A KAM Theorem for the Time Quasi-periodic Reversible Perturbations of Linear Wave Equations Beyond Brjuno Conditions

This paper is concerned with the existence of quasi-periodic response solutions (i.e., solutions that are quasi-periodic with the same frequencies as forcing term) for a class of forced reversible wave equations with derivative nonlinearity. The forcing frequency \(\omega \in \mathbb {R}^2\) $\omega \in {\mathbb{R}}^2$ would be Liouvillean which is weaker than the usual Diophantine and Brjuno conditions. The derivative nonlinearity in the equation also leads to some difficulty in measure estimate. To overcome it, we also use the Töplitz–Lipschitz property of vector field. The proof is based on an infinite dimensional Kolmogorov–Arnold–Moser theorem for reversible systems.