ON SMALL BREATHERS OF NONLINEAR KLEIN-GORDON EQUATIONS VIA EXPONENTIALLY SMALL HOMOCLINIC SPLITTING

The candidate’s outline mirrors the paper’s spatial-dynamics/inner–outer analysis: (i) choice of frequencies ω = √(1/(k(k+ε²))) and the weakly-hyperbolic scale μ = √(1−k²ω²) = √k ε ω; (ii) a leading-order homoclinic for the k-mode −a'' + μ² a − (1/4)a³ = 0, giving u_core(x,t) = 2√(2k) ε ω cosh(√k ε ω x)^{-1} sin(kω t); (iii) algebraic corrections of size O(ε³ k^{-3/2}) with the same cosh(·)^{-1} profile; (iv) a 3k radiative tail controlled by a Stokes constant C_in with exponentially small size e^{−π√(2k)/ε}; and (v) nonexistence of true single-bump breathers when C_in ≠ 0. These points match Theorem 1.2 (nonexistence when C_in ≠ 0; existence of generalized breathers with a two-term bound) and Theorem 1.3 (precise comparison to vh(y)=2√2/ cosh y, splitting formula with sin(3kω t), and tail bounds) in the paper. In particular, the paper’s statements (Theorem 1.2(2)) give the same estimate |u − 2√(2k) ε ω cosh(√k ε ω x)^{-1} sin(kω t)| ≤ M[ε³ k^{-3/2} cosh(·)^{-1} + (√k/ε) e^{−π√(2k)/ε}], and Theorem 1.3(2)(b) shows the Stokes-controlled leading term ∝ C_in e^{−π√(2k)/ε} sin(3kω t), aligning with the model’s tail mechanism and scaling. Minor notational/sign points aside (the model’s mode equation presentation versus the paper’s (1.10)–(1.12)), the logic and estimates agree in substance with the paper’s argument and results .