Fig. 6: Dispersion and stability of travelling waves in the nonlinear NRCH model.

a In a system without number conservation, a possible dynamical steady-state comprises global oscillations of uniform densities (green and blue line). b Number conservation calls for more complicated synchronisation patterns, resulting in a travelling wave state with ϕ_1,2 varying sinuously in space and time (green and blue line); equivalent to θ rotating at all space points with constant phase difference between two points. c Comparison of the dispersion relation for α₀ = 1 and α₁ = − 1 as predicted by theory and as calculated from numerical simulations. The frequency is calculated from simulations with an initial condition \({\phi }_{q}^{{{\rm{w}}}}({{\boldsymbol{r}}},0)\) and a Fourier transform of the time series obtained at a randomly chosen fixed position in space. The blue shaded region indicates values of q/q₀ for which the travelling wave is unstable to linear perturbations and matches results from simulation. d The shaded area represents the unstable region of the phase space for travelling waves with different values of wavenumber q/q₀ (shown in legend) in the (α₀, α₁) plane. e Theoretical predictions in Fig. 6d are checked using numerical simulations with initial conditions slightly perturbed \({\phi }_{q}^{{{\rm{w}}}}({{\boldsymbol{r}}},0)\) [see Eq. (8)]. A heat-map of Δ [see Eq. (16)] shows a confirmation of the theoretical results. The red line is for q = 6π/L, where L is the domain size (in units of \({q}_{0}^{-1}\)).