Small Fluctuations from the Homogeneous Steady State

Abstract

In this chapter we analyze the response to small spatial and temporal fluctuations of the homogeneous steady state. We shall show that under certain conditions, in particular when negative differential conductivity occurs, these fluctuations can grow and lead to the bifurcation of filamentary or domain-like spatial structures or self-sustained oscillations. We show that a variety of electromagnetic modes can exist, depending upon the orientation of the field fluctuation and the wave vector relative to the uniform field. Only some of these modes couple to the g-r instability and are associated with the bifurcation of spatial or temporal dissipative structures. Whether a current filament or an electric-field domain bifurcates, depends upon the details of the uniform current density-field characteristic. In particular, we point out a novel possibility of moving domains associated with an anomalous tilted S-shaped characteristic. Analytical conditions for the Hopf bifurcation of limit-cycle oscillations are also derived. The analysis in this chapter is confined to single-carrier g-r mechanisms, as discussed in Sect. 2.1.