Blow-up solitons at the nonlinear stage of the two-stream instability in quantum plasmas

Abstract

The nonlinear evolution of the quantum two-stream instability in a plasma with counter-streaming electron beams is studied. It is shown that in the long-wave limit the nonlinear stage of the instability can be described by the elliptic nonlinear string equation. We present two types of the nonlinear solutions. The first one is an unstable nonlinear mode that is continuously related with the growing linear solution and the second one is a pulsating soliton. We show that both of these solutions blow up in a finite time.

Short-wavelength soliton in a fully degenerate quantum plasma

“…Recently, a nonlinear string equation (one of the forms of the Boussinesq equation) with the linear dispersion ω 2 ∼ −k 2 + k 4 and quadratic nonlinearity for describing quantum twostream instability has been suggested in Ref. [31]. The corresponding analytical solutions have the form of the blow-up solitons .…”

$N$-soliton solutions of the Fokas-Lenells equation for the plasma ion-cyclotron waves: Inverse scattering transform approach

“…In plasma physics, classical examples of completely integrable equations are the Korteweg-de Vries (KdV) equation (and the modified KdV equation) for the nonlinear ion-acoustic waves, the nonlinear Schrödinger (NLS) equation for the Langmuir waves (both of these equations are also derived for the cases of other branches of plasma oscillations using the reductive perturbation technique), the derivative nonlinear Schrödinger (DNLS) equation describing nonlinear Alfvén waves, and the two-dimensional Kadomtsev-Petviashvili equation, which is a two-dimensional generalization of the KdV equation [2,3]. Slightly less known integrable models (in plasmas) are the Boussinesq equation for the beam instabilities [4] and the nonlinear string equation (elliptic Boussinesq) describing a nonlinear stage of the two-stream instability in quantum plasmas [5], and also the Yajima-Oikawa equations describing the interaction of Langmuir waves with ion-acoustic waves propagating in one direction [6].…”