Critical damping of wake-field oscillations in a dispersive and dissipative medium: Influence of nonlinearity

Abstract

Wake-field oscillations which appear in a dispersive and dissipative medium (plasma) are found to disappear at a critical level of the dissipative effects, such as collisional damping in a plasma. Linear analysis, carried out for the purpose of application of Green's function techniques to the propagation of ultrashort electromagnetic pulses, shows that no wake-oscillations occur for v2/ωp2 ⩾ 0.8, v being the collision frequency and ωp the plasma frequency. For the domain where v2/ωp2 ≈ 0.8 i.e. when the medium-dependent linear term of a renormalized form of the field equation is negligible, the influence of non-linearity should become relatively important, particularly for high fields.Particular nonlinear solutions are given for a nonlinearity of the form βUp, where β and p are constants, and U is the renormalized field variable, assuming that the corresponding linear term can be neglected due to compensation between the dispersive and dissipative effects.It is found that, for p = 3, nonlinear oscillations occur when the coefficient of the nonlinear term has the same sign as the corresponding coefficient (ωp2 - (5/4)v2) of the linear term would have for ωp2 > (5/4)v2. The nonlinear oscillations are described by an elliptic integral of the first kind. The frequency of nonlinear oscillations is found to become proportional to √β and to the amplitude of the nonlinear oscillations.For the opposite sign of the nonlinear term the dissipation dominated solutions correspond to nonoscillatory decaying fields, expressed in terms of nonlinearly transformed variables in space and time. It is furthermore, found that for (5/4)v2 > ωp2 and provided the nonlinear and linear terms have opposite signs, soliton-type solutions exist for p = 3.