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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 

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 , 

Inserting this ansatz, Eqs. (26) and (27) become 

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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 

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 

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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 

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