Damped and Driven Breathers and Metastability

The paper proves existence of damped–driven breathers for finite DNLS by an Implicit Function Theorem (IFT) argument that first solves interior variables given (p1,q1,ε,γ,ω) and then determines p1 and the unique β from two remaining boundary equations, after fixing phase by q1=0. The candidate solution also uses IFT but on an augmented, square system that adds a phase-fixing equation and an energy-balance constraint to remove the U(1) degeneracy; it proves invertibility via block operators L_± and then applies IFT. Both routes are logically sound and compatible with the paper’s equations and auxiliary identities (notably the power-balance relation implicit in the paper’s energy–phase formulation). Hence both are correct but follow different (yet related) formulations.