Non-linear stage of beam instability in the absence of electron capture by a wave field

The system of two electron beams with V1 and V2(V1 > V2) velocities and equal densities (n1=n2) is considered. The energy electron spread for each beam is assumed to be sufficiently small to avoid electron capture by the field of volumetric charge wave. Electrons move in space between two parallel conducting planes in a flat layer of small thickness 2R. Interplane distance 2H is arbitrary. The problem of development of beam instability is solved with provision for dispersion and weak non-linearity. Non-linear equation for a amplitude as a function of distance zeta is derived: S1(d2a)/(d zeta2))=S2a+S3a3. The latter equation assumes solutions of two types: monotonously increasing one (at S3 > 0) and periodical one (at S3 < 0), under conditions of linear instability (s1 and s2 coefficients are positive). In the case of ion compensation of a constant component of volumetric electron charge change of these conditions (S3=0) is realized at p=((v1-v2)/w p=(v sub(1)-v sub(2)/w sub(p)√hH)=1.73, where w sub(p)=(4πl sup(2)n/m)sup(1/2), n0=n1+n2. At lower values of P parameter an explosive instability is observed and at higher values nonlinear stabilization occurs