One-dimensional, weakly nonlinear electromagnetic solitary waves in a plasma.

Abstract

The properties of one-dimensional, weakly nonlinear electromagnetic solitary waves in a plasma are investigated. The solution of the resulting eigenvalue problem shows that the solitary waves have amplitudes which are allowed discrete values only and their vector and scalar potentials are proportional to ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$/${\mathrm{\ensuremath{\omega}}}_{0}$ and (${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$/${\mathrm{\ensuremath{\omega}}}_{0}$${)}^{2}$, respectively, where ${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$ and ${\mathrm{\ensuremath{\omega}}}_{0}$ are the plasma and electromagnetic wave frequencies, respectively. Their widths are comparable to the plasma wavelength ${\ensuremath{\lambda}}_{\mathit{p}}$=2\ensuremath{\pi}c/${\mathrm{\ensuremath{\omega}}}_{\mathit{p}}$ (where c is the velocity of light), except for the lowest-order solitary wave, whose width is large compared with ${\ensuremath{\lambda}}_{\mathit{p}}$, which is a true wakeless solitary wave only in the limit of vanishing amplitude. Simple analytical solutions are derived for higher-order solitary waves, whose vector-potential envelope is highly oscillatory, and are shown to consist, in the group-velocity frame, of two trapped, oppositely traveling waves.