11 of the stripes, shown in Fig. 10. As naively expected, the propagation speed of the stripes increases linearly with v$_{0}$ . An alternative way of displaying the existence of the stripes and quantifying their properties is provided by the two point density correlation function defined in Eq. (23). We have evolved the system starting form a uniform or- dered initial state and evaluated the two-point correla- tion function as a function of r for three directions: 0 $^{◦}$, 45 ◦ and 90 ◦ to the direction of initial orientational order. The result is shown in Fig. 11. When the self propul- sion speed v$_{0}$ < v$_{c}$ (right frame), the correlation decays monotonically in all directions. On the other hand, when v$_{0}$ > v$_{c}$ (left frame), we find well defined oscillations in the correlation function in the directions normal and at 45 ◦ to the direction of motion showing the emergence of the periodic structure associated with the stripe pattern. Also, this clearly indicates that the spatial inhomogeneity develops in the direction of initial orientational ordering even in the region where the linear instability is along a wavevector orthogonal to the ordering direction. Finally we have also investigated the dependence of stripe for- mation on system size and boundary conditions. We find that the width and speed of the stripes remain mostly invariant as we go to larger system sizes. We have also solved our equations with shifted boundary conditions [40] and have found that the striped phase persists. In summary, for values of v$_{0}$ > v$_{c}$ ( ρ ) given by the (black) solid line in Fig. (1), the system develops robust propagating stripes of alternating ordered and disordered regions. The numerically identified transition line follows closely the threshold for the splay instability identified in Eq. (17), goes to zero at the phase transition in agreement with the longitudinal instability identified using Eq. (20) and shows a behavior unlike both of these linear insta- bilities in the intermediate region. Further, for systems initialized in a uniform ordered state, these stripes form along the direction of initial ordering even in the domain where the longitudinal instability is absent. This sug- gests that the pattern selection arises from a complex interplay between the unstable linear modes [41]. Also, the width of the stripes exhibits a scaling behavior consis- tent with the critical wavevectors k$_{c}$ of both instabilities, but is not quantitatively captured by either length scale. A systematic study of the relationship between the linear modes and the patterns observed here will be the focus of a future work. We can show that the nonlinear equations admit a propagating front solution that may correspond to the onset of the stripe phase, although the stability if this solution is yet to be established. In our numerical study we have solved the equations (1) and (2) by systemati- cally dropping various nonlinear terms. We have estab- lished that the terms that are critical for the formation of the striped phase are the homogeneous nonlinearity in the coefficient a$_{2}$ that induces the phase transition, the couplings between density and polarization embodied by the convective terms in the density and polarization The bottom two panels of Fig. 9 show histograms of density and polarization for a fixed self propulsion speed v$_{0}$ > v$_{c}$ and three different values of density. These his- tograms are used to infer the properties of this striped phase. The difference in position of the two peaks in the bimodal histograms indicates the contrast in density and order parameter between the ordered and disordered stripes. The position of the peaks is independent of the self propulsion speed and only weakly dependent on the density, suggesting that the density contrast between the isotropic and ordered stripes is entirely diffusion limited. The height of the high density/high order peaks in the bi- modal histograms is a measure of the width of the ordered stripe with respect to the disordered one. We note from the figure that the height decreases with both increasing v$_{0}$ and ρ$_{0}$ . This indicates that the width of the ordered stripe decreases with increasing values of these two pa- rameters. Further, we measure the speed of propagation polarization histograms. The points shown in the phase diagram are obtained using the density histograms. As noted earlier, this boundary closely tracks the thresh- old line for the onset of the linear instability of splay fluctuations at large ρ$_{0}$ (although this instability arises from spatial gradients normal to those of stripe forma- tion!), and vanishes with v$_{0}$ as ρ$_{0}$ → ρ + c , as predicted by the longitudinal linear instability discussed in Section III. However, there is considerable discrepancy between the linear theory and the numerics in the details of the behavior at low density. FIG. 7: (color online) Plot of C$_{δP}$ ( r , t ) for the case when r is parallel to ˆ p 0 (black/open circles) and when r is perpendicular to it (red/ filled circles). The anisotropy of the correlation functions begins to appear at large r , but larger system sizes are needed to quantify the difference. 10 100 Distance r 0.01 0.1 1 C P (r) Parallel Perpendicular