Energy cascade and Burgers turbulence in the Fermi-Pasta-Ulam-Tsingou chain

Both the paper and the candidate solution derive the k^{-8/3} energy-spectrum scaling at the first shock time for the generalized Burgers equation via the same mechanism: characteristics + an exact 1/(i2πk)-prefactor identity for û_k(τ) followed by a cubic stationary-phase (Airy) evaluation at fold points where Φ′=Φ″=0 and Φ‴≠0. The paper’s equations (52)–(57) match the candidate’s steps and constants (up to a common unimodular phase), including the role of u′_0(x_j) and γ_j=d^3(f∘u_0)/dx^3|_{x_j}, and the normalized FES ∝ k^{-8/3}/ζ_R(8/3) under the stated simplification. The only minor gap is that the model asserts the normalized spectrum without explicitly repeating the paper’s simplifying assumption that the k^{-8/3} form holds for all k≥1 at the shock time; the paper makes this assumption explicit before giving the normalization. Otherwise, the proofs are essentially the same and constants agree. See the paper’s derivation around (52)–(56) and the final normalization discussion leading to (57) and to Ek(ts)/Σm>0 Em(ts)=k^{-8/3}/ζ_R(8/3) .