Large Amplitude Quasi-Periodic Traveling Waves in Two Dimensional Forced Rotating Fluids

We establish the existence of quasi-periodic traveling wave solutions forthe β-plane equation on T2 with a large quasi-periodic traveling wave external force.These solutions exhibit large sizes, which depend on the frequency of oscillations of theexternal force. Due to the presence of small divisors, the proof relies on a nonlinear Nash-Moser scheme tailored to construct nonlinear waves of large size. To our knowledge,this is the first instance of constructing quasi-periodic solutions for a quasilinear PDEin dimensions greater than one, with a 1-smoothing dispersion relation that is highlydegenerate - indicating an infinite-dimensional kernel for the linear principal operator.This degeneracy challenge is overcome by preserving the traveling-wave structure, theconservation of momentum and by implementing normal form methods for the linearizedsystem with sublinear dispersion relation in higher space dimension.