(a) Frequency dispersion relation of the A mode (black lines) and of the O mode (red lines). The Bloch wave vector characterizing the propagating regime is labeled as Re[κ]. Dotted lines: harmonic dispersion. Solid lines: nonlinear dispersion in the softening anharmonic regime for k⁽⁴⁾ = −2 × 10⁸ N/m³. The double arrow indicates the BG amplitude at the 1BZ border in the latter case. The dash-dotted line labels the resonance frequency ν₀ = 9.29 Hz. (b) Continuous black line with full blue circles: difference between the frequency of the O mode at κ = 0 rad/m and the frequency of the A mode at κ = π/L amplitude as a function of the wave amplitude A in the softening regime for k⁽⁴⁾ = −2 × 10⁸ N/m³ corresponding to the BG at the border of the 1BZ. (c) Continuous blue line with full green circles: frequency of the A mode at the 1BZ boundary, ν_A (κ = π/L), as a function of the wave amplitude A in the softening regime for k⁽⁴⁾ = −2 × 10⁸ N/m³. (d) Continuous red line with full blue circles: dimensionless wave vector κ₀L vs. modulus |γ | of the anharmonic parameter.