KP-II Approximation for a Scalar FPU System on a 2D Square Lattice

The paper rigorously proves the KP–II approximation with O(ε^{5/2}) error for waves propagating along the x-axis (Theorem 3.9) on time scales t ∈ [0, ε^{-3}τ0], using a weighted energy and a residual bound of size O(ε^{11/2}) in weighted L2 (equation (17)), not a plain ℓ2 residual bound . For arbitrary propagation angle φ, the authors derive an extended KP–II system, split it into KP–II for A1 and a forced linearized KP–II for A2, and formulate Theorem 5.1 assuming A2 exists with the same smoothness as A1; they also give classes where this can be made rigorous (Y-independent and periodic data with rational φ), but explicitly note that for decaying data on R2 the needed control of ∂X^{-1}∂T^2A2 is “out of reach at the present time” . The candidate model’s outline matches the rotated variables, the amplitude system (34)–(35), and the normal-form choice B1,B2, but it overstates technical points: it asserts a uniform ℓ2 residual of O(ε^{7/2}) and closes an unweighted energy E′ ≤ Cε^3E + Cε^5, whereas the paper’s closure is via a weighted energy and a weighted residual of O(ε^{11/2}) leading to a forcing O(ε^3) . The model also treats well-posedness for A2 more optimistically than the paper allows on R2. Net: the paper provides a conditional extension to arbitrary φ with clear limitations; the model gives a plausible scheme but with technical mismatches and an overreach on well-posedness. Hence both are incomplete.