Quasi-periodic solutions of completely resonant forced wave equations
Massimiliano Berti, Michela Procesi
TL;DR
The paper tackles the existence of small-amplitude quasi-periodic solutions for completely resonant forced nonlinear wave equations with periodic boundary. It develops a variational Lyapunov–Schmidt reduction, pairwise reduced-action functionals, and a linking framework to construct two-frequency quasi-periodic states in two forcing-frequency regimes: rational $\omega_1$ on a Cantor set ${\mathcal{B}}_\gamma$, and irrational $\omega_1$ on ${\mathcal{C}}_\gamma$, with explicit amplitude scalings and a leading-order traveling-wave structure. For the rational case, a finite-dimensional Galerkin-like reduction plus infinite-dimensional linking yields a nontrivial critical point that corresponds to a quasi-periodic solution; for the irrational case, a Lyapunov–Schmidt reduction together with a phase-space ODE analysis provides a nontrivial two-frequency solution and, under certain nondegeneracy conditions, a non-resonant continuation. A nonexistence remark clarifies that, under even-degree leading nonlinearities, small-amplitude quasi-periodic solutions may fail to exist. Overall, the work extends existence theory for completely resonant PDEs by balancing small-divisor issues with variational and topological methods to produce two-frequency quasi-periodic waves.
Abstract
We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number.