Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential

We prove the existence of quasi-periodic solutions for wave equations with a multiplicative potential on T^d and finitely differentiable nonlinearities, quasi-periodically forced in time. The only external parameter is the length of the frequency vector. The solutions have Sobolev regularity both in time and space. The proof is based on a Nash–Moser iterative scheme. The key tame estimates for the inverse linearized operators are obtained by a multiscale inductive argument, which is more difficult than for NLS due to the dispersion relation of the wave equation. We prove the ‘separation properties’ of the small divisors assuming weaker non-resonance conditions than in Bourgain.