Spontaneous Running Waves and Self-Oscillatory Transport in Dirac Fluids

We predict hydrodynamic Turing instability of current-carrying Dirac electron fluids that drives spontaneous self-oscillatory transport. The instability arises near charge neutrality, where carrier kinetics make current dissipation strongly density dependent. Above a critical drift velocity, a uniform electronic flow becomes unstable and undergoes a dynamical transition to a state with coupled spatial modulation and temporal oscillations--an electronic analogue of Kapitsa roll waves in viscous films. The transition exhibits two clear signatures: a nonanalytic, second-order-like onset in the time-averaged current and narrow-band electromagnetic emission at a tunable washboard frequency $f=u/\lambda$. Although reminiscent of sliding charge-density waves, the mechanism is intrinsic and disorder independent. Owing to the small effective mass of Dirac carriers, hydrodynamic time scales translate into emission frequencies in the tens to hundreds of gigahertz range, establishing Dirac materials as a platform for high-frequency self-oscillatory electron hydrodynamics.