Quasi-periodic solutions to nonlinear beam equations on compact Lie groups with a multiplicative potential

The goal of this work is to study the existence of quasi-periodic solutions to nonlinear beam equations with a multiplicative potential. The nonlinearity is required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogeneous manifold with respect to a compact Lie group, which includes standard torus Td, special orthogonal group SO(d), special unitary group SU(d), spheres Sd and the real and complex Grassmannians. The proof is based on a differentiable Nash-Moser iteration scheme.