Small-Amplitude Periodic Traveling Waves in Dimer Fermi–Pasta–Ulam–Tsingou Lattices Without Symmetry

The paper establishes small-amplitude 2π-periodic traveling waves for general dimer FPUT lattices via a Lyapunov–Schmidt (LS) reduction that explicitly manages a two-dimensional kernel (after removing the neutral translation mode) using a gradient/orthogonality structure. Concretely, DφΦc(0, ωc) has a three-dimensional kernel and cokernel spanned by ν0 (translation) and two critical modes νc1, νc2; translation invariance removes ν0, leaving a genuinely two-dimensional kernel Zc = span{νc1, νc2} that must be handled in the finite-dimensional reduction, together with a transversality estimate ⟨L′c[ωc]νc1, νc1⟩ ≠ 0 and the derivative orthogonality property ⟨Φc(φ, ω), φ′⟩ = 0 (proved from the gradient formulation) to close the system. This is the centerpiece of their analysis and is repeatedly emphasized (Theorem 1.1; Section 2 for the linearization; Corollaries 2.3–2.4; Sections 3–4 for the LS/gradient-based bifurcation). In contrast, the model’s solution treats the kernel as one-dimensional by restricting to even functions, thereby discarding the νc2 direction and the translation mode ν0, and proceeds with a scalar Crandall–Rabinowitz-style argument. That restriction is not valid for the general (non-symmetric) dimer studied here, where the operator does not preserve an “even subspace” and symmetry is intentionally not assumed. The model thus omits essential obstructions (the second kernel direction and the neutral mode), uses an unjustified symmetry reduction, and does not incorporate the key derivative orthogonality mechanism the paper exploits to solve the overdetermined bifurcation system. The paper’s proof is internally consistent and carefully addresses these issues, including uniform coercivity on the orthogonal complement and a rigorous transversality bound, whereas the model’s justification for transversality is heuristic. Therefore, the paper is correct for the stated problem, while the model’s proof is not valid in this setting. See Theorem 1.1 for the main result, the dispersion/linearization and critical eigenfunctions νc1, νc2 in Section 2, the gradient/orthogonality machinery in Section 3, and the finite-dimensional solve (ω, γ) in Section 4 that yields the asserted branch (and forces γ = 0) .