Periodically Generated Propagating Pulses

Certain equations with integral constraints have as solutions time-periodic pulses of a field-like unknown while a current-like unknown oscillates periodically with time. A general asymptotic theory of this phenomenon, the generalized Gunn effect, has been found recently. Here we extend this theory to the case of nonlinearities having only one stable zero, which is the case for the usual Gunn effect in n-GaAs. Our ideas are presented in the context of a simple scalar model where the waves can be constructed analytically and explicit expressions for asymptotic approximations can be found. 1. Introduction. The Gunn effect is the periodic oscillation of the current in a passive external circuit attached to a dc-voltage biased semiconductor whose electron drift velocity has a single maximum as a function of the electric field (and therefore the curve of electron velocity versus electric field has negative slope for field values on a certain interval, a fact called negative differential mobility) [22, 25]. During each period of the oscillation, a pulse of the electric field is created at the injecting contact, moves through the semiconductor, and is annihilated at the receiving contact. While originally observed in bulk n-GaAs samples, similar current oscillations, mediated by pulse dynamics in dc voltage biased semiconductors, have been found in many materials , several of which lack negative differential mobility [1]. Instead, other processes (impact ionization at impurities [24], nonlinear capture coefficients [21], nonlinear recombination processes, etc) may be responsible for a current vs. local electric field characteristic curve displaying a local maximum followed by a region of negative slope (negative differential conductivity). Propagation of pulses occurs in many systems of interest in Biology, Physics,. .. : morphogen pulses or spikes in activator-inhibitor reaction-diffusion systems modeling cell development or chemical reactors [16, 12, 13, 23], propagation of nerve impulses along myelinated or unmyelinated fibers [18, 20, 17], pulse propagation through cardiac cells [18], calcium release waves in living cells [9], semiconductor superlattices [8, 26] and oscillatory instabilities of the current in bulk semiconductors with an N-shaped current–field characteristic [1, 22, 25]. These distributed systems can be spatially discrete or continuous and can be described by a variety of model equations. Sometimes a pulse is created from an appropriate initial condition and it reaches a stable shape, moving uniformly until it arrives at a boundary. Sometimes understanding pulse dynamics is the key to describing a more complicated evolution of the system. A …