Breathers from infinity in the anti-continuum limit - Dmitry Pelinovsky

No resonances at other sites n ≠ 0For n ≥ 1, we haveẍ n + x n + N(x n ) = γ(x n+1 − 2x n + x n−1 ),where N(x) = V ′ (x) − x = O(x 3 ) as x → 0.Hper(0, 2 T) is a Banach algebra with respect to multiplication. For allx ∈ B δ (Hper 2 ), there is C(δ) > 0 such that‖N(x)‖ H 2 per≤ C(δ)‖x‖ 3 H 2 per.If linear operator L is invertible and there is C > 0 such that∀f ∈ L 2 per((0, T); l 2 (N)) : ‖L −1 f‖ H 2per≤ C‖f‖ L 2per,the Implicit Function Theorem is applied. Therefore, there is a unique mapHper 2 (0, T) ∋ x 0 ↦→ x ∈ Hper 2 ((0, T); l2 (N)) for small γ > 0 so that for allx 0 ∈ B δ (Hper)2 ∃C(δ) > 0 : ‖x‖ H 2 per≤ C(δ)γ.D.Pelinovsky (McMaster University) Breathers from infinity 11 / 26