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tures of a coarsening process. For v$_{0}$ > v$_{c}$ ( ρ$_{0}$ ) the system orders in a robust striped phase, consisting of traveling high/low density stripes. The high density stripes are or- dered with polarization transverse to the long direction of the stripes, which is also the direction of the stripes’ motion. Both of these phases have been identified in nu- merical studies of the Vicsek model [13, 32]. Here we characterize them in terms of their origins in the hydro- dynamic equations using both analytical and numerical tools. This allows us to derive an understanding that transcends any specific microscopic model and is gener- ically applicable to a large number of self-propelled sys- tems. 

The coupling of density and polarization fluctuations embodied in the convective terms in the hydrodynamic equations of self-propelled systems plays a crucial role in controlling pattern formation. Some of these convec- tive terms reflect the dual role played by the polarization field as vector order parameter and as the mean velocity, resulting in competition between diffusion and convec- tion along the direction of mean local order. There is a qualitative analogy here with sedimentation problems [33, 34], where the interplay of local alignment along the sedimenting direction and diffusion can destabilize the system resulting in convective patterns, although hydro- dynamic interactions, not incorporated here, often also play an important role in sedimenting systems. 

The layout of the paper is as follows. First we intro- duce the hydrodynamic equations that are the starting point of our analysis. Then, we carry out a linear sta- bility analysis about the ordered state and characterize the region of linear stability of the bulk ordered phase 

or homogeneous flock. Next, we report the results of nu- merical solution of the nonlinear hydrodynamic equations and identify and characterize both the fluctuating flock- ing state and the striped phase, as well as the coarsening- like behavior leading to these phases. 

We consider a collection of polar rods of length ℓ mov- ing on an inert substrate characterized by a friction con- stant ζ in two dimensions (2d). Each rod is driven by an internal force F acting along one direction of its long axis, called its head. This force, together with the frictional interaction with the medium, results in a self-propulsion speed v$_{0}$ = F/ζ of constant magnitude. On length scales long compared to ℓ and on time scales long compared to the microscopic interaction times, the dynamics of the system can be described in terms of hydrodynamic fields, namely the conserved densities (here the density ρ ( r , t ) of rods) and the variables associated with possible bro- ken symmetries. A collection of self-propelled polar rod can order in a polarized state, characterized by a finite value of a vector order parameter, P ( r , t ), describing the mean polarization of the rods. The ordered state is also a moving state, with mean velocity ∼ v$_{0}$ P . The dynam- ics of the system is described by coupled equations for density and polarization, given by 

and 

## II. THE MODEL 

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The hydrodynamic equations (1) and (2) have the same form as those first proposed on a phenomenological basis by Toner and Tu [1, 16, 17] to describe the physics of flocking. The parameter a$_{2}$ is chosen to change sign at a characteristic density ρ$_{c}$ , while a$_{4}$ > 0. This guarantees a mean-field continuous transition from an isotropic state with ρ = ρ$_{0}$ and P =0 when a$_{2}$ > 0 to a homogeneous po- larized state with ρ = ρ$_{0}$ and | P$_{0}$ | = √ − a 2 a 4 when a$_{2}$ < 0. These equations have also been derived from specific microscopic models of self-propelled particles on a sub- strate by some of us [23–25] and by Bertin and collabora- tors [21]. Bertin et al obtained hydrodynamic equations by coarse-graining a Vicsek-type model of self-propelled point particles , with a specific aligning rule for the pair interaction. In contrast, Baskaran and Marchetti, con- sidered a model of self-propelled hard rods of finite size 

with excluded volume interaction and analyzed in detail the modifications induced by self-propulsion on the linear and angular momentum exchanged in a binary collision. The equations obtained by Bertin et have precisely the form given in Eqs. (1) and (2), with parameters a$_{2}$ and a$_{4}$ determined by the aligning interaction between particles. In contrast, it was demonstrated in Ref. [23] that steric effects alone are not sufficient to yield a homogeneous bulk polarized state. As a result, the equations derived in [23] for a purely physical model have a$_{4}$ = 0. We note that it was also demonstrated recently in a model of swimmers in a fluid that hydrodynamic interactions among swimmers are equally insufficient to yield a ho- mogeneous polarized state in bulk [26]. These results suggest that genetically and biochemically-regulated sig- naling, or external symmetry-breaking effects, such as 

