Linearly Stable KAM Tori for One Dimensional Forced Kirchhoff Equations under Periodic Boundary Conditions

The paper proves the existence and linear stability of small‑amplitude quasi‑periodic solutions for the 1D forced Kirchhoff equation with periodic boundary conditions and a Fourier multiplier Mξ, via a bespoke infinite‑dimensional KAM scheme tailored to double eigenvalues. The main theorem (Theorem 1) states exactly this result, including the solution’s form u(t,x)=∑n un(ω̄t, ω̃* t)φn(x) and the Cantor set measure estimate meas(O\Oγ)=O(γ) . Crucially, the paper avoids pseudo‑differential calculus and instead builds an abstract KAM normal form with assumptions (A1)–(A7), exploiting a Toeplitz‑Lipschitz structure, a finite‑rank 2×2 block coupling for the ±n modes, and non‑resonance conditions adapted to the double eigenvalue setting . By contrast, the candidate outline proposes the pseudo‑differential/quasi‑linear reducibility route used for response solutions in Montalto [2017], assumes “standard” first/second Melnikov conditions, and claims a smoothing remainder, but does not address the double‑eigenvalue pairing, the finite‑rank non‑diagonal 2×2 normal blocks for |n|≤EK, the Toeplitz‑Lipschitz drift control, or the paper’s precise non‑resonance scheme. The paper itself explicitly distinguishes its method from the pseudo‑differential approach in [42], stressing that it relies only on KAM with refined Toeplitz‑Lipschitz properties in the periodic (double‑eigenvalue) case . Hence the paper’s argument is coherent and aligned with the literature it cites, whereas the model’s outline is not a valid proof for this setting.