.