The Existence of Full-Dimensional KAM tori for one-dimensional nonlinear Klein-Gordon equation

The paper proves Theorem 1.4: for the 1D NLKG with convolution potential, for almost every prescribed frequency vector ω in Π_c (up to a small-measure exclusion), and for small ε, there exists a potential V∈V so that the system admits a linearly stable full-dimensional KAM torus with actions decaying like e^{-2 r ln^σ⌊n⌋}, and the torus’ frequencies are exactly ω. This statement, definitions of V and Π_c, and linear stability are explicitly established (see the precise formulation of Theorem 1.4 and the constructions around (4)–(6), the transformation to complex symplectic coordinates, and the final normal form N* with linear stability) . The nonresonance set is handled via the paper’s tailored conditions (7) and an explicit measure estimate meas R=O(γ^{1/3}), uniform in c≥1, using lemmas that cover both |∑ℓ_n ω_n| and mixed “+ bc” divisors needed by the iteration . The parameter-to-frequency mapping is controlled by an inverse function theorem step that freezes T_n = ω_n^2/c^2 − (c^2+n^2) at each iteration, ensuring one can choose V so that the final normal form has exactly the prescribed ω (see (58)–(60)) . By contrast, the model’s outline replaces the paper’s small-divisor scheme with “first/second Melnikov” conditions and restricts ℓ to |supp ℓ|≤2, introduces 2×2 normal blocks for the ±n degeneracy, and uses a generic a posteriori parameterization/Newton scheme with θ-Fourier smoothing. These choices do not match the paper’s actual obstruction structure: under periodic boundary conditions the paper removes the ±n issue via a stronger zero momentum condition (13), not 2×2 blocks, and its nonresonance conditions (7) and measure construction (R=R1∪R2) cover additional divisor forms (including “+ bc”) absent from the model’s DC set . The model’s measure estimate is asserted by Fubini with generic exponents (τ,b) and summability assumptions that are not justified for full-dimensional tori; crucially, the restriction |supp ℓ|≤2 is not supported by the paper’s scheme and would miss necessary divisors. As a result, the model’s proof outline overlooks core small-divisor and momentum-structure features of this problem. Therefore, the paper’s argument is correct and complete at the level of an arXiv research article, while the model’s is not sound as a proof.