Response Solutions for KdV Equations with Liouvillean Frequency

In this paper, we prove an infinite dimensional KAM (Kolmogorov–Arnold–Moser) theorem, which can be used to the KdV equations u t + ∂ x x x u − ε ∂ x f ( ω t , x , u ) = 0 , where ω = ξ ω ¯ , ω ¯ = ( 1 , α ) 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.