Cell-stable pulse trains destabilized by resonant Bloch modes in a mixed-dispersion nonlinear Schrödinger equation

Co-periodic orbital stability need not persist when the perturbation period is enlarged. For a cubic–quintic nonlinear Schrödinger equation with competing second- and fourth-order dispersion, we prove this cell-to-supercell stability reversal for exact fixed-period dnoidal waves. A Riesz–Bloch reduction relates an isolated rank-six spectral cluster to nonlinear orbital instability on finite supercells. At 𝜉 = 𝑚 2 ⁢ 𝜁 and 𝜆 2 = 𝑚 2 ⁢ 𝑍 , an explicit quadratic-junction polynomial gives a positive root 𝑍 + that connects the reduced inner law 𝜆 ∼ 𝐶 𝛿 ⁢ | 𝜉 | / 𝑚 to the outer law 𝜆 2 ∼ 2 ⁢ ( 1 + 𝛿 ) ⁢ ( 1 + 2 ⁢ 𝛿 ) ⁢ | 𝜉 | . For 𝛿 in any compact 𝐼 ⋐ ( 0 , 1 / 8 ) , fixed 𝐾 𝑎 > 0 and fixed 𝐽 m a x ∈ ℕ , every sufficiently large 𝑁 and every 0 < 𝑚 𝑁 ≤ 𝐾 𝑎 ⁢ 𝑁 − 1 / 2 give a wave orbitally stable on 𝕋 2 ⁢ 𝜋 whose repetition is nonlinearly orbitally unstable on 𝕋 2 ⁢ 𝜋 ⁢ 𝑁 . A positive real eigenvalue occurs at each 𝜉 = 𝑗 / 𝑁 , 1 ≤ 𝑗 ≤ 𝐽 m a x . If 𝑚 𝑁 ∼ 𝜅 ⁢ 𝑁 − 1 / 2 with fixed 0 < 𝜅 ≤ 𝐾 𝑎 , the leading term of its square is 𝜅 2 ⁢ 𝑍 + ⁡ ( 𝑗 / 𝜅 2 ; 𝛿 ) / 𝑁 . Separate fixed-amplitude computations relate spectral growth to modulation frequencies, propagation lengths and detection thresholds under third-order dispersion, Raman response, self-steepening and loss. These calculations distinguish transport on an observation ray from finite-distance detectability. The theorem identifies a long-envelope instability that a co-periodic stability test does not detect.