Fig. 4: State diagram and correlation functions.

a State diagram for the nonlinear NRCH model in two dimensions. Symbols indicate the points where the simulations were carried out and the colours correspond to the dynamical steady-state of the system. The steady-states are summarised in Fig. 1. Within the dashed red line and the y-axis, travelling waves with vanishing wavenumber are linearly unstable to small perturbations and where droplets with reciprocal interactions are stable (see discussion later in the paper) - the result is the effervescent wave. The region between the dashed blue and red lines, waves are unstable for most wave-numbers and droplets once nucleated are marginally stable - resulting in effervescence. b Heat-map for the structure factor S(k) in the effervescent state shows fluctuations occurring at all length scales. c Same as in Fig. 4b but for effervescent waves - S receives contributions from the waves manifesting in a pronounced peak and the isotropic fluctuations. d Power spectrum plotted as a function of the frequency ω for effervescent waves and effervescence for parameters α₀ = α₁ = 4, and α₀ = 2, α₁ = 4.33, respectively. For effervescent waves, S(ω) shows a pronounced peak in a nearly constant background of temporal fluctuations. In the effervescent state, the spectrum is nearly constant and indistinguishable from white noise. In both cases, S(ω) decays at large frequencies. e A heat map of \(\log [S(\omega )]\) as a function of α₁ with a fixed α₀ = 2.5. Note that the steep peak for travelling waves disappears in the effervescence case and reappears for effervescent waves. The dispersion is linear for all waves. The black lines are added to emphasise the sharp boundaries between the dynamical steady-states.