15 discussions. Appendix A: Linearized Equations Here we analyze in more detail the hydrodynamic equations linearized about the homogeneous polar state given in Eq. (6) and (7) in the main text and present a better approximation for the discussion of the splay instability. The hydrodynamic modes deep in the ordered phase were discussed in Section III by entirely neglecting mag- nitude fluctuations δP that decay on microscopic time scales. A better approximation consists of neglecting the rate of change of δP in Eq. ( ?? ) and solve for δP in terms of fluctuations in density and director to lowest order in gradients, with the result We then use this expression to eliminate δ ˜ P from from Eqs. (6) for density and director fluctuations. The eigen- values of the resulting two coupled equations are given the dynamics of density and director fluctuations are then where Again the modes decouple when k = k$_{‖}$ ˆ p$_{0}$ lies along the direction of broken symmetry. The two modes governing This work was supported by NSF grants DMR-075105 and DMR-0806511. We thank S. Ramaswamy for helpful δ ˜ P = 1 2 a$_{20}$ {[ 2 a$_{20}$αP$_{0}$ + ık$_{‖}$ ( v$_{0}$ 2 − ˜ λρ$_{0}$p 2 0 )] δ ˜ ρ ρ$_{0}$ + ık$_{⊥}$ρ$_{0}$λ$_{2}$P 2 $_{0}$δ ˜ p$_{⊥}$ } (A1) s$_{±}$ = 1 2 ( c$_{11}$ + c$_{22}$ ) ± 1 2 √ ( c$_{11}$ − c$_{22}$ ) 2 + 4 c$_{12}$c$_{21}$ (A2) c$_{11}$ = ik$_{‖}$v$_{0}$P$_{0}$ (1 + α ) − k 2 ‖ [ D − 1 2 a$_{20}$ ( v 2 0 2 − ˜ λv$_{0}$ρ$_{0}$P 2 0 )] (A3) c$_{12}$ = ik$_{⊥}$v$_{0}$ρ$_{0}$P$_{0}$ − k$_{‖}$k$_{⊥}$ρ 2 $_{0}$v 2 $_{0}$P 2 0 λ$_{2}$ 2 a$_{20}$ (A4) c$_{21}$ = ik$_{⊥}$ [ v$_{0}$ 2 P$_{0}$ − ρ$_{0}$λ$_{3}$P$_{0}$ (1 + α ) ] − k$_{‖}$k$_{⊥}$ [ ( D$_{s}$ − D$_{b}$ ) − λ$_{3}$ρ$_{0}$ 2 a$_{20}$ ( v$_{0}$ 2 − λρ$_{0}$P 2 0 )] (A5) c$_{22}$ = − ik$_{‖}$λ$_{1}$ρ$_{0}$P$_{0}$ − [ D$_{b}$k 2 ‖ + ( D$_{s}$ − λ$_{3}$λ$_{2}$ρ 2 0 2 a$_{20}$ P 2 0 ) k 2 ⊥ ] (A6) Acknowledgments FIG. 12: Plot of polarization as a function of x as obtained by inverting Eq. 33 for ρ 0 = 0 . 7 and v 0 = 1 . 0. The solution represents a propagating domain boundary between a state with P = 1 and a state with P = 0. For these parameters Λ ∼ 1 . 57. -5 0 5 x 0 0.2 0.4 0.6 0.8 1 P