Large Amplitude Quasi-Periodic Traveling Waves in Two Dimensional Forced Rotating Fluids.
Roberta Bianchini, Luca Franzoi, Riccardo Montalto, Shulamit Terracina
Author Information
Roberta Bianchini: Consiglio Nazionale Delle Ricerche, 00185 Roma, Italy.
Luca Franzoi: Dipartimento di Matematica "Federigo Enriques", Universit�� degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy.
Riccardo Montalto: Dipartimento di Matematica "Federigo Enriques", Universit�� degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy. ORCID
Shulamit Terracina: SISSA, Scuola Internazionale di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy.
We establish the existence of quasi-periodic traveling wave solutions for the -plane equation on with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the frequency of oscillations of the external 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 PDE in dimensions greater than one, with a 1-smoothing dispersion relation that is highly degenerate - indicating an infinite-dimensional kernel for the linear principal operator. This degeneracy challenge is overcome by preserving the traveling-wave structure, the conservation of momentum and by implementing normal form methods for the linearized system with sublinear dispersion relation in higher space dimension.
