Moving modulating pulse and front solutions of permanent form in a FPU model with nearest and next-to-nearest neighbor interaction

The paper’s Theorem 1.1 constructs strictly supersonic traveling fronts of permanent form in a modified FPU lattice with nearest and next-nearest neighbor interactions, with amplitude O(ε), limits O(ε^2) at ±∞, and a sharp NLS-based approximation with sup-norm error O(ε^2 |log ε|). The analysis proceeds via a spatial-dynamics formulation, spectral analysis at c = c*, center-manifold reduction (after factoring out translation and using a first integral), a reversible normal-form calculation that yields a focusing regime (condition (49) holds, in particular when b1,b2>0), and persistence of reversible homoclinics; this is then lifted back to the full lattice and the NLS breather profile is recovered in the approximation. These steps match the candidate solution’s outline essentially point-for-point, aside from minor differences in bookkeeping (the paper reduces from a five-dimensional center manifold to a four-dimensional system via the conserved quantity, whereas the candidate goes directly to a four-dimensional center manifold) and wording (the candidate calls the problem Hamiltonian; the paper uses reversibility and a first integral). See the theorem statement, the NLS ansatz and coefficients, the spatial-dynamics set-up, center manifold/normal-form steps, sign condition, and the final error estimate .