Two-Parametric, Mathematically Undisclosed Solitary Electron Holes and Their Evolution Equation

Abstract

The examination of the mutual influence of the two main trapping scenarios, which are characterized by B and D and which in isolation yield the known sech4 (D=0) and Gaussian (B=0) electron holes, show generalized, two-parametric solitary wave solutions. This increases the variety of hole solutions considerably beyond the two cases previously discussed, but at the expense of their mathematical disclosure, since ϕ(x), the electrical wave potential, can no longer be expressed analytically by known functions. Therefore, they belong to a variety with a partially hidden mathematical background, a hitherto unexplored world of structure formation, the origin of which is the chaotic individual particle dynamics at resonance in the coherent wave particle interaction. A third trapping scenario Γ, being independent of (B, D) and representing the perturbative trapping scenarios in lowest order, provides a broad, continuous band of associated phase velocities v0. For structures propagating near CSEA=1.307, the slowelectronacousticspeed, a Generalized Schamel equation is derived: φτ+[A−B158φ+Dlnφ]φx−φxxx=0, which governs their evolution. A is associated with the phase speed and τ:=CSEAt and φ:=ϕ/ψ≥0 are the renormalized time and electric potential, respectively, where ψ is the amplitude of the structure.