Strong pattern selection and amplitude equation of higher order for ionization waves in a neon glow discharge

Motivated by recent experiments and numerical simulations of the positive c olumn of a neon glow discharge we investigate the Eckhaus instability of tr aveling waves. Compared to the classical results the plasma system shows so me peculiarities, e.g., an asymmetric stability region and strong selection of periodic patterns. These complex phenomena may be explained by a transi tion from supercritical to subcritical Hopf bifurcation near the critical p oint In the weak nonlinear region the wave dynamics is approximated by a qu intic Ginzburg-Landau equation supplemented by nonlinear gradient terms. St arting from a hydrodynamic model the coefficients of this equation, which d epend on the plasma parameters, are calculated. The stability properties of plane wave solutions are discussed for an infinitely long discharge as wel l as for finite ones. The theoretical results show mast of the properties t hat are observed in real experiments.