(color online) (a) versus k/N for the Toda model at different times t with N = 1024, ε = 0.01, and α = 1.0. (b) /ε vs. k/N for the Toda model with uniform on-site energies at different times t for N = 1024, ε = 0.01, α = 1.0. for all n. (c) /ε versus k/N for the Toda model with disordered on-site energies at different times t for N = 1024, ε = 0.01, and α = 1.0. V_n are chosen randomly from the interval (0.5, 1.5). (d) The red curve is the result of with the same parameters as those in panel (a). The green curve is the result of with the same parameters as those in panel (b). The blue curve is the result of with the same parameters as those in panel (c). For all cases, the lowest 10% of frequency modes are initially excited.