0.01 0.1 1 r/L 1 1 - C (r/L) v$_{0}$ = 0.1 v$_{0}$ = 0.2 v$_{0}$ = 0.3 1 10 100 1000 r 0 0.2 0.4 0.6 0.8 1 C (r) 1 10 100 1000 r 0 0.1 0.2 0.3 0.4 0.5 C P (r) v$_{0}$ = 0.0 v$_{0}$ = 0.1 v$_{0}$ = 0.2 v$_{0}$ = 0.3 0.01 1 100 r/L 0 0.2 0.4 C P (r/L) 12 FIG. 8: (color online) Late-time two point correlation functions of density, C$_{ρ}$ ( r ), and order parameter, C$_{δP}$ ( r ), fluctuations. The data are obtained from an initial ordered configuration for ρ 0 = 0 . 7, L = 512 and various values of v 0 , in the region corresponding to the fluctuating flocking state. Inset (top panel) shows (1 − C$_{ρ}$ ( r/L ( t ))) vs. the scaled distance r/L ( t ), for various values of v 0 . The dashed straight line shows the cuspy nature of two-point correlation function with cusp exponent α = 0 . 6. Inset (bottom panel) shows the scaled two-point order parameter correlation function, C$_{δP}$ ( r/L ( t )), vs. the scaled distance r/L ( t ), for several v 0 . The dashed straight line shows the logarithmic decay of C$_{δP}$ ( r/L ( t )). The density equation can be formally integrated by pos- tulating an isotropic state at x = ∞ ( P ( ∞ , t ) = 0) and a polar state at x = −∞ ( P ( −∞ , t ) = 1). This gives us the ratio of the density in in a polarized region to the density an isotropic region as equations ( v$_{0}$ ∇ ρ P and v$_{0}$ ( ∇ ρ ) /ρ , respectively) and the convective nonlinearity controlled by the parameter λ$_{3}$ . The longitudinal instability arises from the interplay of a$_{2}$ and the convective terms while the splay instability is controlled by the term proportional to λ$_{3}$ . It is useful then to consider a simplified description of the nonlin- ear dynamics where diffusion is neglected and only terms essential for the pattern formation are retained, given by The density contrast is infinitely sharp when the front propagates at a speed v$_{0}$P commensurate with the de- gree of ordering in the stripes. Diffusion, neglected here, will smooth the density crossover between the two re- gions. This is in agreement with our numerical results that indicated that the contrast between the two regions is insensitive to the parameters and is indeed diffusion limited. Writing the density from Eq. (30) as ρ = 1 / ( U − v$_{0}$P ), and substituting in the order parameter equation, we can formally integrate Eq. (31) to obtain a solution of the form Denoting by x the direction of broken symmetry of the putative HP state, we postulate a solution of these equa- tions in the form of a front uniform in y and propagating along x with a yet undetermined constant speed U , where for simplicity we have assumed U ∼ v$_{0}$ and we have introduced a dimensionless friction constant, ζ$_{R}$ = ( a$_{2}$ + a$_{4}$ ) and a dimensionless length scale, Λ = λ 3 v 0 ζ $_{R}$. This formal solution cannot be inverted analytically. A plot is shown in Fig. 12 and clearly displays that the Inserting this ansatz, Eqs. (26) and (27) become ρ$_{pol}$ ρ$_{iso}$ = U U − v$_{0}$P . (32) x = − Λ ( P − 1) + Λ a$_{4}$ ζ$_{R}$ ln [ a$_{4}$ ( P − 1) $^{2}$(1 + P ) ] + v$_{0}$ 2 a$_{2}$ ln( P ) , (33) ∂$_{t}$ρ = − ∇ · ( ρv$_{0}$ P ) (26) ∂$_{t}$ P = − [ a$_{2}$ ( ρ ) + P $^{2}$a$_{4}$ ( ρ ) ] P − v$_{0}$ 2 ρ ∇ ρ + λ$_{3}$P$_{i}$ ∇ ρP$_{i}$ (27) ρ ( x, y, t ) = ρ ( x − Ut ) , (28) P ( x, y, t ) = P ( x − Ut ) ˆx . (29) ∂$_{x}$ ln ρ = v$_{0}$ ( U − v$_{0}$P ) ∂$_{x}$P , (30) ( a$_{2}$ + a$_{4}$P 2 ) P − ( λ$_{3}$P 2 − v$_{0}$ 2 ρ ) ∂$_{x}$ρ − ( U + λ$_{3}$ρP ) ∂$_{x}$P = 0 . (31)