Figure 1 shows a sequence of ensemble averaged spectra for a lattice of length and nonlinearity strength at times , , , , , , , , , displaced for ease of viewing. The evolution proceeds from a set of initially excited modes to a superharmonic cascade to all wave numbers with exponentially decreasing energy by . By the initial band has transferred much of its energy to intermediate wavenumbers, forming a slight hump. Thereafter, this hump of energy rolls back via an inverse cascade to low wavenumbers. At , the energy spectrum exhibits a plateau at low wavenumbers and an exponential falloff at higher wavenumbers. This last spectrum is the motivation for the term ``knee'', below which the waves are in equipartition and above which they are not substantially excited [#!ac:tdsrb!#,#!lg:peels!#]. After the spectrum evolves over much longer time scales, eventually arriving at equipartition throughout.
Figure 2 shows a similar experiment with a larger ensemble (20 realizations), and . The initial excitation band is very much smaller, including only 20 waves. Again spectra are displaced and in this example are shown at . Energy is driven first to an intermediate range of wavenumbers which saturate. Subsequently, an inverse cascade of energy extends the band of equilibration backward to lower wavenumbers until , at which point only the lowest wavenumber has yet to reach equipartition. At the spectrum is quasi-stationary, equipartition being achieved among all wavenumbers less than . The energy in larger wavenumbers decreases rapidly. At the highest modes begin to acquire energy. Eventually the whole spectrum will arrive at equipartition; this process is outside of the scope of the current work.