Sobolev quasi-periodic solutions of multidimensional wave equations with a multiplicative potential (Q2918661)

The authors prove the existence of quasi-periodic solutions with the forced frequencies for wave equations with a multiplicative potential on \(\mathbb T^d\), and finitely differentiable nonlinearities, quasi-periodically forced in time. The solutions have Sobolev regularity both in time and space. The proof is based on 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 NLS duo to the dispersion relation of the wave equation.